Motivic Integration

This monograph focuses on the geometric theory of motivic integration, which takes its values in the Grothendieck ring of varieties. This theory is rooted in a groundbreaking idea of Kontsevich and was further developed by Denef & Loeser and Sebag. It is presented in the context of formal schemes over a discrete valuation ring, without any restriction on the residue characteristic. The text first discusses the main features of the Grothendieck ring of varieties, arc schemes, and Greenberg schemes. It then moves on to motivic integration and its applications to birational geometry and non-Archimedean geometry. Also included in the work is a prologue on p-adic analytic manifolds, which served as a model for motivic integration. With its extensive discussion of preliminaries and applications, this book is an ideal resource for graduate students of algebraic geometry and researchers of motivic integration. It will also serve as a motivation for more recent and sophisticated theories that have been developed since.

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Progress in Mathematics 325

Antoine Chambert-Loir Johannes Nicaise Julien Sebag

Motivic Integration Ferran Sunyer i Balaguer Award winning monograph

Progress in Mathematics Volume 325

Series Editors Antoine Chambert-Loir, Université Paris-Diderot, Paris, France Jiang-Hua Lu, The University of Hong Kong, Hong Kong SAR, China Michael Ruzhansky, Imperial College, London, UK Yuri Tschinkel, Courant Institute of Mathematical Sciences, New York, USA

More information about this series at http://www.springer.com/series/4848

Antoine Chambert-Loir Julien Sebag



Johannes Nicaise

Motivic Integration

Antoine Chambert-Loir Département de Mathématiques Université Paris-Sud Orsay Orsay, France

Johannes Nicaise Department of Mathematics University of Leuven Heverlee, Belgium

Julien Sebag Département de Mathématiques Université de Rennes 1 Rennes, France

ISSN 0743-1643 ISSN 2296-505X (electronic) Progress in Mathematics ISBN 978-1-4939-7885-4 ISBN 978-1-4939-7887-8 (eBook) https://doi.org/10.1007/978-1-4939-7887-8 Library of Congress Control Number: 2018940430 Mathematics Subject Classification (2010): 14E18, 14G22 © Springer Science+Business Media, LLC, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Science+Business Media, LLC part of Springer Nature. The registered company address is: 233 Spring Street, New York, NY 10013, U.S.A.

Ferran Sunyer i Balaguer (1912–1967) was a selftaught Catalan mathematician who, in spite of a serious physical disability, was very active in research in classical mathematical analysis, an area in which he acquired international recognition. His heirs created the Fundació Ferran Sunyer i Balaguer inside the Institut d’Estudis Catalans to honor the memory of Ferran Sunyer i Balaguer and to promote mathematical research. Each year, the Fundació Ferran Sunyer i Balaguer and the Institut d’Estudis Catalans award an international research prize for a mathematical monograph of expository nature. The prizewinning monographs are published in this series. Details about the prize and the Fundació Ferran Sunyer i Balaguer can be found at http://ffsb.espais.iec.cat/en This book has been awarded the Ferran Sunyer i Balaguer 2017 prize.

The members of the scientific commitee of the 2017 prize were: Rafael de la Llave Georgia Institute of Technology Jiang-Hua Lu The University of Hong Kong Joan Porti Universitat Autònoma de Barcelona Eero Saksman University of Helsinki Yuri Tschinkel Courant Institute of Mathematical Sciences, New York University

Ferran Sunyer i Balaguer Prize winners since 2005: 2006

Xiaonan Ma and George Marinescu Holomorphic Morse Inequalities and Bergman Kernels, PM 254

2007

Rosa Miró-Roig Determinantal Ideals, PM 264

2008

Luis Barreira Dimension and Recurrence in Hyperbolic Dynamics, PM 272

2009

Timothy D. Browning Quantitative Arithmetic of Projective Varieties, PM 277

2010

Carlo Mantegazza Lecture Notes on Mean Curvature Flow, PM 290

2011

Jayce Getz and Mark Goresky Hilbert Modular Forms with Coefficients in Intersection Homology and Quadratic Base Change, PM 298

2012

Angel Cano, Juan Pablo Navarrete and José Seade Complex Kleinian Groups, PM 303

2013

Xavier Tolsa Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory, PM 307

2014

Veronique Fischer and Michael Ruzhansky Quantization on Nilpotent Lie Groups, Open Access, PM 314

2015

The scientific committee decided not to award the prize

2016

Vladimir Turaev and Alexis Virelizier Monoidal Categories and Topological Field Theory, PM 322

To Jan Denef and François Loeser, as a testimony of admiration and recognition Voor Jan Denef en François Loeser, met bewondering en erkentelijkheid À Jan Denef et François Loeser, en témoignage d’admiration et de reconnaissance

INTRODUCTION

The story of motivic integration began with a famous lecture in Orsay by Kontsevich in 1995 where he proved that birationally equivalent complex Calabi–Yau varieties have the same Hodge numbers. This result had a strong significance for mirror symmetry. Indeed, it is predicted that if two smooth Calabi–Yau manifolds form a “mirror pair,” their Hodge numbers should satisfy some symmetry. However, the varieties that are produced by the duality of models A and B in string theory are only defined up to birational equivalence, so that a result such as Kontsevich’s theorem at least provides soundness of the prediction of mirror symmetry. The starting point for Kontsevich’s theorem was the proof by Batyrev (1999a) that birationally equivalent complex Calabi–Yau varieties have the same Betti numbers. Batyrev’s proof was based on a reduction to the case where both Calabi–Yau varieties, as well as the birational morphism relating them, are defined over a p-adic field and have good reduction over the residue field. In that case, he used p-adic integration to show that these reductions have the same Hasse–Weil zeta function. An application of the Weil conjectures allowed him to conclude that the considered Calabi–Yau varieties have the same Betti numbers. Kontsevich’s remarkable insight was that one can upgrade p-adic integration to a geometric integration theory, which he called motivic integration. It avoids the reduction to positive characteristic by replacing the ring of padic integers by the ring of complex formal power series. It also produces a stronger result, namely, an equality between the classes of birationally equivalent Calabi–Yau varieties in a suitable Grothendieck ring of virtual varieties. While the precise significance of this equality is not precisely understood, it implies readily that the varieties share similar motivic invariants. For example, not only are their rational singular cohomology groups are isomorphic (coincidence of Betti numbers), but the underlying Hodge structures are isomorphic as well. In particular, they have the same Hodge numbers. ix

x

INTRODUCTION

The key to the theory of motivic integration is an understanding of the fundamental geometric nature of p-adic integration. By algebraic manipulations, the computation of a large class of p-adic integrals is reduced to the counting of rational points on suitable varieties over finite fields. In a nutshell, Kontsevich’s revolutionary idea was to use the algebraic structure of the Grothendieck ring of varieties to turn these varieties (and not only their number of points over a finite field) into a definition of the integrals. While the finiteness of the residue field of a p-adic field crucially underlies the definition of the p-adic measure and the point counting over the residue field, it becomes totally redundant in this approach. As has been understood afterwards, one can even set up a uniform theory for all complete discretely valued fields (with perfect residue field in the mixed characteristic case). Following Kontsevich’s lecture, the theory of motivic integration was developed in several directions and has found a wide array of applications, ranging from singularity theory and birational geometry to the Langlands program. The pioneering role was played by Denef and Loeser (1999), who gave Kontsevich’s ideas solid foundations and generalized them to singular algebraic varieties over a field of characteristic zero. In particular, their change of variables theorem for motivic integrals is the key result behind most of the applications in birational geometry. In his Bourbaki talk, Looijenga (2002) proposed a generalization of the theory to varieties over a ring of formal power series of characteristic zero. A further generalization was worked out by Sebag (2004a) in his PhD thesis where he defined motivic integrals on formal schemes over complete discrete valuation rings. This theory was then used in subsequent work of Loeser and Sebag (2003) to define motivic integrals of volume forms on non-Archimedean analytic spaces. These works constitute the geometric approach to motivic integration, and they form the main theme of this book. There exists a different (but related) strand based on the model theory of valued fields: the theories of Cluckers and Loeser (2008) and Hrushovski and Kazhdan (2006). These beautiful theories are in some sense much more powerful than the geometric motivic integration. For example, they furnish a geometric understanding of integrals involving additive characters, and their behavior in families, that proved crucial for some applications to the Langlands program. However, both theories fall outside of the scope of this volume; in fact, each of them deserves a book of their own. Moreover, while these theories are theoretically independent, their very structure is strongly influenced by that of classical motivic integration. Let us now give an overview of the content of this book. In the prologue, we present the classical theory of p-adic integration: manifolds over non-Archimedean locally compact fields (p-adic fields and fields of Laurent series over a finite field), volume forms on such manifolds and their integrals. The results presented in this chapter mainly predate the theory of motivic integration, but have served as a driving force behind its development. They will in fact be present throughout the whole book. We

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present three main applications: Batyrev’s theorem on the equality of Betti numbers for birationally equivalent Calabi–Yau varieties, the very starting point of motivic integration; Igusa’s local zeta functions, and the monodromy conjecture; Serre’s invariant of p-adic manifolds. In chapter 2, we define the Grothendieck ring of varieties and some of its variants. These Grothendieck rings play an essential role in the theory of motivic integration: they are the rings where motivic integrals take their values. So, in order to understand the geometric information contained in motivic integrals, it is of crucial importance to understand the structure and properties of the Grothendieck rings, and the geometric invariants that can be extracted from them through various realization morphisms. The Grothendieck ring of varieties is also an important object of study in its own right, because it governs the piecewise geometry of algebraic varieties. Many of its properties remain mysterious, and we will discuss several interesting open questions surrounding this object. In Kontsevich and Denef–Loeser’s initial theories of motivic integration, the arc schemes of varieties over a field are the measure spaces over which motivic integrals are defined, and their constructible subsets are the basic measurable sets. In the theories of Looijenga and Sebag, these measure spaces are the Greenberg schemes of schemes over a complete discrete valuation ring. While the construction of Greenberg schemes recovers the arc schemes as a particular case, the importance of classical motivic integration justifies an autonomous presentation. We thus proceed to the study of arc schemes in chapter 3. These spaces are “infinite dimensional” varieties parameterizing formal arcs on schemes which are constructed through a process of Weil restriction. They were originally introduced by Nash (1995) to study the structure of singularities and their resolutions. In this chapter, we also give a detailed exposition of a theorem of Grinberg and Kazhdan (2000) and Drinfeld (2002) that provides “finite dimensional models” of arc spaces around a reasonable arc. When we work with schemes (or formal schemes) over complete discrete valuation rings, the natural generalizations of arc schemes are Greenberg schemes, which we construct in chapter 4. In the equal characteristic case, their construction is a rather straightforward variant of the Weil restriction considered in the preceding chapter. In the mixed characteristic case, subtle representability issues make it much more intricate. For motivic integration, fine properties of Greenberg schemes (or arc schemes) are needed; their detailed analysis is the object of chapter 5. We first present the Artin–Greenberg approximation theorem and its consequences for the Boolean algebra of constructible subsets of Greenberg schemes. The second part of the chapter compares Greenberg schemes under a morphism of formal schemes. These two aspects are the geometric input for the definition of the motivic measure and the change of variables formula for motivic integrals.

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In chapter 6, we combine the result of the first four chapters and construct the theory of motivic integration on a formal scheme of finite type over a complete discrete valuation ring. We end the chapter with the change of variables formula, one of the central theorems of the book. These two chapters are written in the framework of formal schemes; we insist that their results comprise in particular the case of arc schemes of varieties. A reader who would be mainly interested in this case will find a typographical dictionary at the beginning of each chapter. Finally, we discuss several applications of motivic integration in chapter 7. We begin with the proof of Kontsevich’s theorem and Denef and Loeser’s motivic Igusa zeta functions, as well as the definition of stringy Hodge invariants of singular varieties. We then discuss the interaction between singularity theory and the geometry of arc spaces, such as the study of the log-canonical threshold following Mustaţă (2002), as well as the Nash problem. While it is logically independent from the theory of motivic integration, we devote at this point a section to the motivic zeta function defined by Kapranov (2000). We conclude the chapter by developing the theory of motivic integration on non-Archimedean analytic spaces. In particular, we construct the motivic Serre invariant defined by Loeser and Sebag (2003) and explain the non-Archimedean interpretation of the motivic Igusa zeta function, following Nicaise and Sebag (2007b). The appendix reviews some basic material on constructible sets, birational geometry, formal schemes, and non-Archimedean analytic spaces that is used in the other chapters. The main chapters of the book are numbered from 2 to 7; they are preceded by the prologue on p-adic integration, which is chapter 1; the appendix is referred to as A. Paragraphs and statements are then numbered as chapter/section.subsection.paragraph. When we refer to a statement of the same chapter, we omit the chapter symbol. Acknowledgments. — Jan Denef and François Loeser introduced the three authors to the topic of motivic integration. We are honored to dedicate this book to them. The authors thank the institutions which hosted them during the preparation of this book, namely (by chronological order) université de Bordeaux 1, KU Leuven, université de Rennes 1, Institute for advanced study (Princeton), université Paris-Sud, université Paris-Diderot, and Imperial College (London). We thank the following colleagues for their advice or corrections: Margaret Bilu, David Bourqui, Emmanuel Bultot, Serge Cantat, Javier Fresán, Florian Ivorra, Bernard Le Stum, Husein Mourtada, Fabrice Orgogozo, Silvain Rideau. We especially warmly thank François Loeser for his support at many phases of the preparation of this book. The authors are grateful to the Ferran Sunyer i Balaguer Foundation, and to its director Manuel Castellet, for their gracious hospitality during our visit to Barcelona.

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The research of ACL was partially supported by the Agence nationale de la recherche projects ANR-2010-BLAN-0119-01 (Positive), ANR-13-BS01-0006 (Valcomo) and ANR-15-CE40-0008 (Défigéo), the Institut universitaire de France, as well as by a grand of the National Science Foundation under agreement No. DMS-0635607. The research of JN was partially supported by the ERC Starting Grant MOTZETA (project 306610) of the European Research Council, the research grant G.0415.10 of the Fund for Scientific Research – Flanders (FWO), and the KU Leuven research grant OT/11/069. The research of JS was partially supported by the Agence nationale de la recherche project ANR-15-CE40-0008 (Défigéo).

CONTENTS

1. Prologue: p-Adic Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 1. Analytic Manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Local Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Analytic Manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Hensel’s Lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Differential Forms and Measures. . . . . . . . . . . . . . . . . . . . . . . . . 1.5. Classification of Compact K-analytic Manifolds. . . . . . . . . . 1.6. K-analytic Manifolds Associated with Smooth Schemes . . § 2. The Theorem of Batyrev–Kontsevich. . . . . . . . . . . . . . . . . . . . . . . 2.1. Calabi–Yau Varieties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Hodge Numbers and Hasse–Weil Zeta Functions . . . . . . . . . 2.3. From Complex Numbers to p-adic Numbers. . . . . . . . . . . . . . § 3. Igusa’s Local Zeta Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. The Local Zeta Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Denef’s Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. The Topological Zeta Function of Denef–Loeser. . . . . . . . . . 3.4. The Monodromy Conjecture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Poincaré Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 6 9 12 16 18 25 25 26 30 32 32 40 43 46 50

2. The Grothendieck Ring of Varieties. . . . . . . . . . . . . . . . . . . . . . . . § 1. Additive Invariants on Algebraic Varieties. . . . . . . . . . . . . . . . . . 1.1. Definition and Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. The Grothendieck Group of Varieties. . . . . . . . . . . . . . . . . . . . 1.3. Constructible Subsets and Additive Invariants . . . . . . . . . . . 1.4. Piecewise Isomorphisms and Additive Invariants . . . . . . . . . § 2. Motivic Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Definition of Motivic Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The Ring Structure on K0 (VarS ). . . . . . . . . . . . . . . . . . . . . . . . 2.3. Piecewise Trivial Fibrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 56 56 57 59 63 67 67 68 69 xv

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2.4. Some Classes in K0 (VarS ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Spreading-Out and Applications. . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Variants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Cohomological Realizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Grothendieck Rings of Categories. . . . . . . . . . . . . . . . . . . . . . . . 3.2. Mixed Hodge Theory and Motivic Measures . . . . . . . . . . . . . 3.3. Hodge Realization over a Base. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Étale Cohomology and Motivic Measures. . . . . . . . . . . . . . . . 3.5. Étale Realization over a Base. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. The Crystalline Realization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Motivic Homotopic Realizations. . . . . . . . . . . . . . . . . . . . . . . . . . 4. Localization, Completion, and Modification. . . . . . . . . . . . . . . . . 4.1. Dimensional Filtration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Localization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Completion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. A Modified Grothendieck Ring of Varieties. . . . . . . . . . . . . . . 5. The Theorem of Bittner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Bittner’s Presentation of K0 (VarS ). . . . . . . . . . . . . . . . . . . . . . 5.2. Application to the Construction of Motivic Measures . . . . 5.3. Motives and Motivic Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . 6. The Theorem of Larsen–Lunts and Its Applications. . . . . . . . . 6.1. The Theorem of Larsen–Lunts. . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Other Examples of Motivic Measures. . . . . . . . . . . . . . . . . . . . 6.3. The Cut-and-Paste Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Zero Divisors in the Grothendieck Ring of Varieties. . . . . . 6.5. Algebraically Independent Classes. . . . . . . . . . . . . . . . . . . . . . . .

71 74 77 80 80 85 90 94 98 103 108 109 109 110 111 113 120 120 127 129 133 133 137 139 144 147

3. Arc Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 1. Weil Restriction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Reminders on Representability. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. The Weil Restriction Functor. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Representability of a Weil Restriction: The Affine Case . . 1.4. Representability: The General Case. . . . . . . . . . . . . . . . . . . . . . § 2. Jet Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Jet Schemes of a Variety. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Truncation Morphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 3. The Arc Scheme of a Variety. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Arcs on a Variety. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Relative Representability Properties. . . . . . . . . . . . . . . . . . . . . 3.3. Representability of the Functor of Arcs. . . . . . . . . . . . . . . . . . 3.4. Base Point and Generic Point of an Arc. . . . . . . . . . . . . . . . . . 3.5. Constant Arcs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. Renormalization of Arcs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Differential Properties of Jets and Arc Schemes. . . . . . . . . .

153 153 154 157 160 161 162 162 165 166 168 169 171 172 177 180 181 183

§

§

§

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§ 4. Topological Properties of Arc Schemes. . . . . . . . . . . . . . . . . . . . . . 4.1. Connected Components of Arc Schemes. . . . . . . . . . . . . . . . . . 4.2. Irreducible Components of Arc Schemes. . . . . . . . . . . . . . . . . . 4.3. Kolchin’s Irreducibility Theorem. . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Application of the Valuative Criterion. . . . . . . . . . . . . . . . . . . . 4.5. Irreducible Components of Constructible Subsets in Arc Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 5. The Theorem of Grinberg–Kazhdan–Drinfeld. . . . . . . . . . . . . . . 5.1. Formal Completion of the Space of Arcs. . . . . . . . . . . . . . . . . 5.2. Weierstrass Theorems for Power Series. . . . . . . . . . . . . . . . . . . 5.3. Reduction to the Complete Intersection Case . . . . . . . . . . . . 5.4. Proof of the Theorem of Grinberg–Kazhdan–Drinfeld . . . . 5.5. Gabber’s Cancellation Theorem and Consequences. . . . . . .

188 188 189 191 194 195 197 197 200 202 204 207

4. Greenberg Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 1. Complete Discrete Valuation Rings. . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Witt Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Complete Discrete Valuation Rings and Their Extensions 1.3. The Structure of Complete Discrete Valuation Rings. . . . . § 2. The Ring Schemes Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Construction: The Equal Characteristic Case. . . . . . . . . . . . 2.2. Construction: The Mixed Characteristic Case. . . . . . . . . . . . 2.3. Basic Properties of the Ring Schemes Rn . . . . . . . . . . . . . . . . 2.4. The Ideal Schemes Jnm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 3. Greenberg Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Greenberg Schemes as Functors. . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Representability of the Greenberg Schemes. . . . . . . . . . . . . . . 3.3. Greenberg Schemes of Formal Schemes. . . . . . . . . . . . . . . . . . . 3.4. Néron Smoothenings of Formal Schemes. . . . . . . . . . . . . . . . . 3.5. Néron Smoothening and Greenberg Schemes. . . . . . . . . . . . . § 4. Topological Properties of Greenberg Schemes. . . . . . . . . . . . . . . 4.1. Irreducible Components of Greenberg Schemes. . . . . . . . . . . 4.2. Constructible Subsets of Greenberg Schemes. . . . . . . . . . . . . 4.3. Thin Subsets of Greenberg Schemes. . . . . . . . . . . . . . . . . . . . . . 4.4. Order Functions and Constructible Sets. . . . . . . . . . . . . . . . . .

211 212 212 218 221 225 225 226 234 236 240 240 246 248 251 253 255 255 256 257 260

5. Structure Theorems for Greenberg Schemes. . . . . . . . . . . . . . § 1. Greenberg Approximation on Formal Schemes. . . . . . . . . . . . . . 1.1. Fitting Ideals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Greenberg Schemes of Smooth Formal Schemes . . . . . . . . . . 1.3. The Singular Locus of a Formal Scheme. . . . . . . . . . . . . . . . . . 1.4. An Application of Hensel’s Lemma. . . . . . . . . . . . . . . . . . . . . . . 1.5. Greenberg’s Approximation Theorem. . . . . . . . . . . . . . . . . . . . § 2. The Structure of the Truncation Morphisms. . . . . . . . . . . . . . . . 2.1. Principal Homogeneous Spaces and Affine Bundles. . . . . . .

263 264 264 265 266 270 271 277 277

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2.2. Truncation Morphisms and Principal Homogeneous Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. The Images of the Truncation Morphisms. . . . . . . . . . . . . . . . § 3. Greenberg Schemes and Morphisms of Formal Schemes . . . . . 3.1. The Jacobian Ideal and the Function ordjacf . . . . . . . . . . . . 3.2. Description of the Fibers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Codimension of Constructible Sets in Greenberg Spaces. . 3.4. Example: Contact Loci in Arc Spaces. . . . . . . . . . . . . . . . . . . .

278 282 288 288 293 297 300

6. Motivic Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 1. Motivic Integration in the Smooth Case. . . . . . . . . . . . . . . . . . . . 1.1. Working with Constructible Sets. . . . . . . . . . . . . . . . . . . . . . . . . 1.2. The Change of Variables Formula in the Smooth Case . . . § 2. The Volume of a Constructible Subset of a Greenberg Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. What Is a Motivic Volume?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Reduction to the Reduced Flat Case. . . . . . . . . . . . . . . . . . . . . 2.3. A Dimensional Lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Volume of Thin Constructible Subsets. . . . . . . . . . . . . . . . . . . 2.5. Existence of the Volume of a Constructible Subset . . . . . . . § 3. Measurable Subsets of Greenberg Schemes. . . . . . . . . . . . . . . . . . R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Summable Families in M X0 3.2. Definition of Measurable Subsets. . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Existence and Uniqueness of the Volume of Measurable Subsets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Countable Additivity of the Measure μ∗X . . . . . . . . . . . . . . . . . 3.5. Negligible Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. C-Measurable Subsets of Gr(X). . . . . . . . . . . . . . . . . . . . . . . . . . § 4. Motivic Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Integrable Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Direct and Inverse Images of Measurable Subsets . . . . . . . . 4.3. The Change of Variables Formula. . . . . . . . . . . . . . . . . . . . . . . . 4.4. An Example: The Blow-Up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 5. Semi-algebraic Subsets of Greenberg Schemes. . . . . . . . . . . . . . . 5.1. Semi-algebraic Subsets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Semi-algebraic Subsets of Greenberg Schemes. . . . . . . . . . . . 5.3. Measurability of Semi-algebraic Subsets. . . . . . . . . . . . . . . . . . 5.4. Rationality of Motivic Power Series. . . . . . . . . . . . . . . . . . . . . .

305 307 307 309

7. Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 1. Kapranov’s Motivic Zeta Function. . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Symmetric Products of Varieties. . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Definition of Kapranov’s Motivic Zeta Function. . . . . . . . . . 1.3. Motivic Zeta Functions of Curves. . . . . . . . . . . . . . . . . . . . . . . . 1.4. Motivic Zeta Functions of Surfaces. . . . . . . . . . . . . . . . . . . . . . .

363 364 364 374 377 380

311 311 311 312 314 316 318 319 320 323 326 330 331 333 334 336 340 342 345 345 347 351 353

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§

§

§

§

§

§

1.5. Rationality of Kapranov’s Zeta Function of Finite Dimensional Motives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Valuations and the Space of Arcs. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Divisorial Valuations and Discrepancies. . . . . . . . . . . . . . . . . . 2.2. Valuations Defined by Algebraically Fat Arcs. . . . . . . . . . . . 2.3. Minimal Log Discrepancies and the Log Canonical Threshold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Arc Spaces and the Log Canonical Threshold. . . . . . . . . . . . 2.5. The Nash Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Motivic Volume and Birational Invariants. . . . . . . . . . . . . . . . . . . 3.1. Motivic Power Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The Jacobian Ideal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Motivic Igusa Zeta Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Stringy Invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. The Theorem of Batyrev–Kontsevich. . . . . . . . . . . . . . . . . . . . . 4. Denef–Loeser’s Zeta Function and the Monodromy Conjecture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Motivic Zeta Functions Associated with Hypersurfaces. . . 4.2. The Motivic Nearby Fiber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Lefschetz Numbers of the Monodromy. . . . . . . . . . . . . . . . . . . 4.4. The Motivic Monodromy Conjecture. . . . . . . . . . . . . . . . . . . . . 5. Motivic Invariants of Non-Archimedean Analytic Spaces. . . . 5.1. Néron Smoothening for Formal R-schemes Formally of Finite Type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Motivic Integration of Volume Forms on Rigid Varieties . 5.3. The Motivic Serre Invariant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Comparison with p-adic Integration. . . . . . . . . . . . . . . . . . . . . . 5.5. The Trace Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Motivic Zeta Functions of Formal Schemes and Analytic Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Definition of the Motivic Zeta Function. . . . . . . . . . . . . . . . . . 6.2. Bounded Differential Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Resolution of Singularities for Formal Schemes. . . . . . . . . . . 6.4. Néron Smoothening After Ramification. . . . . . . . . . . . . . . . . . 6.5. A Formula for the Motivic Zeta Function. . . . . . . . . . . . . . . . 6.6. Comparison with Denef and Loeser’s Motivic Zeta Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7. Motivic Zeta Functions of Calabi–Yau Varieties. . . . . . . . . . 7. Motivic Serre Invariants of Algebraic Varieties. . . . . . . . . . . . . . 7.1. Weak Néron Models of Algebraic Varieties. . . . . . . . . . . . . . . 7.2. Motivic Integrals and Motivic Serre Invariants for Smooth Algebraic Varieties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Motivic Serre Invariants of Open and Singular Varieties. . 7.4. The Trace Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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384 386 386 388 391 393 400 403 403 405 407 414 417 419 419 423 424 427 427 428 429 434 435 437 439 439 440 441 444 446 449 451 451 452 455 458 460

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Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 1. Constructibility in Algebraic Geometry. . . . . . . . . . . . . . . . . . . . . 1.1. Constructible Subsets of a Scheme. . . . . . . . . . . . . . . . . . . . . . . 1.2. The Constructible Topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Constructible Subsets of Projective Limits. . . . . . . . . . . . . . . § 2. Birational Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Blow-Ups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Resolution of Singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Weak Factorization Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Canonical Divisors and Resolutions. . . . . . . . . . . . . . . . . . . . . . 2.5. K-equivalence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. A Birational Cancellation Lemma. . . . . . . . . . . . . . . . . . . . . . . . § 3. Formal and Non-Archimedean Geometry. . . . . . . . . . . . . . . . . . . 3.1. Formal Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Morphisms of Finite Type and Morphisms Formally of Finite Type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Smoothness and Differentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Formal Schemes over a Complete Discrete Valuation Ring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Non-Archimedean Analytic Spaces. . . . . . . . . . . . . . . . . . . . . . .

465 465 465 467 468 469 469 470 471 472 474 476 478 478

Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

499

Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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483 486 488 491

CHAPTER 1 PROLOGUE: p-ADIC INTEGRATION

Motivic integration and some of its applications take they very inspiration from results of p-adic integration, that is, integration on analytic manifolds over non-Archimedean locally compact fields. This prologue aims at presenting this theory in a mostly self-contained manner, as well as three main results. The first one (theorem 1.5.3) is the classification by Serre (1965) of p-adic compact manifold, that led to the definition by Loeser and Sebag (2003) of the motivic Serre invariant of an analytic space. We then give the proof of Batyrev’s theorem that birational Calabi–Yau varieties have the same Betti numbers, the very theorem that inspired Kontsevich’s definition of motivic integration and his theorem that birational Calabi–Yau varieties have the same Hodge numbers. Nevertheless, we also explain an alternative deduction by Wang (2002); Ito (2003), using p-adic Hodge theory. Finally, we introduce Igusa’s local zeta function in §3 and its application to the definition of an invariant of complex singularities, the topological zeta function of Denef and Loeser (1992). We also discuss Igusa’s monodromy conjecture and survey the main known results in its direction.

§ 1. ANALYTIC MANIFOLDS 1.1. Local Fields (1.1.1) Valued Fields. — Let K be a field. An absolute value |·| on K is a map |·| : K → R0 satisfying the following properties: – |0| = 0, |1| = 1; – for every a, b ∈ K, |a + b|  |a| + |b| (triangle inequality); – for every a, b ∈ K, |ab| = |a||b|. © Springer Science+Business Media, LLC, part of Springer Nature 2018 A. Chambert-Loir et al., Motivic Integration, Progress in Mathematics 325, https://doi.org/10.1007/978-1-4939-7887-8_1

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CHAPTER 1. PROLOGUE: p-ADIC INTEGRATION

A field with an absolute value is called a valued field. The most obvious examples are probably the field of real numbers and the field of complex numbers, together with the usual absolute values. There is a “trivial” example that will often have to be excluded, namely that of the trivial absolute value for which |a| = 1 for every a ∈ K × . In any case, we shall be mostly interested in this book by examples of a different nature, where the absolute value satisfies the ultrametric inequality: |a + b|  max(|a|, |b|); these absolute values and the corresponding valued fields are called ultrametric, or non-Archimedean. Example 1.1.2. — Let p be a prime number. The p-adic absolute value on the field Q of rational numbers is defined as follows: every nonzero a ∈ Q can be written as a fraction pu m/n, where u ∈ Z and m, n are integers that are not divisible by p; the p-adic absolute value of a is then defined by |a|p = p−u . It is well defined, by uniqueness of the decomposition in prime factors. The first and third axiom in the definition of an absolute value are obvious to check. The second one is verified by the following simple computation: if a = pu m/n and b = pv s/t, with u  v and m, n, s, t prime to p, then a + b = pu (mt + pv−u sn)/nt. The denominator nt of the fraction is prime to p but the numerator mt + pv−u sn might be divisible by p. In any case, one can write a + b in the form pU M/N , for some prime to p integers M and N , and some integer U  u; as a consequence, |a + b|p = p−U  p−u = max(|a|p , |b|p ). This shows that |·|p is indeed an absolute value on Q, and that it satisfies the ultrametric inequality. The corresponding topology (see below) will be called the p-adic topology. (1.1.3). — In the general context of a valued field, we can talk about disks in K: for a ∈ K and r ∈ R0 , the “open disk” D(a, r) and the “closed disk” E(a, r) are the sets of points x ∈ K such that |x − a| < r and |x − a|  r, respectively. The absolute value |·| defines a metric on K, defined by d(x, y) = |x − y|, and we can consider the associated metric topology on K. The open disk D(a, r) is open in this topology, and the closed disk E(a, r) is closed. If the absolute value is trivial, then the topology is discrete: every open disk D(a, 1) is reduced to a point. Let us assume that K is non-Archimedean. In this case, the topology of K is totally discontinuous: the only non-empty connected subsets are singletons. Indeed, it follows from the ultrametric inequality that if two disks intersect, one must be contained in the other: let x ∈ E(a, r) ∩ E(b, s) and assume that r  s; then |x − a|  r and |x − b|  s, so that |a − b| = |(a − x) − (b − x)|  s; moreover, for every y ∈ E(a, r), then |y − b| = |(y − a) − (a − b)|  s, hence y ∈ E(b, s). Similarly, every point of an open or closed disk is a center of the disk. This also implies that the closed disk E(a, r) is open in K and that the open disk D(a, r) is closed in K, all a in K and all r > 0.

§ 1. ANALYTIC MANIFOLDS

3

Definition 1.1.4. — Let K be a valued field. a) One says that K is complete if its metric topology is complete, that is, if every Cauchy sequence converges. b) One says that K is a local field if its absolute value is nontrivial and its metric topology is locally compact.  (1.1.5). — Every valued field K has a completion, which is a valued field K,  coincides containing K as a dense subset on which the absolute value of K with that of K (Bourbaki 1985, chap. 6, §5, no 3). Remark 1.1.6. — Let K be a local field. a) Let us show that every closed disk E(0, s) in K is compact. Let thus s ∈ R0 . Let B be a compact neighborhood of the origin in K; by definition, there exists r > 0 such that E(0, r) ⊂ B; then E(0, r) is compact, as a closed subset of the compact set B. Let also a ∈ K × be such that |a| = 1; up to considering a−1 , we assume that 0 < |a| < 1. Then the map x → am x is a homeomorphism from the closed disk E(0, s) to the disk E(0, |a|m s). For m large enough, one has |a|m s  r, and E(0, |a|m s) is a closed subset of the compact disk E(0, r), hence is compact. b) The field K is complete. Indeed, a Cauchy sequence (xn ) in K is bounded, hence possesses a limit value x, because K is locally compact. Then one has x = lim(xn ). Example 1.1.7. — Let p be a prime number. The field of rational numbers is not complete for the p-adic topology. For instance, the sequence n 2 ( i0 pi )n∈N is a Cauchy sequence in Q that has no limit in Q. The completion of Q with respect to the p-adic absolute value is called the field of p-adic numbers and is denoted by Qp ; it is endowed with the unique absolute value which extends the p-adic absolute value. By the ultrametric inequality, the closed unit disk E(0, 1) is a subring of Qp ; we denote it by Zp and call it the ring of p-adic integers. Moreover, the open unit disk D(0, 1) coincides with the ideal (p) of Zp generated by p. For each n  0, the quotient ring Zp /(pn ) is isomorphic to Z/pn Z; in particular, this quotient is finite. A pigeonhole argument implies that every sequence in Zp has a converging subsequence, and thus that Qp is locally compact. Example 1.1.8. — Let k be a field and let K = k((T )) be the field of Laurent series with coefficients in k. Let us fixa real number q > 1. The elements of K are formal series of the form a = n∈Z an T n , with coefficients an ∈ k, such that an = 0 if n is sufficiently small. If a = 0, one sets |a| = q −n , where n is the smallest integer such that an = 0. This defines an ultrametric absolute value on K, called the T -adic absolute value with base q, for which K is complete. The valued field K is locally compact if and only if the field k is finite.

4

CHAPTER 1. PROLOGUE: p-ADIC INTEGRATION

(1.1.9). — Let K be a complete valued field and let L be an algebraic extension of K. One can prove, using Hensel’s lemma (lemma 1.3.2 below), that the absolute value on K can be extended in a unique way to an absolute value on L; see Bourbaki (1985, chap. 6, §8, no 7). Assume, moreover, that L is a finite extension of K; since L is a finite dimensional K-vector space, its topology is complete, and is locally compact if and only if K is locally compact. In particular, finite extensions of Qp , or of Fp ((T )) are local fields in a canonical way. As we now explain, these are essentially the only local fields. (1.1.10) Archimedean Valued Fields. — Let K be a valued field whose absolute value is Archimedean. If K is complete, then it follows from theorems of Gelfand–Mazur and Ostrowski that K = R or K = C, with the absolute value a power |·|e of their usual absolute value with exponent e ∈ ]0, 1], (Bourbaki 1985, chap. 6, §6, no 4, théorème 2). (1.1.11) Non-Archimedean Valued Fields. — Let K be a non-Archimedean valued field. Then the closed unit disk R = E(0, 1) is a subring of K, and the open unit disk m = D(0, 1) is the unique maximal ideal of R (one says that R is a local ring). The quotient field k = R/m is called the residue field of K. Note that R satisfies the property of being a valuation ring: for every x in K × , either x or x−1 belongs to R; in particular, the fraction ring of R is equal to K. For this reason, the ring R is called the valuation ring of K. If the absolute value on K is trivial, then R = K, m = (0) and k = K. (1.1.12) Non-Archimedean Local Fields. — Let K be a non-Archimedean valued field; let R be its valuation ring, m be its maximal ideal and k be its residue field. Then K is locally compact if and only if m is a principal ideal and k is a finite field. Indeed, the quotient k is both compact (as a quotient of the compact space R) and discrete (because m is open), hence k is finite. Moreover, the ideal m is principal and R is a discrete valuation ring. Indeed, m is also compact, hence there exists  ∈ m such that || = supx∈m |x|; one has 0 < || < 1; let q = 1/||. Let us show that such an element  generates m (one says that it is a uniformizer). Let x be any nonzero element of K and let n ∈ Z be maximal such that |x|  ||n , so that x/n ∈ R. Then |x| > ||n+1 , hence || < |x/n |  1. The definition of  implies that |x/n | = 1, and u = x/n is a unit in R. Consequently x = un ; define vK (x) = n = − log(|x|)/ log(q), so that |x| = q −vK (x) . Then x ∈ R if and only if vK (x)  0. Moreover, x ∈ m if and only if vK (x)  1, which implies that x ∈ (), as claimed. The map vK : K × → Z is a normalized discrete valuation on K. If K has characteristic zero, then K is a finite extension of the p-adic field Qp , and there exists a positive real number r > 0 such that the absolute value on K is the unique extension of |·|rp (Bourbaki 1985, chap. 6, §9, no 3, théorème 1).

§ 1. ANALYTIC MANIFOLDS

5

If K has characteristic p > 0, then K is a finite extension of the field Fp ((T )) of Laurent series with coefficients in the finite field Fp , and there exists a real number q > 1 such that the absolute value on K is the unique extension of the T -adic absolute value with base q. (1.1.13) Haar Measure. — Let K be a local field. By a general theorem on locally compact groups, see for example (Bourbaki 1963, chap. 7, §1, no 2, theorem 1), there exists a Haar measure μ on the additive group (K, +), unique up to multiplication by a positive real number. For every a ∈ K × , the map Ω → μ(aΩ) is again a Haar measure on (K, +), hence there exists a nonnegative real number modK (a) such that μ(aΩ) = modK (a)μ(Ω) for every bounded measurable subset Ω of K. If we set modK (0) = 0, then the map modK : K → R0 satisfies modK (ab) = modK (a) modK (b) for all a, b ∈ K; it is called the modulus of K and is independent of the choice of the Haar measure μ. Let us show how the modulus is related to the absolute value. If K = R then modK is the usual absolute value, and if K = C it is the square of the usual absolute value. Now assume that K is non-Archimedean. Let R be its valuation ring, m its maximal ideal and k the residue field. Let  ∈ m be any uniformizer. Let (ax )x∈k be a system of representatives of k in R: for each x ∈ k, ax is an element of R whose residue class modulo m equals x. Then the closed unit disk E(0, 1) decomposes as the disjoint union of the open unit disks D(ax , 1); since the measure μ is invariant by translation, one has  μ(D(ax , 1)) = Card(k)μ(D(0, 1)). μ(E(0, 1)) = x∈k

Moreover, D(0, 1) = m = E(0, ||), which implies that 1 μ(E(0, 1)). μ(E(0, ||)) = Card(k) By induction, one deduces that for every integer n ∈ Z μ(E(0, ||n )) = Card(k)−n μ(E(0, 1)). Let c = − log(||)/ log(Card(k)) be the unique positive real number such that || = Card(k)−c . Then for every a ∈ K, one has μ(E(0, |a|)) = |a|c μ(E(0, 1)). Thus the modulus is given by modK : K → R0 ,

a → modK (a) = |a|c .

This is again an absolute value on K, which defines the same topology on K as |·|. We say that the absolute value |·| is normalized if c = 1, i.e., if |·| = modK . From now on, we will normalize as follows the Haar measure on K: if K is non-Archimedean, with valuation ring R, then we consider the unique Haar measure μ on K such that μ(R) = 1. If K = R, resp. K = C, then we consider the unique Haar measure μ on K such that the set modK (z)  1 has measure 2, resp. 2π; these are the usual Lebesgue measures on R and C.

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1.2. Analytic Manifolds (1.2.1) Convergent Power Series. — The rudiments of the theory of analytic functions of several variables can be worked out over any complete valued field. A nice introduction can be found in Igusa (2000); we will review some basic properties and theorems. Let K be a complete valued field. For every a ∈ K d and every r ∈ R0 , we write D(a, r) and E(a, r) for the open and closed polydisks of radius r centered at a in K d : D(a, r) = {x ∈ K d ; |xi − ai | < r for all i}, E(a, r) = {x ∈ K d ; |xi − ai |  r for all i}. When a = (0, . . . , 0), we also write Dd (0, r) and E d (0, r) to avoid ambiguities. As in the case d = 1, one observes that if two closed polydisks meet, then one of them is contained in the other.  We consider power series f (T ) = n∈Nd fn T1n1 · · · Tdnd in d variables with coefficients in K. We shall use the standard multi-index notation, writing n = T1n1 · · · Tdnd for n ∈ Nd . Then we have |n| = n 1 + · · · + nd and T f (T ) = n∈Nd fn T n . The radius of convergence of f is defined by ρ(f ) = sup{r  0 ; the sequence (|fn |r|n| ) is bounded} = sup{r  0 ; the sequence (|fn |r|n| ) converges to 0}  −1 =

lim sup|fn |1/|n|

.

|n|→∞

One also has ρ(f )  sup{r  0 ; the series



fn xn converges for all x ∈ D(0, r)},

n

and equality holds if |K × | is dense in R>0 . In particular, the series f (x) =  n d n fn x defines a continuous function on the open polydisk D (0, ρ(f )), and this function determines the power series when ρ(f ) > 0 and the valuation of K is nontrivial. Power series with a positive radius of convergence will be called convergent power series. For every a ∈ Dd (0, ρ(f )), we can expand f (a + T ) as a power series in T . When K is ultrametric, one can check that ρ(f (a + T )) = ρ(f ): in the particular case where |K × | is dense in R>0 , this follows from the fact that Dd (0, r) = D(a, r) for all r  ρ(f ). If f is convergent and g1 , . . . , gn are power series in e variables U1 , . . . , Ue , with constant term 0, then the formal power series f (g1 (U ), . . . , gd (U )) is convergent.

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7

(1.2.2) Analytic Functions. — Let U be an open subset of K d . One says that a function f : U → K is K–analytic (1) if, for every point a ∈ U , there exist a convergent power series fa ∈ K[[T ]] such that f (x) = fa (x − a) for all x ∈ D(a, ρ(fa )) ∩ U . Every K-analytic function is continuous. It is clear that the K-analytic functions U → K form a K-algebra with the operations of pointwise addition and multiplication; moreover, if a K-analytic function on U does not vanish anywhere, then its inverse is K-analytic as well. If m is a positive integer, a function f : U → K m,

u → (f1 (u), . . . , fm (u)),

is said to be K-analytic if the coordinate functions f1 , . . . , fn are K-analytic on U . Let U be an open subset of K d , V be an open subset of K e and p be a positive integer. Let f : U → V and g : V → K p be K-analytic functions. Then the composition g ◦ f : U → K p is K-analytic. (1.2.3) Differential Calculus. — The partial derivatives of a power series are  defined as usual: if f = n∈Nd fn T n , then  ∂f ni+1 = ni fn T1n1 . . . Tini −1 Ti+1 . . . Tdnd . ∂Ti d n∈N

Their radii of convergence are at least equal to that of f , and actually equal if K has characteristic zero. Now, let f be a K-analytic function on an open subset U of K d . Its partial derivatives are themselves defined as usual, by the formula f (a + tεi ) − f (a) ∂f , (a) = lim t→0 ∂xi t for i ∈ {1, . . . , d}, where (ε1 , . . . , εd ) is the standard basis of K d . They are themselves K-analytic. Indeed, if f is defined by a convergent power series fa in an open neighborhood of fa , then ∂f /∂xi is defined by the convergent power series ∂fa /∂Ti in this neighborhood. As in classical differential geometry, we define the Jacobian matrix of a Kanalytic map f : U → K d as its first derivative; it is thus the matrix-valued analytic map on U given by Df (a) = (∂fi /∂xj (a)) . Its determinant defined an analytic map Jf on U , called the Jacobian determinant of f . Theorem 1.2.4 (Implicit function theorem). — Let m and n be positive integers and set X = (X1 , . . . , Xm ) and Y = (Y1 , . . . , Yn ). Let

(1) Such functions are sometimes called locally analytic; they should not be confused with the rigid analytic functions.

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F1 (X, Y ), . . . , Fn (X, Y ) be elements of K[[X, Y ]] such that Fi (0, 0) = 0 for all i and such that  ∂Fi (0, 0) = 0. det ∂Yj 1i,jn a) There exists a unique family f = (f1 , . . . , fn ) of elements in K[[X]] such that fi (0) = 0 and Fi (X, f (X)) = 0 for all i. If Fi is a convergent power series for all i, then fi is convergent for all i. If a is sufficiently close to 0 in K m , then (a, f (a)) lies in the domain of convergence of each of the Fi and Fi (a, f (a)) = 0. If (a, b) is sufficiently close to (0, 0) in K m+n and F (a, b) = 0, then b = f (a). b) Assume, moreover, that each of the Fi has coefficients in R and that

∂Fi det (0, 0) 1i,jn ∈ m. ∂Yj Then each of the fj has coefficients in R. Proof. — Part a) is proven is (Igusa 2000, 2.1.1). For part b), see point (i) of Igusa (2000, 2.2.1). To be precise, the property is only stated there under the assumption that the valuation on K is discrete, but the proof works in general. In any case, we will only use this property when K is local, and thus discretely valued. Theorem 1.2.5 (Local inversion). — Let U be an open subset of K d , and let f : U → K d be a K-analytic function. Let a be a point in U such that the Jacobian matrix Df (a) of f at a does not vanish. Then there exist an open neighborhood Ua of a in U such that f (Ua ) is an open neighborhood of f (a) in K d , and a K-analytic function g : f (Ua ) → Ua such that g ◦ f = idUa and f ◦ g = idf (Ua ) . Proof. — This follows from the implicit function theorem, see Igusa (2000, theorem 2.2.1). (1.2.6) Analytic Manifolds. — Let M be a topological space. A ddimensional chart on M is the datum (U, ϕ) of an open subset U ⊂ M and a homeomorphism ϕ from U to an open subset of K d . Two d-dimensional charts (U, ϕ) and (V, ψ) are called compatible if the homeomorphism ϕ(U ∩ V ) → ψ(U ∩ V ) defined by composing ψ and the inverse of ϕ is K-analytic, as well as the map ψ(U ∩ V ) → ϕ(U ∩ V ) defined by composing ϕ and the inverse of ψ. A d-dimensional K-analytic atlas on M is a set of mutually compatible charts (Ui , ϕi ) such that the union of the sets Ui covers M . A K-analytic manifold of dimension d is a topological space M together with a d-dimensional K-analytic atlas on M , up to the following equivalence relation: two atlases are identified if their union is again an atlas on M , in other words if every chart of the first atlas is compatible with every chart of the second. If one likes, one can consider all possible charts at once and

§ 1. ANALYTIC MANIFOLDS

9

define a K-analytic manifold as a topological space together with a maximal atlas. In this context, a function f : U → K defined on an open subset U of M , is said to be K-analytic if, for every chart (V, ψ) such that V ⊂ U , the map f ◦ ψ −1 : ψ(V ) → K is K-analytic. It suffices that this property holds for a family (Vi , ψi ) of charts such that Vi = U . Standard notions from differential geometry can be copied almost verbatim. A morphism f : M → N of K-analytic manifolds is a continuous map such that for every open subset V of N and every K-analytic function u on V , the function u ◦ f on f −1 (V ) is K-analytic. One also defines an isomorphism of K-analytic manifolds as a bijective morphism whose inverse is K-analytic. As in differential geometry, this theory can also be described using the point of view of sheaves. Observe that the K-analytic functions on M form a subsheaf OM in local K-algebras of the sheaf of K-valued functions on M , so that a K-analytic manifold can be naturally viewed as a locally K-ringed space. Conversely, one can define a K-analytic manifold of dimension d to be a locally K-ringed space which is locally isomorphic to the polydisk E d (0, 1) with its sheaf of K-analytic functions. One shows in a standard way that this definition is equivalent to the previous one. The affine space K d is a K-analytic manifold. An open subset U of a Kanalytic manifold has a canonical structure of K-analytic manifold, defined by restricting the open sets of the charts of the atlas (or taking the charts of a maximal atlas that are contained in U , or by restricting the structure sheaf to U ). The product of two K-analytic manifolds admits a canonical structure of a K-analytic manifold. The graph of a morphism of K-analytic manifolds is a K-analytic manifold. It is often necessary to restrict oneself to K-analytic manifolds which are paracompact, in particular Hausdorff. Example 1.2.7. — Let X be a smooth K-scheme, purely of dimension d. We shall explain in section (1.6) below how the set M = X(K) can be endowed with a canonical structure of a K-analytic manifold. 1.3. Hensel’s Lemma (1.3.1). — Let R be a local ring, that is, a ring possessing a unique maximal ideal; let m be its maximal ideal and let k be its residue field. One says that the ring R is henselian if for every polynomial f ∈ R[T ] and every a ∈ R such that f (a) ∈ m and f  (a) ∈ m, there exists a unique element b ∈ R such that f (b) = 0 and b ≡ a (mod m). This is equivalent to saying that every simple root of f in k lifts uniquely to a root of f in R.

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Lemma 1.3.2 (Hensel’s lemma in one variable) Let R be a complete discrete valuation ring. Let f ∈ R[T ] be a polynomial. Let n  e  0 be integers and let a be an element of R such that f (a) ≡ 0 (mod mn+e+1 )

and

f  (a) ≡ 0 (mod me+1 ).

Then there exists a unique element b in R such that f (b) = 0

and

b ≡ a (mod mn+1 ).

In particular, R is a henselian ring. Proof. — Let  be a uniformizer in R. We will find the desired element b by writing b = a + n+1 u and solving the equation f (a + n+1 u) = 0 in the unknown u over the ring R. By the Taylor expansion of f , there exists a unique polynomial g ∈ R[T ] such that f (a + U ) = f (a) + f  (a)U + U 2 g(U ), so that our equation can be rewritten as f (a) + f  (a)n+1 u + 2n+2 u2 g(n+1 u) = 0. By assumption, there exists u1 ∈ R such that f (a) = u1 n+e+1 . Moreover, since f  (a) is not divisible by e+1 , the ratio u2 = e /f  (a) belongs to R. Our equation can then be rewritten as u=−

f (a) f  (a)n+1



2n+2 2 u g(n+1 u) = −u1 u2 − 2n+2−e u2 u2 g(n+1 u). f  (a)

Since n  e  0, one has 2n  e, hence u ≡ −u1 u2 (mod m). Let us thus set a1 = a − n+1 u1 u2 = a −

f (a) . f  (a)

Then a1 is an element of R such that a ≡ a1 (mod mn+1 ) and f (a1 ) ≡ 0 (mod mn+e+2 ). Moreover, every element of R satisfying these two properties is congruent to a1 modulo mn+2 . Define a sequence (am )m∈N by a0 = a and am+1 = am − f (am )/f  (am ) for every m  0 (Newton’s iteration). What precedes shows that for every integer m  0, one has am ≡ am+1 (mod mn+m+1 ), and that f (am ) ∈ mn+m+e+1 . Since the ring R is complete, the sequence (am ) converges to an element b ∈ R; one has b ≡ a (mod mn+1 ) and f (b) = 0. Moreover, one proves by induction on m that every element b of R such that b ≡ a (mod mn+1 ) and f (b ) = 0 satisfies b ≡ am (mod mn+m+1 ); consequently, b is the only such element. Hensel’s lemma can be generalized to formal power series in several variables. Lemma 1.3.3 (Hensel’s lemma). — Let R be a complete discrete valuation ring and let m be its maximal ideal. We fix integers r   0

§ 1. ANALYTIC MANIFOLDS

11

and n  e  0. Let f = (f1 , . . . , f ) be elements of R[[T1 , . . . , Tr ]]. Let a = (a1 , . . . , ar ) ∈ (m)r be such that fi (a) ≡ 0 (mod mn+e+1 ) for every i in {1, . . . , }. Assume that the principal ( × )-minor

 ∂fi Δ = det (a) ∂Tj i,j=1,..., of the Jacobian matrix Df (a) does not belong to me+1 . Then there exists a unique element b = (b1 , . . . , br ) of Rr such that f (b) = 0, aj ≡ bj (mod mn+1 ) for j ∈ {1, . . . , }, and bj = aj for j ∈ { + 1, . . . , r}. Proof. — Adjoining to (f1 , . . . , f ) the equations fj = Tj − aj for j ∈ { + 1, . . . , r}, we may assume that = r. Let A = Df (a) be the Jacobian matrix of f at a and let B ∈ Mr (R) be its adjugate; one has AB = BA = ΔIr . Let us choose a uniformizer  of R. We seek a solution b in the form b = a + n+1 u, with u ∈ Rr . We shall show that there exists an element u ∈ Rr , unique modulo m, such that b ≡ a (mod mn+1 ) and f (b) ≡ 0 (mod mn+2+e ). There exists a unique family g = (g1 , . . . , gr ) of power series in R[[T1 , . . . , Tr ]] belonging to the ideal (T1 , . . . , Tr )2 such that f (a1 + T1 , . . . , ar + Tr ) = f (a1 , . . . , ar ) + A · (T1 , . . . , Tr ) + g(T1 , . . . , Tr ). As a consequence, there exists a family h = (h1 , . . . , hr ) of power series in R[[T1 , . . . , Tr ]] such that f (a + n+1 u) = f (a) + n+1 A · u + 2n+2 h(u). Let v = −f (a)/n+1 Δ. By assumption, each coordinate of f (a) is divisible by n+e+1 , and e+1 does not divide Δ, hence one has v ∈ Rr . By what precedes, one has f (a + n+1 Bv) = f (a) − Δ−1 A · B · f (a) + 2n+2 h(Bv) ≡ 0

(mod 2n+2 ).

Since n  e, the congruence f (a + n+1 Bv) ≡ 0 (mod mn+e+e ) holds in particular. Conversely, the equation f (a + n+1 u) ≡ 0 (mod mn+e+2 ) is equivalent to the equation f (a) + n+1 A · u ≡ 0 (mod mn+e+2 ). Since e+1 does not divide Δ, it implies the equation u ≡ v (mod m)n+2 . By induction on m, it follows that for every m  1, there exists an element a(m) ∈ Rr , unique modulo mm+n+1 , such that f (a(m) ) ≡ 0 (mod mm+e+n+1 ) and a(m) ≡ a (mod mn+1 ). The sequence (a(m) ) converges to an element a) = 0. Moreover, every such element is congruent a ˜ ∈ Rr such that f (˜ to a(m) modulo mn+m+1 for every m, hence is equal to a ˜. The following corollary is a variant of lemma 1.3.3 for restricted power series, namely for formal power series f ∈ R[[T1 , . . . , Tr ]] whose coefficients tend to 0. The radius of convergence of such a series f satisfies ρ(f )  1. This implies in particular that for every a ∈ Rr , f can be evaluated at T = a, just

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by substituting ai for Ti in the expansion of f . The restricted power series form a noetherian R-algebra, denoted R{T1 , . . . , Tr }. Corollary 1.3.4. — The analogous statement holds when the power series fi are restricted power series and a is an element of Rr . Proof. — Indeed, we reduce to the previous situation by applying the coordinate change Tj = Tj − aj . 1.4. Differential Forms and Measures (1.4.1) Integration on the Affine Space. — In this section, we assume that K is a local field. We recall that the modulus modK : K → R0 is defined by the property that μ(aΩ) = modK (a)μ(Ω) for every bounded measurable subset Ω ⊂ K and every a ∈ K. We endow the affine space K d with the tensor product measure dμ(x) = dμ(x1 ) ⊗ · · · ⊗ dμ(xd ). This is a Haar measure on the locally compact group (K d , +). In the sequel, we shall devote much interest to a geometric understanding of the analytic properties of integrals of particular functions on K d , see examples 3.1.5 and 3.1.6. Theorem 1.4.2 (Change of variables formula) Let U be an open set in K d and let f : U → K d be an injective K-analytic map whose Jacobian Jf does not vanish on U . Then, for every measurable positive (resp. integrable) function ϕ : f (U ) → R   ϕ(y) dμ(y) = ϕ(f (x)) modK (Jf (x)) dμ(x). f (U )

U

Proof. — A proof can be found, for instance, in Igusa (2000, 7.4.1). Proposition 1.4.3. — Let Ω be an open subset of K d . a) Let M a submanifold of Ω which has codimension  1 at every point. Then M has measure zero with respect to the measure on K d . b) Let f be a K-analytic function on Ω and let Z be its zero locus. If Z has empty interior, then Z has measure zero. Proof. — a) The question is local on Ω. Using the implicit function theorem and the change of variables formula, we may assume that Ω = E d (0, 1) and M = E d−c (0, 1) × {(0, . . . , 0)}, for some positive integer c. Since {0} has measure zero in K, it follows that M has measure zero. b) Again, we may assume that Ω = E d (0, 1) and that f is defined by a power series F which converges on Ω. Assume that Z has empty interior; then F = 0. After some linear change of variables by a matrix in GLd (R), the Weierstrass preparation theorem, see for example (Igusa 2000, theorem 2.3.1), reduces the question to the case where F = Tdm +am−1 Tdm−1 +· · ·+a0 , where

§ 1. ANALYTIC MANIFOLDS

13

a0 , . . . , am−1 are power series in T1 , . . . , Td−1 which converge on E d−1 (0, 1). Then the conclusion follows from the Fubini theorem. (1.4.4) Differential Forms. — Let U be an open subset of K d and let p be a nonnegative integer. A differential form of degree p on U is an element of the p-th exterior power of the free O(U )-module with basis (dx1 , . . . , dxd ). In other words, it is an object of the form  ωI dxi1 ∧ · · · ∧ dxip , ω= 1i1 0. Moreover, f∗ Q = Q , because f is a universal homeomorphism. Consequently, the Leray spectral sequence in étale cohomology with proper supports for the morphism f degenerates at E1 . This implies

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j j that the functoriality morphism f ∗ from Hét,c (Y, Q ) to Hét,c (X, Q ) is an isomorphism, as claimed. c) Let p be the characteristic of k. For every power q of p, let X (q) be the base change of X by the Frobenius morphism a → aq on k, and let Φ(q) : X → X (q) the relative Frobenius morphism. By Kollár (1997, proposition 6.6), there exist a power q of p and a morphism g : Y → X (q) such that g◦f = Φ(q) . Since Φ(q) induces an isomorphism on rigid cohomology, f ∗ is injective and g ∗ is surjective. On the other hand, g is also a universal homeomorphism, so that g ∗ is an isomorphism. Consequently, f ∗ is an isomorphism as claimed. (6)

Corollary 4.4.5. — Let k be a field. a) If k is finite, the motivic measures given by point counting and the zeta function factor through K0uh (Vark ). b) Let be a prime number which is invertible in k. The motivic measures given by the -adic étale realization χét , the Euler–Poincaré polynomial, and the Euler characteristic on Vark vanish on the ideal Ikuh and factor through the modified Grothendieck group of varieties K0uh (Vark ). c) Assume that k is a perfect field of positive characteristic. Then the motivic measure given by the rigid realization vanishes on the ideal I uh and factors through K0uh (Vark ). Proposition 4.4.6. — Let f : X → Y be a morphism of finite type between noetherian Q-schemes. If f is a universal homeomorphism, then f is a piecewise isomorphism. Proof. — Since fred : Xred → Yred is still a universal homeomorphism (ÉGA IV2 , §2.4.3 (vi)), we may assume that X and Y are reduced. By noetherian induction, we also assume that for every closed subscheme Z  Y , the morphism fZ : X ×Y Z → Z deduced from f (which is a universal homeomorphism, by assumption) is a piecewise isomorphism. It remains to show that there exists a dense open subset U of Y such that the morphism fU : X ×Y U → Y deduced from f by base change is an isomorphism. For this, we may assume that Y is irreducible. Then X is irreducible as well, because it is homeomorphic to Y . Let ηY be the generic point of Y ; then its inverse image in X consists of a unique point ηX , which is the generic point of X. The residue field κ(ηX ) is a purely inseparable extension of the residue field κ(ηY ) of ηY . Since these fields have characteristic zero, the morphism f induces an isomorphism of fields κ(ηX ) ∼ = κ(ηY ). It follows that there exists a dense open subset U of Y such that f induces an isomorphism from f −1 (U ) to U . This concludes the proof. Corollary 4.4.7. — For every Q-scheme S, one has ISuh = (0) so that the projection K0 (VarS ) → K0uh (VarS ) is an isomorphism. (6) We

thank B. Le Stum for having communicated us this proof.

§ 4. LOCALIZATION, COMPLETION, AND MODIFICATION

117

Remark 4.4.8. — Let k be a field of characteristic p > 0 and let S = Spec(k). Then we do not know whether ISuh = (0). If k is an imperfect field and k  is a finite nontrivial purely inseparable extension of k, then the canonical morphism from Spec(k  ) to Spec(k) is a universal homeomorphism, so that euh (Spec(k  )) = 1 in K0uh (Vark ). However, we do not know whether e(Spec(k  )) = 1 in K0 (Vark ). Similarly, let E be an elliptic curve over k whose j-invariant does not belong to Fp ; let E (p) = E ×ϕ k, where ϕ : k → k is the Frobenius morphism, given by x → xp . The absolute Frobenius morphism from E to itself factors through a k-morphism F : E (p) → E which is a universal homeomorphism. However, we do not know whether the classes of E and E (p) in K0 (Vark ) are equal or not. Remark 4.4.9. — Let k be a field. Then the Grothendieck ring K0uh (Vark ) of varieties modulo universal homeomorphisms has a natural interpretation in terms of model theory. Let L be the language of rings over k, and let ACFk be the theory of algebraically closed fields in the language L . Rapidly, formulas in L consist in combinations of polynomial equalities with coefficients in k with quantifiers (∀ “for all,” ∃ “there exists”) and Boolean connectors (∧ “and,” ∨ “or,” ¬ “negation”). One may also use logical connectors, such as → “implies,” ↔ “is equivalent to,” or ∃! “there exists a unique,” which are abbreviations; for example, if ϕ and ψ are two formulas, then ϕ → ψ is a shortcut for ¬ϕ ∨ ψ. A formula without free variables in the language L belongs to ACFk if, equivalently, (i) it can be proved from the axioms stating that a field is an algebraically closed extension of k, or (ii) it holds in any algebraically closed extension of k. Two formulas of the language L , say ϕ(x1 , . . . , xm ) and ψ(y1 , . . . , yn ), are said to be equivalent (modulo ACFk ) if there exists a formula η(x1 , . . . , xm , y1 , . . . , yn ) of L such that the following formula ∀y1 . . . ∀yn ψ(y1 , . . . , yn ) → ∃!x1 . . . ∃!xm η(x1 , . . . , xm , y1 , . . . , yn ) ∧ ϕ(x1 , . . . , xm ) belongs to the theory ACFk . By quantifier elimination in the theory ACFk , aka Chevalley’s constructibility theorem, every formula in L is equivalent (modulo ACFk ) to a quantifier free formula. In particular, a formula without free variables belongs to ACFk if and only if it holds in some algebraically closed extension of k. The Grothendieck group K0 (ACFk ) of the theory ACFk is the quotient of the free abelian group generated by equivalence classes of formulas in L by the subgroup generated by objects of the form [ϕ ∧ ψ] + [ϕ ∨ ψ] − [ϕ] − [ψ]

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whenever ϕ and ψ are formulas in L in the same free variables. This group is endowed by the unique ring structure for which [ϕ(x1 , . . . , xm )] · [ψ(y1 , . . . , yn )] = [ϕ(x1 , . . . , xm ) ∧ ψ(y1 , . . . , yn )], whenever ϕ and ψ are formulas in disjoint sets of free variables. There is a unique morphism of groups λ from K0 (Vark ) to K0 (ACFk ) such that for every family (pi )1im of polynomials in k[T1 , . . . , Tn ], the class e(V (p1 , . . . , pm )) of the closed subscheme of Ank defined by the polynomials pi is mapped to the class of the formula p1 (x1 , . . . , xn ) = 0 ∧ · · · ∧ pm (x1 , . . . , xn ) = 0. This morphism of groups λ is a morphism of rings. It follows from quantifier elimination that this morphism λ is surjective. To show that λ induces an isomorphism from K0uh (Vark ) to K0 (ACFk ), one first observes that the kernel of λ contains the ideal Ikuh . Indeed, if f : X → Y is a universal homeomorphism of (affine) k-varieties, it induces a bijection X(K) → Y (K), for every algebraically closed field extension K of k, so that the formulas of ACFk defining X and Y are equivalent. In ¯ : K uh (Vark ) → K0 (ACFk ). One conparticular, λ factors as a morphism λ 0 ¯ cludes the proof by showing that one can explicitly construct an inverse of λ by associating with the class of a formula ϕ in m free variables, the class in K0uh (Vark ) of the constructible set defined by ϕ. We refer to Nicaise and Sebag (2011, 5.3.13) for details. (4.4.10). — As for the rings K0 (VarS ) and MS , the relative dimension of S-varieties induces an increasing filtration on the rings K0uh (VarS ) and MSuh . Precisely, for every integer d ∈ Z, we define Fd K0uh (VarS ) as the subgroup generated by classes euh (X/S) of S-varieties X such that dim(X/S)  d. Similarly, let Fd MSuh be the subgroup generated by elements of the form euh (X/S)L−m S , where X is an S-variety and m ∈ Z is an integer such that dim(X/S) − m  d. In other words, the dimension filtration of MSuh is the image of the corresponding filtration of MS . As in sections 4.3.1 and 4.1.1, the filtration F• MSuh induces a topology on the ring MSuh which can be defined by an ultrametric semi-norm. Its separated completion is a non-Archimedean complete normed ring. (4.4.11). — Let f : T → S be a morphism of schemes. The class of universal homeomorphisms is stable by base change. Consequently, the (semi)ring morphisms f ∗ : K0+ (VarS ) → K0+ (VarT ), f ∗ : K0 (VarS ) → K0 (VarT ), and f ∗ : MS → MT pass to the quotient and give rise to (semi)ring morphisms f ∗ : K0+,uh (VarS ) → K0+,uh (VarT ), f ∗ : K0uh (VarS ) → K0uh (VarT ) and f ∗ : MSuh → MTuh . If f is finitely presented, every T -scheme of finite presentation can be viewed as an S-scheme of finite presentation. If X and Y are universally homeomorphic S-schemes, then they are universally homeomorphic as T -schemes. This gives rise to a morphism of additive monoids

§ 4. LOCALIZATION, COMPLETION, AND MODIFICATION

119

f! : K0uh (VarT ) → K0uh (VarS ) and to morphisms of abelian groups f! : K0uh (VarT ) → K0uh (VarS ) and f! : MTuh → MSuh . Proposition 4.4.12. — Let S be a scheme, let X, Y , and Z be S-varieties, and let f : X → Y be a morphism of S-schemes. Assume that for every perfect field F and every point y ∈ Y (F ), the F -schemes Xy = X ×Y Spec(F ) and Z ×S Spec(F ) are universally homeomorphic. Then e+,uh (X/S) = e+,uh (Y /S) · e+,uh (Z/S) in K0+,uh (VarS ). Proof. — Let us first assume that S is a noetherian scheme. Then X, Y, Z are noetherian schemes as well. By noetherian induction, we assume that for every closed subscheme Y   Y , one has e+,uh ((X ×Y Y  )/S) = e+,uh (Y  /S) e+,uh (Z/S). It is then enough to show the existence of a nonempty open subscheme U of Y such that e+,uh (X ×Y U/S) = e+,uh (U/S) · e+,uh (Z/S) in K0+,uh (VarS ). We may replace X and Y by their largest reduced subschemes; we may also assume that Y is affine and integral. Let B be the ring of regular functions on Y , so that Y  Spec(B), let κ(Y ) = Frac(B) be the function field of Y , and let F be the perfect closure of the field κ(Y ). By assumption, the F -schemes X ×Y Spec(F ) and Z ×S Spec(F ) are universally homeomorphic. Consequently, there exist an F -variety V and two F -morphisms p : V → X ×Y Spec(F ) and q : V → Z ×S Spec(F ) which are universal homeomorphisms. The B-algebra F is the direct limit of its finitely generated sub-B-algebras. Hence, by (ÉGA IV3 , §8.8.2 and §8.10.5), there exist a finitely generated subB-algebra B  of F , a B  -scheme V  , and morphisms of B  -schemes p : V  → X  ×Y Spec(B  ),

q  : V  → Z  ×S Spec(B  )

which are finitely presented and universal homeomorphisms and which induce the morphisms p and q by base change from Spec(B  ) to Spec(F ). Since the morphism Spec(B  ) → Y is purely inseparable over the generic point of Y , and the generic point of Y is the projective limit of the dense open subschemes of Y , it follows from (ÉGA IV3 , §8.10.5) that there exists a dense open subscheme U of Y such that the second projection Spec(B  ) ×Y U → U is a universal homeomorphism. Consequently, in the commutative diagram of S-schemes all arrows represent universal homeomorphisms of S-schemes. This implies the following equality e+,uh (X ×Y U/S) = e+,uh (V  ×Spec(B  ) U/S) = e+,uh (Z ×S U ) in K0+,uh (VarS ). This concludes the proof under the assumption that S is noetherian.

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V ⬘ ´ Spec(B⬘) U p ¢ ´ Id U

q ¢ ´ Id U

X ´ Y Spec(B ¢ ) ´ Y U

Z ´ S (Spec(B ¢ ) ´ Y U )

X ´YU

Z ´S U

In the general case, there exists a scheme S  of finite type over Z, a morphism u : S → S  , S  -varieties X  , Y  , Z  and an S  -morphism f  : X  → Y  such that X = X  ×S  S, Y = Y  ×S  S, Z = Z  ×S  S, and f = fS . Moreover, for every perfect field F and every point y  ∈ Y  (F ), one has an isomorphism Xy  = Xy of F -schemes, where y ∈ Y (F ) is deduced from y  . This shows that S  , X  , Y  , Z  , f  satisfy the hypothesis of the proposition. Consequently, one has the equality e+,uh (X  /S  ) = e+,uh (Y  /S  ) · e+,uh (Z  /S  ) in K0+,uh (VarS  ), and the desired formula follows by applying the base-change morphism u∗ : K0+,uh (VarS  ) → K0+,uh (VarS ).

§ 5. THE THEOREM OF BITTNER Let k be a field of characteristic 0 and let S be a k-variety. In this section, we give an alternative presentation, due to Bittner (2004) of the Grothendieck group K0 (VarS ) which only involves smooth k-varieties which are projective over S. We then deduce some important applications. 5.1. Bittner’s Presentation of K0 (VarS ) (5.1.1). — Let P be the property of an S-variety to be irreducible, smooth over k, and projective over S. Let VarPS be the full subcategory of VarS consisting of S-varieties that satisfy P. Isomorphism classes [X] of objects X in VarPS form a set, denoted VarPS . (5.1.2). — Assume moreover that S = Spec(k), the fiber product X ×k Y of two smooth projective irreducible varieties is again smooth and projective, hence is a disjoint union of smooth projective irreducible varieties. P This endows the free abelian group Z(Vark ) with a natural ring structure: if (Z1 , . . . , Zn ) is the family of irreducible components of X ×k Y , one sets [X] · [Y ] =

n  i=1

[Zi ].

§ 5. THE THEOREM OF BITTNER

121

Theorem 5.1.3 (Bittner 2004). — Let k be a field of characteristic zero and let S be a k-variety. Let P be the property of an S-variety to be irreducible, smooth over k, and projective over S. Let P

β : Z(VarS ) → K0 (VarS ) be the group homomorphism that maps the isomorphism class of an S-variety X ∈ VarPS to its class e(X/S) in K0 (VarS ). Then β is a surjective morphism of abelian groups, and its kernel is generated by the elements of the form [BlY (X)] − [X] − [E] + [Y ]

(“blow-up relations”)

where X is an irreducible projective k-smooth, S-variety, Y is a k-smooth integral closed subvariety of X, BlY (X) is the blow-up of X along Y , and E is its exceptional divisor. When S = Spec(k), then β is a morphism of rings. P

Proof. — Let G be the subgroup of Z(VarS ) generated by the elements of the form [BlY (X)] − [X] − [E] + [Y ], as in the statement of the theorem. We set P

A = Z(VarS ) /G; this is an abelian group. Since the blow-up relations hold in K0 (VarS ) (example 2.4.3), the morphism β factors through a unique morphism of groups, β˜ : A → K0 (VarS ), and we need to prove that β˜ is an isomorphism of groups. For every projective S-variety X which is k-smooth, let us write ˜e(X/S) for the sum of the classes in A of the irreducible components of X; by con˜ e(X/S)) = e(X/S). Let, moreover, Y be a closed struction, one has β(˜ ˜ be the blow-up of X along Y , subscheme of X which is k-smooth, let X ˜ and E are k-smooth, and and let E be its exceptional divisor; then X reasoning irreducible component by irreducible component, one sees that ˜ = ˜e(X) + ˜e(E) − ˜e(Y ). ˜e(X) We want to construct an inverse γ˜ to the group morphism β˜ : A → K0 (VarS ). This map γ˜ will be an A-valued additive invariant on VarS whose composition with β˜ induces the identity on K0 (VarS ). Let P denote the property of a S-variety to be smooth over k and quasi-projective over S. Since the canonical morphism ϕP : K0 (VarPS ) → K0 (VarS ) is an isomorphism, it is sufficient to construct the additive invariant γ˜ on VarPS . Let X be a k-smooth, projective S-variety, and let D be a divisor with strict normal crossings in X. Denoting by (Di ) the family of its irreducible components, let  γ˜ (X, D) = (−1)Card(J)˜e(DJ /S), J⊂I

where, for every subset J of I, DJ =

 j∈J

Dj .

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Lemma 5.1.4. — Let X be a projective S-variety, smooth over k, let D be an effective divisor with strict normal crossings in X, and let (Di )i∈I be the family of its irreducible components. Let Y be an irreducible closed subscheme of X. Let K be the set of all i ∈ I such that Y ⊂ Di ; we assume  that i∈K Di meets Y transversally. ˜ for the blow-up BlY (X) of X along Y , E for its exceptional We write X ˜ i for the strict transform of Di . divisor, and, for every i ∈ I D ˜ ˜ is projective, k-smooth; the divisor D ˜ = E + a) The S-variety X i∈K Di has simple normal crossings. ˜ D) ˜ = γ˜ (X, D); b) If Y ⊂ D, then γ˜ (X, c) Otherwise, D ∩ Y is a divisor with strict normal crossings in Y and one has ˜ D) ˜ = γ˜ (X, D) − γ˜ (Y, D ∩ Y ). γ˜ (X, ˜ → X be the blowing-up morphism. Being the blow-up Proof. — Let p : X of a k-smooth, projective S-variety along a k-smooth closed subscheme, the ˜ is projective and k-smooth. S-variety X The transversality assumption on Y and D implies moreover that the ˜ ˜ ˜ = E+ divisor D i∈J Di has strict normal crossings in X. Separating a ˜ subset of the set of irreducible components of D according to whether it contains E or not, we thus observe that   ˜ J /S) − ˜ J /S). ˜ D) ˜ = (−1)Card(J)˜e(D ˜e(E ∩ D γ˜ (X, J⊂I

J⊂I

˜ J of DJ identifies with the For every subset J of I, the strict transform D ˜ J identifies with its exceptional blow-up of DJ along Y ∩ DJ , and E ∩ D divisor. Consequently, one has ˜ J /S) = ˜e(DJ /S) − ˜e(Y ∩ DJ /S), ˜ J /S) − ˜e(E ∩ D ˜e(D hence the relation 

˜ J /S) − ˜e(E ∩ D ˜ J /S) ˜ D) ˜ = (−1)Card(J) ˜e(D γ˜ (X, J⊂I

=



(−1)Card(J) (˜e(DJ /S) − ˜e(Y ∩ DJ /S))

J⊂I

= γ˜ (X, D) − 





(−1)Card(J)˜e(Y ∩ DJ /S).

J⊂I

Let D = i∈K Di . The transversality assumption of Y with respect to D implies that Y ∩ D is a strict normal crossings divisor in Y , with irreducible components the irreducible components of Y ∩ Di , for i ∈ K. Let J be a subset of I and let J  = J ∩ K. Since Y ⊂ Di for i ∈ I K, one has ˜J = E ∩ D ˜ J  . Consequently, Y ∩ DJ = Y ∩ DJ  and E ∩ D

§ 5. THE THEOREM OF BITTNER

˜ D) ˜ − γ˜ (X, D) = γ˜ (X,



(−1)Card(J)˜e(Y ∩ DJ /S)

J⊂I

=

123



J  ⊂K



(−1)Card(J )˜e(Y ∩ DJ  )





(−1)Card(J ) .

J  ∈I K



For every finite set S, one has s⊂S (−1)Card(s) = (1 − 1)Card(S) = 1 if S = ∅ and 0 otherwise. Assume that Y ⊂ D. Since Y is irreducible, there exists i ∈ I such that ˜ D) ˜ = γ˜ (X, D). Y ⊂ Di ; in particular, K = I. Consequently, γ˜ (X, Assume now that Y ⊂ D, hence K = I and  ˜ D) ˜ − γ˜ (X, D) = (−1)Card(J)˜e(Y ∩ DJ ). γ˜ (X, J⊂I

For every i ∈ I, Y ∩ Di is a smooth divisor in Y , possibly not irreducible; these divisors meet transversally. Decomposing each divisor Y ∩ Di as a sum of smooth disjoint divisors, we see that  (−1)Card(J)˜e(Y ∩ DJ ) = γ˜ (Y, Y ∩ D). J⊂I

This concludes the proof of the lemma. Let us now return to the proof of theorem 5.1.3. So let X be a ksmooth, quasi-projective, S-variety. Taking an embedding of X in a projective space PnS and considering the closure of X in PnS , we see that there exists a projective S-variety X  containing a dense open subscheme isomorphic to X. By the theorem of Hironaka (1964) on strong resolution of singu¯ which is smooth over k, larities, there thus exists a projective S-variety X, ¯ ¯ D is isoand a divisor D ⊂ X with strict normal crossings such that X morphic to X. Let (Di )i∈I be the family  of irreducible components of D. For every subset J of I, we set DJ = j∈J Dj . Then DJ is a closed sub¯ which is k-smooth; in particular, DJ is projective over S. By scheme of X example 1.3.7, one has ¯ e(X/S) = e(X/S) − e(D/S)  ¯ (−1)Card(J)−1 e(DJ /S) = e(X/S) − =



∅=J⊂I Card(J)

(−1)

e(DJ /S).

J⊂I

In particular, one has ˜ γ (X, ¯ D)) = e(X/S). β(˜

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¯ D) only depends on X = X ¯ D We will show that this element γ˜ (X, ¯ and that the induced map X D → γ˜ (X, D) defines an additive invariant on Varksm,qproj with values in A. By the weak factorization theorem of Abramovich et al. (2002), any two ¯ D) and (X ¯  , D ) as above are linked by a chain of blowcompactifications (X, ups (or blow-downs) as in lemma 5.1.4 along k-smooth closed integral subschemes meeting the boundary transversally and disjoint from X. By induction, this implies that ¯ D) = γ˜ (X ¯  , D ). γ˜ (X, Let us denote this element of A by γ(X). Let now Y be a k-smooth closed subset of X, and let Y¯ be its Zariski ¯ By resolution of singularities, we may assume that Y¯ is kclosure in X. ˜ be the blow-up of X ¯ smooth and meets the boundary D transversally. Let X ¯ ˜ along Y , let D be the strict transform of D, and let E be the exceptional divisor. With the notation of lemma 5.1.4, ˜ D ˜ + E) γ(X Y ) = γ˜ (X, ¯ D) − γ˜ (Y¯ , D ∩ Y¯ ) = γ˜ (X, = γ(X) − γ(Y ). This proves that the map X → γ(X) is an additive invariant on Varksm,qproj ˜ such that β(γ(X)) = e(X/S) for every k-smooth, projective S-variety X. The final assertion then follows from the definitions of the map β and of P the ring structure on the abelian group Z(Vark ) . This concludes the proof of theorem 5.1.3. Remark 5.1.5. — The statement remains valid if one replaces the adjective “projective” by “proper.” The proof is the same. Corollary 5.1.6. — Let k be a field of characteristic zero. Let A be an abelian group and let χ : Varksm,proj → A be a map such that χ([BlY (X)]) = χ([X]) − χ([Y ]) + χ([E]), whenever X is a projective smooth k-scheme, Y a closed smooth subscheme, BlY (X) is the blow-up of X along Y and E is its exceptional divisor. Then there is a unique additive invariant χ ˜ on Vark which extends χ. If, moreover, A is a ring and χ([X × X  ]) = χ([X])χ([X  ]) for all smooth projective k-schemes X and X  , then χ ˜ is a morphism of rings. Corollary 5.1.7. — Let k be a field of characteristic zero. The abelian group Mk is generated by the elements e(X)/Lp , where X ranges over projective smooth irreducible varieties, with relations: a) For every closed smooth subscheme Y of X with blow-up BlY (X) and exceptional divisor E, and every integer p  0, one has e(BlY (X))/Lp − e(E)/Lp = e(X)/Lp − e(Y )/Lp .

§ 5. THE THEOREM OF BITTNER

125

b) For every smooth projective irreducible k-variety X and every integer p  0, e(X × P1 )/Lp+1 = e(X)/Lp + e(X)/Lp+1 . Proof. — Let A be the free abelian group generated by Varkproj,sm,irr ×N, and let R be the subgroup of A generated by elements of the form [BlY (X), p] − [E, p] − [X, p] + [Y, p] and [X × P1 , p + 1] − [X, p] − [X, p + 1], where X ranges over a set of representatives of isomorphism classes of projective smooth irreducible k-varieties, Y range over irreducible smooth closed subschemes of X, and p ranges over nonnegative integers. There exists a unique group homomorphism α : A → Mk such that α([X, p]) = e(X)/Lp for every pair (X, p); it is surjective and its kernel contains the subgroup R. We need to show that the resulting morphism from A/R to Mk is an isomorphism. To that aim, we construct its inverse explicitly. Observe that the group A has a natural ring structure such that [X, p][Y, q] =

m 

[Zi , p + q],

i=1

if X, Y are projective, smooth, and connected k-varieties, p, q ∈ N, and Z1 , . . . , Zm are the irreducible components of X ×k Y ; moreover, the subgroup R is an ideal, and the map α is a morphism of rings. In the ring A/R, one has [X, p + 1]([P1 , 0] − [1, 0]) = [X ×k P1 , p + 1] − [X, p + 1] = [X, p]. In particular, [1, 1] is the inverse of [P1 , 0] − [1, 0]. By theorem 5.1.3, there exists a unique morphism of rings β : K0 (Vark ) → A/R such that β(e(X)) = [X, 0] for every proper, smooth, and connected kvariety. One has, β(L) = β(P1 ) − β(1) = [P1 , 0] − [1, 0], so that β(L) is invertible. Consequently, there exists a unique morphism of rings β˜ : Mk → A/R which extends β. By definition, p −p ˜ ˜ ˜ ) = β(e(X))β(L) = [X, 0][1, p] = [X, p], β(α([X, p])) = β(e(X)/L

for every projective, smooth, connected k-variety and every integer p  0. This implies that β˜ ◦ α is the identity morphism. In particular, α is injective, which concludes the proof. Corollary 5.1.8. — Let k be a field of characteristic zero, and let Mk be the localization K0 (Vark )[L−1 ] of K0 (Vark ) with respect to L. There is a unique ring homomorphism D : Mk → Mk − dim(X)

such that D(e(X)) = L e(X) for every projective smooth integral kscheme X. Moreover, D is an involution: D ◦ D = idMk .

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CHAPTER 2. THE GROTHENDIECK RING OF VARIETIES

Proof. — Let X be a smooth projective k-scheme, and let Y be a smooth closed subscheme of X. By example 2.4.3, equation (2.4.3.3), one has L− dim(X) e(BlY (X)) − L− dim(E) e(E) = L− dim(X) e(X) − L− dim(Y ) e(Y ). Existence and uniqueness of D then follow at once from corollary 5.1.6. One has D(1) = 1 and D(e(P1 )) = L−1 e(P1 ) = 1 + L−1 , hence D(L) = −1 L . It follows that for every projective smooth integral k-scheme X, one has D(D(e(X))) = D(L− dim X e(X)) = D(L)− dim X D(e(X)) = Ldim X L− dim X e(X) = e(X). Another application of corollary 5.1.6 then implies that D ◦ D is the identity. Remark 5.1.9. — The morphism D is a motivic incarnation of Poincaré duality. a) Let be a prime number invertible in k. The functor V → V ∨ in RepGk Q induces an involutive ring morphism K0 (RepGk Q ) which we still denote by an exponent ∨ . Let us then show that for every object x ∈ Mk , one has χét (D(x)) = χét (x)∨ .

(5.1.9.1)

Let X ∈ Vark ; let d = dim(X). First assume that X is smooth, connected, and proper. In this case, the Poincaré duality theorem in étale coho2d−i i mology furnishes a canonical isomorphism Hét (X ⊗k k s , Q )  Hét (X ⊗k s ∨ k , Q ) (−2d) in RepGk Q . Since χét (L) = [Q (−1)], this implies the formula χét (D(X)) = χét (e(X)L−2d ) = χét (X)χét (L)−2d =

2d 

(−1)i [H i (X ⊗k k s , Q )](2d)

i=0

=

2d 

(−1)i [H 2d−i (X ⊗k k s , Q )∨ ]

i=0

= χét (X)∨ in K0 (RepGk Q ). Moreover, D(L) = L−1 , χét (D(L)) = χét (L−1 ) = χét (L)−1 = Q (1) = χét (L)∨ . This implies that the given formula holds when x = [X]L−p . Since the elements of this form span Mk , this formula holds in general. b) When k = C, a similar argument shows that the analogous formula (5.1.9.2)

χHdg (D(x)) = χHdg (x)∨

holds in K0 (pHS), for every x ∈ K0 (pHS).

§ 5. THE THEOREM OF BITTNER

127

5.2. Application to the Construction of Motivic Measures Example 5.2.1 (Arapura and Kang 2006). — As a first application of theorem 5.1.3, let us show how it allows to construct the Hodge invariant χHdg : K0 (VarC ) → K0split (pHS) in the Grothendieck ring K0split (pHS) of the additive category of polarizable Hodge structures, without using mixed Hodge theory. For every smooth projective complex variety X, the Hodge structure on n (X, Q) is polarizable (see §3.2.8); let h(X) be the class of Hsing 

2 dim(X) n (−1)n [Hsing (X, Q)]

i=0

in the ring K0 (pHS). Let Y be a smooth nowhere dense closed subscheme of X, let BlY (X) be the blow-up of X along Y , and let E be its exceptional divisor. Write p : BlY (X) → X for the canonical morphism, q : E → Y and q  : BlY (X) E → X Y their restrictions; let i : Y → X and j : E → BlY (X) be the canonical closed immersions; let i : X Y → X and j  : BlY (X) E → BlY (X) be the complementary closed immersions. These morphisms sit in the following commutative diagram j

E



BlY (X)

q

p i

Y





X

E

BlY (X) X

q⬘

Y.

Then, (part of) the long exact sequences of cohomology with compact support associated with the closed immersions i and j write : Hcn (X _~

i ¢*

Y) (q ¢ ) *

Hcn (BlY (X)

H n (X)

i*

E)

H n (BlY (X))

Hcn +1 (X

q*

p* j ¢*

H n (Y )

j*

H n (E)



Y)

(q ¢ ) *

Hcn +1 (BlY (X)

E)

n (To shorten notation, we wrote H n (X) for Hsing (X, Q), and similarly for cohomology with compact supports.) Let us justify the isomorphisms or injections which were indicated on the diagram: since q  is an isomorphism, the map (q  )∗ induced on cohomology is an isomorphism too; moreover, p : BlY (X) → X is surjective, hence induces an injection on cohomology. Let us now deduce by diagram chasing that for every integer n, these maps induce an exact sequence of pure polarizable Hodge structures of weight n: (p∗ ,i∗ )

−j ∗ +q ∗

0 → H n (X) −−−−→ H n (BlY (X)) ⊕ H n (Y ) −−−−−→ H n (E) → 0.

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The injectivity of the map (p∗ , i∗ ) is obvious, since p∗ is injective on H n (X). Moreover, for every α ∈ H n (X), one has (−j ∗ +q ∗ )◦(p∗ , i∗ )(α) = −j ∗ ◦p∗ (α)+q ∗ ◦i∗ (α) = −(p◦j)∗ (α)+(i◦q)∗ (α) = 0 since i ◦ q = p ◦ j. Let now α ∈ H n (BlY (X)) and β ∈ H n (Y ) be such that −j ∗ (α) + q ∗ (β) = 0. Then 0 = ∂(−j ∗ (α) + q ∗ (β)) = −∂(q ∗ (β)) = −(q  )∗ (∂(β)); since (q  )∗ is an isomorphism, we have ∂(β) = 0. Consequently, there exists γ ∈ Hcn (X) such that β = i∗ (γ), hence j ∗ (α) = q ∗ (β) = q ∗ (i∗ (γ)) = j ∗ (p∗ (γ)) since i ◦ q = p ◦ j. Consequently, α − p∗ (γ) comes from Hcn (BlY (X) E). Since (q  )∗ is an isomorphism, there exists δ ∈ Hcn (X Y ) such that α − p∗ (γ) = (q  )∗ (j∗ (δ)) = p∗ (i∗ (δ)). Consequently, α = p∗ (γ + i∗ (δ)), while β = i∗ (γ) = i∗ (γ + i∗ (δ)). This shows that Ker(−j ∗ + q ∗ ) = Im(p∗ , i∗ ). Let us finally show that the map −j ∗ + q ∗ is surjective. Let α ∈ H n (E). Since (q  )∗ is an isomorphism, there exists β ∈ Hcn+1 (X Y ) such that ∂(α) = (q  )∗ (β). Then, p∗ (i∗ (β)) = (j  )∗ (q  )∗ (β) = (j  )∗ (∂(α)) = 0. Since p∗ is injective, i∗ (β) = 0, and there exists γ ∈ H n (Y ) such that β = ∂(γ). Then, ∂(α) = (q  )∗ (∂(γ)) = ∂(q ∗ (γ)), so that α − q ∗ (γ) ∈ Ker(∂). Consequently, there exists δ ∈ H n (BlY (X)) such that α − q ∗ (γ) = j ∗ (δ). This shows that α = q ∗ (γ) + j ∗ (δ) belongs to Im(−j ∗ + q ∗ ), as claimed. This implies the following equality [H n (BlY (X)))] = [H n (X)] + [H n (E)] − [H n (Y )] in the Grothendieck group K0 (pHS). However, since every exact sequence of pure polarizable Hodge structures admits a splitting, this equality also holds in the Grothendieck group K0split (pHS). By theorem 5.1.3, this shows that there exists a unique morphism of groups hn from K0 (VarC ) to K0 (pHS) which maps the class e(X) of a projective smooth complex variety to the class [H n (X)] in K0 (pHS).  It follows that the additive invariant X → h(X) = (−1)n hn (X) is an additive invariant, as well. Moreover, the Künneth formula implies that h is a motivic measure. Example 5.2.2 (Ekedahl 2009, Theorem 3.4). — Let k be a field of ¯ characteristic zero, let k¯ be an algebraic closure of k, and let Gk = Gal(k/k) be its absolute Galois group. Fix a prime number . For every smooth proper k-variety X and every integer p, denote by Ap (Xk¯ ) the group of p-codimensional cycles on Xk¯ modulo rational equiva¯ Z ) by the cycle class lence, and let NSp (Xk¯ ) be its image in H 2p (X ⊗k k, map. This is a finitely generated abelian group endowed with a linear action of Gk , and this action factors through the Galois group Gal(K/k) of a finite extension K of k. Such objects form an additive category ModGk Z, of which we consider the Grothendieck group K0split (ModGk Z).

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We claim that there is a unique additive invariant ns : Mk → K0split (ModGk Z),

e(X)/Lp → [NSp (Xk¯ )].

By corollary 5.1.7 to Bittner’s theorem, it suffices to establish the following two properties: (i) For every closed smooth subscheme Y of X with blow-up BlY (X) and exceptional divisor E, and every integer p  0, one has NSp (BlY (X)) ⊕ NSp (Y ) = NSp (E) ⊕ NSp (X). (ii) For every smooth proper k-variety X and every integer p  0, NSp+1 (X × P1 ) = NSp (X) ⊕ NSp+1 (X). These properties follow from the computation of the Chow groups of a blowup, resp. of X × P1 , of their étale cohomology, and of the compatibility of the cycle class. The arguments are analogous to those explained in the proof of example 5.2.1, and we refer the reader to (Ekedahl 2009) for details. 5.3. Motives and Motivic Measures (5.3.1). — Let k be a field. We mentioned above Grothendieck’s letter dated 16 August 1964, and published in Colmez and Serre (2001), where he introduced the group K0 (Vark ). In the same letter, Grothendieck introduced the notion of motive (7) , and he speculated on the relation between the Grothendieck group of varieties and the Grothendieck group of motives: “[. . . ] on trouve un homomorphisme naturel L(k) → M(k), qui est d’ailleurs un homomorphisme d’anneaux [. . . ]. La question générale qui se pose est alors de savoir ce qu’on peut dire sur cet homomorphisme, est-il loin d’être bijectif ? [. . . ]” (8) Assume that k has characteristic zero. When M(k) is the Grothendieck group of the category of Chow motives over k, the existence of such a ring morphism was first proved by Gillet and Soulé (1996) and Guillén, Navarro Aznar (2002); see also Bondarko (2009). In fact, these authors proved a much finer result: with every k-scheme of finite type, they assign a complex of Chow motives which is well-defined up to homotopy.

(7) The

English word motive has two meanings, either a factor inducing to act in a certain way or, especially in art, a distinctive feature, a structural principle, a pattern. Grothendieck referred to the latter meaning. (8) “[. . . ] one obtains a natural homomorphism L(k) → M(k), which is actually a homomorphism of rings [. . . ]. The general question that can be asked is to know what one can say on this homomorphism, is it far from being bijective ? [. . . ]”

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In this section, we explain how theorem 5.1.3 leads to a quick and easy proof of the existence of this motivic measure (in characteristic zero). In positive characteristic, no such map is known; however, Gillet and Soulé (2009) proved the existence of a Euler characteristic in a slightly different category of motives. (5.3.2). — We begin by quickly reviewing the category of pure motives. Let k be a field. Let R be a commutative ring, and let ∼ be an adequate equivalence relation on cycles with coefficients R; important examples are given by rational equivalence (which is the finest adequate equivalence relation) and numerical equivalence (the coarsest one). If X is a projective integral smooth k-variety and n ∈ N, we write An∼,R (X) for the quotient of the free R-module generated by integral closed subschemes of X of codimension n modulo the equivalence relation ∼. The category EffMot∼,R (k) of effective pure motives (over k, with coefficients R, for the adequate equivalence relation ∼) is an R-linear pseudoabelian tensor category. We refer to Manin (1968); Kleiman (1968); Demazure (1969–1970); André (2004); Murre et al. (2013) for comprehensive presentations of this construction. Let us just say that this category is built from the category of projective smooth varieties, by adding correspondences (modulo ∼) as morphisms and then adding images and kernels of projectors. Its tensor structure is induced by the product of k-varieties. With every projective smooth k-variety X is associated its “motive” M∼,R (X), and the assignment X → M∼,R (X) is a functor from the category Varksm,proj to the category EffMot∼,R (k). We also mention the important theorem of Jannsen (1992): if R is a field of characteristic zero, then the category EffMot∼,R (k) is a semisimple abelian category if and only if the chosen adequate equivalence relation ∼ coincides with numerical equivalence. Proposition 5.3.3. — Let k be a field of characteristic zero, let ∼ be an adequate equivalence relation, and let R be a ring. There exists a unique additive invariant χ∼,R on Vark with values in the Grothendieck group K0split (EffMot∼,R (k)) such that for every projective smooth kvariety X, χ∼,R (X) is the class of M∼,R (X) in K0split (EffMot∼,R (k)). Moreover, χ∼,R is a motivic measure. Proof. — By theorem 5.1.3, it suffices to prove that for every connected smooth projective k-variety X and every connected smooth closed integral subvariety Y of X, one has (5.3.3.1)

M∼,R (BlY (X)) − M∼,R (X) = M∼,R (E) − M∼,R (Y )

where BlY (X) is the blow-up of X along Y and E is the exceptional divisor. In fact, this equality follows immediately from the computation of the motive of a blow-up, cf. (Manin 1968, §9), itself a consequence of the computation of the cycle groups (modulo ∼) of a blow-up, (Fulton 1998, §6.7), and of Manin’s “identity principle.”

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By the definition of the tensor structure on EffMot∼,R (k), one has M∼,R (X ×k Y ) = M∼,R (X) ⊗ M∼,R (Y ). Consequently, χ∼,R is a motivic measure. (5.3.4). — This ring morphism χ∼,R is called the (effective) motivic Euler characteristic with coefficients in R. If R = Z, we will speak of the integral motivic Euler characteristic. If R = Q, then we call χ∼,R the rational motivic Euler characteristic. (5.3.5). — One defines the Lefschetz motive Lmot ∈ EffMot∼,R (k) as the kernel of the morphism M∼,R (P1k ) → M∼,R (pt) associated with any k-rational point of P1k . The structural morphism P1k → Spec(k) furnishes a splitting of this morphism, so that M∼,R (P1k ) = M∼,R (pt) ⊕ Lmot . It follows that χ∼,R (A1 ) = χ∼,R (P1k ) − χ∼,R (pt) = [Lmot ]. Let Mot∼,R be the category of pure motives with coefficients in R (for the adequate equivalence relation ∼); it is obtained from EffMot∼,R by inverting Lmot . In particular, this class is invertible in K0split (Mot∼,R ), so that the morphism χ∼,R gives rise to a ring morphism (5.3.5.1)

χ∼,R : Mk → K0split (Mot∼,R ),

which is still called the motivic Euler characteristic. Remark 5.3.6. — In his letter quoted above, Grothendieck also asked whether the motivic Euler characteristic is far from being bijective. While it seems in fact plausible that this morphism is surjective, we mention that the rational motivic Euler characteristic is not injective. Indeed, two isogenous abelian k-varieties A and B have isomorphic effective Chow motives with Qcoefficients but have different classes in K0 (Vark ), unless they are isomorphic (example 6.2.1). Thus, if A is isogenous, but not isomorphic, to B, then e(A) − e(B) is a nontrivial element in the kernel of χrat,Q . To the best of our knowledge, the question of the injectivity of χrat,Z is still open. Ivorra and Sebag (2012) have given another example of a nontrivial element in the kernel of χ∼,Q : K0 (Vark ) → K0 (EffMotR (k)). Assume for simplicity that k = C. Guletski˘ı and Pedrini (2002) showed that the motive of a certain complex Godeaux surface X (the quotient of the Fermat quintic in P3C by the natural Z/5Z-action), which is of general type, has a decomposition in Motrat,Q of the form M∼,Q (X) = L2mot ⊕ 9Lmot ⊕ 1. This is also the decomposition of the motive of a surface obtained by blowingup 8 distinct points in P2C . However, we cannot have e(X) = e(Y ) in K0 (VarC ), by proposition 6.3.8.

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(5.3.7). — Let ∼ be an adequate equivalence relation, and let M ∈ Mot∼,Q be a rational pure motive. Let n  0 be an integer. The symmetric group Sn acts naturally on the nth tensor power M ⊗n , hence so act the canonical projectors 1  1  σ and ε(σ)σ n! n! σ∈Sn

σ∈Sn

of the group algebra Q[Σn ], where ε(σ) denotes the signature of a permutation σ. By definition, the symmetric power Symn (M ) and the antisymmetric power ∧n (M ) are the images of these projectors acting on M ⊗n . We refer to André (2004, §2.2) for more details. Definition 5.3.8. — Let ∼ be an adequate equivalence relation, and let M ∈ Mot∼,Q be a pure motive. a) One says that M is evenly finite dimensional if there exists an integer n ∈ N∗ such that ∧n M = 0; the dimension of M is then defined to be the maximal integer m such that ∧m M = 0. b) One says that M is oddly finite dimensional if there exists an integer n ∈ N∗ such that Symn (M ) = 0; the dimension of M is then defined to be the maximal integer m such that Symm (M ) = 0. c) One says that M is finite dimensional if there exist an evenly finite dimensional motive M + and an oddly finite dimensional motive M − in M∼,Q such that M ∼ = M − ⊕ M + ; the dimension of M is the integer defined by the formula dim(M ) := dim(M + ) + dim(M − ). Remark 5.3.9. — One can check that finite dimensional motives in Mot∼,Q are stable under direct sums and direct factors. Example 5.3.10. — The full subcategory of Motrat,Q formed by the finite dimensional motives is stable under direct sums and direct factors. It contains the motives of smooth projective k-curves and of abelian varieties. By the blow-up formula (5.3.3.1) and the weak factorization theorem A/2.3.3, it follows that finite dimensionality of the associated Chow motives in Motrat,Q is a birational invariant for smooth projective k-varieties of dimension at most 3. Proposition 5.3.11 (Ivorra and Sebag 2012, proposition 4) Let M, N ∈ Motrat,Q such that [M ] = [N ] in K0 (Motnum,Q ). Let us assume that M, N are finite dimensional. Then M ∼ = N in Motrat,Q . Proof. — By Jannsen (1992), the category Motnum,Q (k) is semisimple; hence there exists an isomorphism of motives (5.3.11.1)

Mnum,Q (X)  Mnum,Q (Y ).

Since rational equivalence is finer than numerical equivalence, we can lift this isomorphism to a morphism of motives for rational equivalence: (5.3.11.2)

Mrat,Q (X) → Mrat,Q (Y ).

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By assumption, the Chow motives Mrat,Q (X) and Mrat,Q (Y ) are finite dimensional. It follows from Kimura (2005) that the morphism (5.3.11.2) is in fact an isomorphism. (5.3.12). — Many important conjectures shed light on the category of Chow motives. One of them, independently due to Kimura (2005, 7.1) and O’Sullivan (2005) asserts that Chow motives of projective and smooth k-varieties are finite dimensional (see also André (2004, §12)). Under this assumption, we prove below that two projective smooth k-varieties which define the same class in K0 (Vark ) have isomorphic Chow motives and isomorphic Chow groups. Proposition (Ivorra and Sebag 2012, propositions 4.3 and 4.4) Let k be a field of characteristic zero, let X and Y be projective and smooth k-varieties such that e(X) = e(Y ) in Mk . Assume that the motives Mrat,Q (X) and Mrat,Q (Y ) are finite dimensional. Then the k-varieties X and Y have isomorphic rational Chow motives and isomorphic rational Chow groups. Proof. — Applying the motivic Euler characteristic for numerical equivalence, we see that (5.3.12.1)

[Mnum,Q (X)] = [Mnum,Q (Y )]

in K0 (Motnum,Q (k)). Then, we conclude by proposition 5.3.11 that Mrat,Q (X) ∼ = Mrat,Q (Y ). The rest of the proposition follows from the formula: HomMotrat,Q (Lnmot , Mrat,Q (X)) = An (X)Q . Example 5.3.13. — Let k be an algebraically closed field of characteristic zero. Let X and Y be smooth projective k-surfaces such that e(X) = e(Y ) in Mk . Then the rational Chow motives of X and Y are isomorphic (Ivorra and Sebag 2012).

§ 6. THE THEOREM OF LARSEN–LUNTS AND ITS APPLICATIONS 6.1. The Theorem of Larsen–Lunts Let k be a field of characteristic zero. Definition 6.1.1. — Two integral k-varieties X and Y are said to be stably n birational if there exist positive integers m and n such that X×Pm k and Y ×Pk are birational.

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(6.1.2). — This is an equivalence relation. It is strictly coarser than birational equivalence; for example, P1k is stably birational to a point. There are even examples of pairs of integral k-varieties of the same dimension which are stably birational but not birational (see §6.3). However, if X and Y are stably birational integral k-varieties of the same dimension, and if the Kodaira dimension of X (or Y ) is nonnegative, then X and Y are birational. See theorem A/2.6.3 and (Liu and Sebag 2010, theorem 2). (6.1.3). — Let SBk be the set of equivalence classes of integral k-varieties by the stably birational relation. For every projective smooth integral kvariety X, we denote by sb(X) its class in SBk . Let Z(SBk ) be the free abelian group on the set SBk ; we identify an element of SBk with the corresponding element of Z(SBk ) . If X is a smooth  projective k-variety, possibly not integral, we define sb(X) to be the sum j sb(Xj ) of the classes of its irreducible components Xj . There is a unique ring structure on Z(SBk ) such that sb(X) sb(Y ) = sb(X ×k Y ) whenever X and Y are projective smooth k-schemes. Lemma 6.1.4. — Let k be a field of characteristic zero. Let X and Y be proper smooth k-varieties. If they are stably birational, then e(X) ≡ e(Y ) (mod L). Proof. — By definition of the stably birational equivalence, there exist inten n gers m, n  0 such that X ×k Pm k is birational to Y ×k Pk . Since e(Pk ) is congruent to 1 modulo L, we may assume that X is birational to Y . By the weak factorization theorem, we can relate X and Y by a sequence of blow-ups and blow-downs with smooth centers. Thus it suffices to show that such a blow-up does not affect e(X) modulo L, a property which follows from example 2.4.3. Theorem 6.1.5 (Larsen and Lunts 2003). — Let k be a field of characteristic zero. a) There exists a unique additive invariant sb with values in Z(SBk ) which associates with every projective smooth connected k-variety X its class sb(X). b) This additive invariant is a motivic measure. The associated ring morphism sb : K0 (Vark ) → Z(SBk ) is surjective; its kernel is the ideal generated by L. Proof. — a) By Bittner’s theorem 5.1.3, it suffices to prove that sb(BlY (X)) − sb(X) = sb(E) − sb(Y ) for every projective smooth connected k-variety X and every (strict) irreducible smooth closed subscheme Y of X, where BlY (X) is the blow-up of X along Y and E is the exceptional divisor. Since BlY (X) and X are birational, the left hand side of the expression is zero. Moreover, E is the projectivized normal bundle to Y . Let r = codim(Y, X). Since X and Y are smooth, E

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is a Zariski-locally trivial fibration with fiber the projective space Pr−1 . In particular, E is birational to Y × Pr−1 . This implies that E and Y are stably birational, whence sb(E) = sb(Y ). This proves the existence and uniqueness of the additive invariant sb. b) For every pair (X, Y ) of projective smooth connected k-varieties, one has sb(X ×k Y ) = sb(X) sb(Y ), by the very definition of the ring structure on Z(SBk ) . This means that the additive invariant sb is a motivic measure. By Hironaka’s theorem on resolution of singularities, every integral kvariety is birational to a smooth, projective, connected k-variety. Consequently, sb is surjective. Since P1k is stably birational to Spec(k), we have 1 = sb(Spec(k)) = sb(e(P1k )) = sb(L + 1) = sb(L) + 1 in Z(SBk ) , hence sb(L) = 0. Let us denote by sb : K0 (Vark )/(L) → Z(SBk ) the ring morphism induced by sb. It is surjective. By lemma 6.1.4, the classes in K0 (Vark ) of two smooth, projective, connected k-varieties which are birational are equal modulo L. Consequently, there exists a unique group homomorphism from Z(SBk ) to K0 (Vark )/(L) which, for every smooth, projective, connected k-variety X, maps sb(X) to the class of e(X) modulo L. It is an inverse of sb, hence sb is an isomorphism. Proposition 6.1.6. — Let k be a field of characteristic zero. Let X be a proper, smooth, and connected k-variety. Then the ring morphism sb : K0 (Vark ) → Z(SBk ) sends e(X) to the stably birational class of X. Proof. — According to Chow’s lemma and Hironaka’s theorem on resolution of singularities, X is birational to a projective, smooth, and connected kvariety X  . By lemma 6.1.4, e(X  ) = e(X) (mod L), hence sb(e(X  )) = sb(e(X)). On the other hand, sb(e(X  )) = sb(X  ), which also is the stably birational class of X. Corollary 6.1.7. — Two smooth, proper, and connected k-varieties are stably birational if and only if their classes in K0 (Vark ) are congruent modulo L. Corollary 6.1.8. — Let X and Y be proper and smooth k-varieties such that e(X) = e(Y ) in K0 (Vark ). Assume that Y is equidimensional and that no irreducible component of X is uniruled. Then X and Y are birational. Proof. — Let X1 , . . . , Xm be the irreducible components of X; let Y1 , . . . , Yn be the irreducible components of Y . By the theorem of Larsen and Lunts (theorem 6.1.5), one has m  i=1

sb(Xi ) =

n  j=1

sb(Yj )

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in the free abelian group Z(SBk ) . Consequently, m = n, and we may assume that for every i, Xi , and Yi are stably birational. Since e(X) = e(Y ), we have dim(X) = dim(Y ), hence dim(Yi ) = dim(Y )  dim(Xi ) for every i. It then follows from corollary A/2.6.4 that for every i, Xi , and Yi are birational. This concludes the proof. Corollary 6.1.9. — Let k be a field of characteristic zero. Let μ be a motivic measure on Vark . The following assertions are equivalent: (i) One has μ(L) = 0; (ii) Two smooth, proper, and connected k-varieties which are birational have the same image under μ (“μ is a birational invariant”); (iii) Two smooth, proper, and connected k-varieties which are stably birational have the same image under μ; (iv) The motivic measure μ factors through sb. Proof. — The implications (iv)⇒(iii)⇒(ii) are obvious. If μ is a birational invariant, then one has μ(P2k ) = μ(BlP (P2k )), where P is any k-rational point of P2k . Since e(BlP (P2 )) = e(P2 ) + L, we get that μ(L) = 0, so that (ii) implies (i). Assume finally that μ(L) = 0, so that μ factors through K0 (Vark )/(L). By theorem 6.1.5, μ factors through sb, hence (iv). Corollary 6.1.10. — Let k be a field of characteristic zero. Let X and Y be proper and smooth k-varieties such that e(X) ≡ e(Y ) (mod L) in K0 (Vark ). The following properties hold: a) The variety X has a rational point if and only if Y has a rational point; b) The greatest common divisor (resp. the minimum) of the degrees of closed points coincides for X and Y ; c) The Chow groups of 0-cycles A0 (X) and A0 (Y ) are isomorphic. Proof. — Let X1 , . . . , Xm be the connected components of X; let Y1 , . . . , Yn be the connected components of Y ; they are proper and smooth. By the existence of the motivic measure sb (theorem 6.1.5), the equality e(X) ≡ e(Y ) (mod L) implies the following equality in the abelian group Z(SBk ) : m  i=1

[Xi ] = sb(e(X)) = sb(e(Y )) =

n 

[Yj ],

j=1

where brackets indicate that we take the stably birational class of a proper, smooth, irreducible k-variety. Consequently, m = n, and we may assume that for every integer i, Xi and Yi are stably birational. We may also assume that X and Y are irreducible, in which case we have shown that they are stably birational. n Let m and n be integers such that X ×k Pm k is birational to Y ×k Pk . If m X has a k-point, then so has X ×k Pk has a k-point; since X is smooth,

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Nishimura’s lemma (lemma 6.1.11 below) implies that Y ×k Pnk has a k-point; hence Y has k-point. This proves a). Applying this argument for every finite extension of k, assertion b) follows readily. m The projection X ×k Pm k → X induces an isomorphism from A0 (X ×k Pk ) n to A0 (X); similarly, A0 (Y ×k Pk ) is isomorphic to A0 (Y ). Moreover, two proper, smooth, and birational varieties have isomorphic Chow groups of 0-cycles. Consequently, A0 (X) and A0 (Y ) are isomorphic, that is, c). Lemma 6.1.11 (Nishimura). — Let k be a field; let X and Y be connected k-varieties. Assume that X has a smooth k-rational point and that Y is proper. If there is a rational map from X to Y , then Y has a k-rational point. Proof. — (9) We argue by induction on n = dim(X). Replacing X by its smooth locus, we assume that X is smooth. Let x ∈ X(k) be a rational point. If dim(X) = 0, then every rational map is defined at x, and its ˜ be the blow-up of X at x, let image is a k-rational point of Y . Let X ˜ p : X → X be the canonical morphism, and let E = p−1 (x) be the exceptional ˜ is smooth and that E  Pn−1 . Moreover, the given divisor. Observe that X k ˜ to Y . rational map from X to Y can be viewed as a rational map f from X ˜ ˜ Since Y is proper and X is smooth, there exists an open subset U of X ˜ ˜ ˜ codim(X U, X)  2 on which the map f is defined. Since codim(E, X) = 1, one has E ∩ U = ∅. Consequently, the restriction of f to E ∩ U defines a , the k-variety E has a smooth rational map from E to Y . Since E  Pn−1 k rational point, and its dimension is smaller than that of X. By induction, Y has a k-rational point. , When k is infinite, the proof can be simplified as follows. Since E  Pn−1 k its rational points are Zariski dense; hence there is a rational point e ∈ E(k) in the domain of definition of f |E . Its image in Y is a rational point. 6.2. Other Examples of Motivic Measures Theorem 6.1.5 may also be used to construct new motivic measures. Let us give some examples. Example 6.2.1 (Poonen 2002). — Let k be a field of characteristic zero. We denote by AVk the set of isomorphism classes of abelian k-varieties. The product of abelian varieties defines a monoid structure on AVk , and we denote by Z(AVk ) the associated monoid ring. To every smooth, proper, and connected k-variety, we can associate its Albanese variety Alb(X). It is well-known that the Albanese variety is invariant under stably birational equivalence. Theorem 6.1.5 implies that there (9) This proof is due to J. Kollár and E. Szabó, see the appendix of Reichstein and Youssin (2000).

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exists a unique ring morphism alb : K0 (Vark ) → Z(AVk ) that maps e(X) to the class of Alb(X) for every smooth, proper, and connected k-variety X. Corollary 6.2.2. — Let k be a field of characteristic zero. Two abelian kvarieties have the same class in K0 (Vark ) if and only if they are isomorphic. Proof. — Indeed, an abelian variety is its own Albanese variety. Example 6.2.3. — The fundamental group π1 (X) of a connected complex variety X is a group of finite presentation. (Since we will only be interested in the fundamental group up to isomorphism, we neglect the choice of a base point.) The projective line P1C is simply connected. Let X, Y be smooth, proper, and connected complex varieties. If X and Y are birational, then π1 (X)  π1 (Y ). One also has an isomorphism π1 (X × Y )  π1 (X) × π1 (Y ). This shows that the fundamental group defines a stably birational invariant. Let FpGp be the set of isomorphism classes of finitely presented groups. The product of groups endows the set FpGp with the structure of a monoid; let Z(FpGp) be the associated monoid ring. Then there exists a unique ring morphism K0 (VarC ) → Z(FpGp) which maps e(X) to the class of π1 (X), for every smooth, proper, and connected complex variety X. Corollary 6.2.4. — If two proper smooth connected complex varieties have the same class in K0 (VarC ), then their fundamental groups are isomorphic. Remark 6.2.5. — Let k be an algebraically closed field of characteristic zero. By similar arguments, one can prove that two proper smooth connected k-varieties which have the same class in K0 (Vark ) have isomorphic étale fundamental groups. Example 6.2.6. — Let us assume that k = C. Ekedahl (2009) extended the motivic measure alb (example 6.2.1) to an additive invariant pic on MC with values in the Grothendieck group G of the additive category of complex algebraic group schemes whose connected component is an abelian variety and whose group of connected components is finitely generated. The formula is as follows: for every smooth projective complex variety X and every integer p  0, c,p pic(e(X)/Lp ) = [A0,p X ] + [AX ],

where A0,p X is Weil’s intermediate Jacobian and 2p+2 (X, Z) ∩ H p+1,p+1 (X, C). Ac,p X =H

§ 6. THE THEOREM OF LARSEN–LUNTS AND ITS APPLICATIONS

139

2p+1 By definition, A0,p (X, C)/F p+1 H 2p+1 (X, C) + X is the real manifold H 2p+1 H (X, Z), but its complex structure is the one induced by the Weil operator which acts by multiplication by ia−b on H a,b (X, C). Then Ac,p X is a complex torus, and one can check, using the Riemann bilinear relations, that A0,p X is indeed an abelian variety. 0,0 is the Jacobian of X, and Ac,0 For p = 0, AX X is the Néron-Severi group of X, so that pic(e(X)) is the class in G of the algebraic group Pic(X). Using the arguments of theorem 6.4.1, this motivic measure may be used to prove that MC contains nontrivial zero divisors of the form e(A) − e(B), where A and B are complex abelian varieties. We refer the reader to Ekedahl (2009), corollary 3.5 for more details.

6.3. The Cut-and-Paste Property (6.3.1). — Let k be a field of characteristic zero. When k = C, Larsen and Lunts (2003) have raised the question whether two k-varieties which have the same class in K0 (Vark ) are piecewise isomorphic. This amounts to the question whether the canonical morphism from K0+ (Vark ) to K0 (Vark ) is injective. The following theorem of Borisov (2014) will imply that it is not the case; it will also prove that L is a zero divisor in the ring K0 (Vark ) (see corollary 6.3.5). Theorem 6.3.2 (Borisov 2014). — Let k be a field of characteristic zero. There exists a pair (X, Y ) of smooth projective Calabi–Yau k-varieties which are not stably birational and such that (e(X) − e(Y ))L6 = 0. Proof. #2 ∨ — Let V be a finite dimensional complex vector space, and let W ⊂ V be a vector subspace; a point w ∈ W is viewed as an alternate 2form ϕw on V . We assume that dim(V ) = dim(W ) = 7 and that W is chosen generically. Let XW ⊂ Gr(2, V ) be the subvariety of the Grassmannian variety of 2-planes of V consisting of planes P such that ϕw |P = 0 for every w ∈ W . On the other hand, let YW ⊂ P(W ) be the subvariety consisting of lines Cw generated by an element w ∈ W whose associated form ϕw has rank < 6. The choice of a generic W assures that for every nonzero w ∈ W , the corresponding form ϕw has rank 4 if Cw ∈ YW and rank 6 otherwise. Moreover, XW and YW are smooth projective Calabi–Yau manifolds of dimension 3 with Picard group isomorphic to Z, not isomorphic one to the other; see Rødland (2000); Borisov and Căldăraru (2009) for more details. Let us prove by contradiction that XW and YW are not stably birational. Assume that they are. Since XW is a Calabi–Yau variety, it is not uniruled;

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hence corollary A/2.6.4 implies that XW and YW are birational. By examp q ple A/2.5.4, they are related by a K-equivalence XW ← −Z− → YW . This Kequivalence induces a bijection between their sets of integral divisors, hence an isomorphism of divisor groups, compatibly with linear equivalence. Since YW has Picard group Z, the image of an ample divisor by this bijection is ample, and both classes define the same projective embedding. Consequently, XW and YW are isomorphic, a contradiction. (Alternatively, one can use theorem 4.9 of Kollár (1989) asserting that XW and YW would be linked by a sequence of flops, which is prevented by the fact that both varieties have Picard group Z.) By theorem 6.1.5, one has e(XW ) = e(YW ) in K0 (Vark ). Let us now prove that (e(XW ) − e(YW ))L6 = 0. Let Z ⊂ Gr(2, V ) × P(W ) be the Cayley hypersurface corresponding to pairs (P, Cw) where P is a 2-plane in V and Cw is a line in W generated by a form such that ϕw |P = 0. Let us analyze the first projection p : Z → Gr(2, V ). Let x ∈ Gr(2, V ) and let Px ⊂ Vκ(x) be the corresponding plane. If x ∈ XW , then, by definition of XW , one has p−1 (x) = P(W )κ(x)  P6κ(x) . On the other hand, if x ∈ XW , then the fiber p−1 (x) is a hyperplane in P(W )κ(x) , hence p−1 (x)  P5κ(x) . This implies that over XW , the projection p induces a piecewise trivial fibration with fiber P6k , while over Gr(2, V ) XW , it induces a piecewise trivial fibration with fiber P5k . Consequently, e(Z) = (e(Gr(2, V )) − e(XW )) e(P5 ) + e(XW ) e(P6 ) = e(Gr(2, V )) e(P5 ) + e(XW )L6 . Let us now analyze the second projection q : Z → P(W ). Let y ∈ P(W ) and let w ∈ Wκ(y) be a generator of the line y. If y ∈ YW (resp. y ∈ YW ), then ϕw has rank 4 (resp. 6); hence q −1 (y) is a Lagrangian variety of 2planes associated with a rank 4 (resp. rank 6) alternate form. Every two alternate forms on a vector space are equivalent (Darboux’s theorem), so that q −1 (y) = LGr4 (2, V )κ(y) for y ∈ YW and q −1 (y)  LGr6 (2, V )κ(y) otherwise, where, for r ∈ N, LGrr (2, V ) denotes the subvariety of Gr(2, V ) consisting of 2-planes in V which are isotropic with respect to a given alternate form of rank r on V . Therefore, e(Z) = (e(P(W )) − e(YW )) e(LGr6 (2, V )) + e(YW ) e(LGr4 (2, V )) = e(P6 ) e(LGr6 (2, V )) + e(YW )(LGr4 (2, V ) − LGr6 (2, V )). The Grassmannian variety Gr(2, V ) and its subvarieties LGr4 (2, V ) and LGr6 (2, V ) admit a partition in locally closed subsets isomorphic to affine spaces. Precisely, let us identify V with C7 . As in example 2.4.5, any plane P in V is generated by a unique pair (v1 , v2 ) of vectors such that the associated 7 × 2 matrix is in reduced column echelon form with 2 pivot rows indices a, b with 1  a < b  7. Fixing a pair (a, b) of row indices defines a locally closed subspace of Gr(2, V ) which is an affine space Sa,b of dimension (b − a − 1) +

§ 6. THE THEOREM OF LARSEN–LUNTS AND ITS APPLICATIONS

2(7 − b). Consequently, e(Gr(2, V )) =



141

Lb−a−1+2(7−b)

1a 0. The unique ring morphism ϕξ : Z[T ] → Mk which sends T to ξ is injective. In particular, K0 (Vark ) has characteristic zero. Proof. — Let d = deg(EP(ξ)). The composition EP ◦ϕξ is the ring morphism Z[T ] → Z[t, t−1 ] that maps T to EP(ξ). Since EP(ξ) is not constant, it is not algebraic in Q(t); hence the morphism ϕξ is injective. This concludes the proof. Proposition 6.5.2. — Let k be a field. If k has characteristic zero, then Card(K0 (Vark )) = Card(k). In the general case, one has Card(N)  Card(K0 (Vark ))  max(Card(N), Card(k)). Proof. — For each integer n  1, the cardinality of k[T1 , . . . , Tn ] is equal to max(Card(N), Card(k)). By Hilbert’s Finite basis theorem, the set of ideals in k[T1 , . . . , Tn ] also has cardinality max(Card(N), Card(k)), so that the set of k-subschemes of Ank has cardinality max(Card(N), Card(k)). Since K0 (Vark ) is generated, as a group, by the classes of affine k-varieties, this implies the second inequality. The first one follows from the fact that K0 (Vark ) contains Z. Let us assume that k has characteristic zero. For any j ∈ k, there exist an elliptic curve over k with j-invariant equal to j, and elliptic curves of distinct j-invariant are not isomorphic. By example 6.2.1, non-isomorphic elliptic curves over k have distinct classes in K0 (Vark ). This furnishes a subset of K0 (Vark ) of cardinality Card(k), hence the proposition.

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(6.5.3). — The remainder of this section is devoted to constructing large families (ai )i∈I of elements of K0 (Vark ) which are algebraically independent, that is, such that the unique ring morphism Z[(Ti )i∈I ] → K0 (Vark ),

Ti → ai ,

is injective. We begin with characteristic zero, where the existence of such examples is much easier to establish. (6.5.4). — Let k be a field of characteristic zero. Let IsoAVk and sIsoAVk be the sets of isogeny classes of abelian k-varieties and of simple abelian k-varieties, respectively. Let Z(IsoAVk ) be the free abelian group on the set IsoAVk . The product of (isogeny classes of) abelian varieties endows it with the structure of a ring. According to Poincaré’s complete reducibility theorem, the category of abelian k-varieties up to isogeny is semisimple, so that this ring Z(IsoAVk ) is isomorphic to the ring of polynomials in the family of indeterminates indexed by the set sIsoAVk of isogeny classes of simple abelian k-varieties. Let isog : Z(AV) → Z(IsoAVk ) be the canonical morphism of abelian groups from Z(AVk ) to Z(IsoAVk ) ; it is in fact a morphism of rings. Proposition 6.5.5. — For every abelian variety A over k, the element isog ◦ alb(e(A)) is the isogeny class of A. In particular, the morphism isog ◦ alb : K0 (Vark ) → Z(IsoAVk ) is surjective. Proof. — An abelian variety being its own Albanese variety, one has alb(e(A)) = [A] in Z(AVk ) ; the proposition follows from the definition of the morphism isog. Corollary 6.5.6 (Liu and Sebag 2010). — Let k be a field of characteristic zero. Let (Ai )i∈I be a family of nonzero abelian varieties over k such that for i = j, Hom(Ai , Aj ) = 0. Then the classes e(Ai ), for i ∈ I, are algebraically independent in K0 (Vark ). Remark 6.5.7. — Let k be a field of characteristic zero. It is well known that the set sIsoAVk of isogeny classes of simple abelian varieties over k is infinite. For example, when p ranges over the set of all prime numbers, elliptic curves with j-invariants 1/p are pairwise non-isogenous over Q, because they have bad reductions at distinct sets of primes. Then they are also nonisogenous over k. Since every morphism between abelian varieties is the composition of a translation and of a morphism of abelian varieties, two abelian k-varieties which are isomorphic as k-varieties are isomorphic as abelian varieties. Consequently, one has Card(AVk )  Card(Vark ) = max(Card(k), N) = Card(k). On the other hand, two elliptic curves with distinct j-invariants are not isomorphic, so that Card(AVk )  Card(k). Consequently, Card(AVk ) = Card(k).

§ 6. THE THEOREM OF LARSEN–LUNTS AND ITS APPLICATIONS

149

Since an abelian k-variety is isogenous to at most countably many other abelian k-varieties, one has Card(AVk )  Card(IsoAVk ) Card(N), which implies Card(IsoAVk ) = Card(k).

(6.5.7.1)

Moreover, every abelian k-variety is isogenous to a product of simple abelian k-varieties. If Card(k) > Card(N), this implies the equality Card(sIsoAVk ) = Card(k). On the other hand, we have seen that Card(sIsoAVk )  Card(sIsoAVQ ) = Card(N), hence the relation Card(sIsoAVk ) = Card(k).

(6.5.7.2)

Corollary 6.5.8 (Liu and Sebag 2010). — Let k be a field of characteristic zero. Then the ring K0 (Vark ) is not noetherian. Proof. — By the preceding remark, the ring Z[sIsoAVk ] is a polynomial ring in infinitely many indeterminates, hence is not noetherian. By proposition 6.5.5, this ring is a quotient of the ring K0 (Vark ). Lemma 6.5.9. — Let α1 , . . . , αn be nonzero complex numbers which are multiplicatively independent. Then the functions d → α1d , . . . , d → αnd on N∗ are algebraically independent over C. Proof. — Let F ∈ C[T1 , . . . , Tn ] be a polynomial such that F (α1d , . . . , αnd ) = 0  for every d ∈ N∗ . Write F = m∈Nn cm T m . Then, the rational function Φ defined by Φ(U ) =



cm

m∈Nn



= ⎝c0 −

∞ 

α1dm1 . . . αndmn U d

d=1

 0=m∈Nn

⎞ cm ⎠ +

 0=m∈Nn

1−

cm . . . αnmn U

α1m1

is identically 0. On the other hand, the hypothesis that α1 , . . . , αn be multiplicatively independent implies that all quantities α1m1 . . . αnmn are pairwise distinct, when m runs over Nn . By uniqueness of the expansion in partial fractions, we get cm = 0 for every m ∈ Nn {0} and c0 = 0 as well. This concludes the proof of the lemma. Proposition 6.5.10 (Krajíček and Scanlon (2000)) Let k = Fp be an algebraic closure of the finite field with p elements. Let E1 , . . . , En be elliptic curves over k, ordinary, and pairwise non-isogenous. Then for every extension K of k, the classes L, e(E1 ), . . . , e(En ) are algebraically independent in MK .

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Proof. — If these classes were algebraically dependent, there would exist a finite extension Fq of Fp over which E1 , . . . , En are defined and such that the classes L, e(E1 /Fq ), . . . , e(En /Fq ) are already algebraically dependent in MFq . Let F ∈ Z[T0 , T1 , . . . , Tn ] be a nonzero polynomial such that 

F (L, e(E1 /Fq ), . . . , e(En /Fq )) = 0 ;

m write F = m∈Nn+1 cm T . For every integer d > 0, we can apply the counting measure over Fqd , hence

F (q d , Card(E1 (Fqd )), . . . , Card(En (Fqd ))) = 0. By the theory of elliptic curves over finite fields, for every i ∈ {1, . . . , n}, there exists an algebraic integer αi such that the eigenvalues of Frobq acting on H 1 (Ei ⊗Fq k, Q ) are αi and q/αi ; then, Card(Ei (Fqd )) = (1 − αid )(1 − (q/αi )d ) for every integer d > 0. Moreover, αi is imaginary quadratic and |αi |2 = q. Consequently, for every integer d > 0, we have F (q d , (1 − α1d )(1 − q d /α1d ), . . . , (1 − αnd )(1 − q d /αnd )) = 0. Expanded, this relation furnishes a relation of the form  cm q dm0 α1dm1 . . . αndmn = 0, m∈Zn+1

cm ;

moreover, for every element m of Nn+1 such that T m for some integers is a maximal monomial of F , one has c(m0 ,−m1 ,...,−mn ) = cm = 0. As a consequence, the functions d → q d , d → α1d , . . . , d → αnd are algebraically dependent. By lemma 6.5.9, the algebraic numbers q, α1 , . . . , αn are multiplicatively dependent. Let m0 , . . . , mn ∈ Z be integers, not all zero, such that q m0 α1m1 . . . αnmn = 1. Taking absolute values, we get 2m0 + m1 + · · · + mn = 0. Consequently, the algebraic numbers 1 = α12 /q, . . . , n = αn2 /q are multiplicatively dependent and have absolute value 1 and degree  2. Moreover, the minimal polynomial of i is reciprocal, so that i and 1/i are conjugate, for every i. Since Ei is ordinary, i is not a root of unity. (If id = 1, then αi2d = q d , and Card(Ei (Fq2d )) = (1 − q d )2 is prime to p. This implies that Ei (k) has no point of order p, hence is supersingular.) Since |i | = 1, it follows that [Q(i ) : Q] = 2. Recall also that the elliptic curve Ei has complex multiplication by an order of the quadratic field Q(i ) and that two such elliptic curves are isogenous over k. Consequently, for i = j, the quadratic fields Q(i ) and Q(j ) are distinct.

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151

Let us now derive a contradiction by induction on n. For this, we may assume by induction that 1 , . . . , n−1 are multiplicatively independent and consider a nontrivial multiplicative dependence relation 1m1 . . . nmn = 1 between 1 , . . . , n . By assumption, mn = 0. Moreover, since n is not a root of unity, there exists an integer i ∈ {1, . . . , n − 1} such that mi = 0 (in particular, n  2); without loss of generality, we assume i = 1. Since Q(n ) = Q(1 ) by assumption, there exists an element σ ∈ Gal(Q/Q) such that σ(1 ) = 1 and σ(n ) = n . Let us apply σ to the previous multiplicative dependence relation; this gives 1m1 σ(2 )m2 . . . σ(n )mn = 1. Since i and 1/i are conjugate, one has σ(i ) = i±1 . Taking the product of these relations, we thus obtain  i2mi = 1, i∈I

where I is the set of all integers i ∈ {1, . . . , n} such that σ(i ) = i . By construction, I = ∅ and n ∈ I. Consequently, 1 , . . . , n−1 are multiplicatively independent, a contradiction.

CHAPTER 3 ARC SCHEMES

This chapter is devoted to the study of the arc schemes associated with schemes X defined over arbitrary base schemes S. Informally speaking, an arc on a scheme X is a formal germ of a curve on X, and the arc scheme L∞ (X/S) parameterizes the arcs in the fibers of X → S. The arc scheme was originally defined by Nash (1995) to obtain information about the structure of algebraic singularities and their resolutions. It also takes the spotlight in the theory of motivic integration, as the measure space over which functions are integrated. In section 2, we construct the spaces of jets, which are approximate arcs up to finite order. The construction consists of a process of restriction of scalars à la Weil, presented in section 1. We then explain in section 3 why arc schemes exist and how to recover them as limits of jet schemes. We study their topology in section 4 and their differential properties in section 3. Finally, in section 5 we explain a local structure theorem for arc schemes due to Grinberg and Kazhdan (2000) and Drinfeld (2002).

§ 1. WEIL RESTRICTION Let S be a scheme, and let S  be an S-scheme; in the sequel, we will mainly consider the case where S  is finite and locally free over S. By base change, every S-scheme Y gives rise to an S  -scheme Y ×S S  , in a functorial way. The Weil restriction is the right adjoint of this functor: under some hypotheses on X or S  , it associates with an S  -scheme X an S-scheme RS  /S (X) together with functorial bijections: HomS  (Y ×S S  , X)  HomS (Y, RS  /S (X)). Our presentation is inspired by Bosch et al. (1990, Chapter 7, §6) and consists in first defining the Weil restriction of X as a Zariski sheaf and then giving sufficient conditions for its representability as a scheme. © Springer Science+Business Media, LLC, part of Springer Nature 2018 A. Chambert-Loir et al., Motivic Integration, Progress in Mathematics 325, https://doi.org/10.1007/978-1-4939-7887-8_3

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1.1. Reminders on Representability (1.1.1). — Let S be a scheme, and let F be a presheaf in sets on the category of S-schemes, that is, a contravariant functor F : Schop S → Sets. Explicitly, this amounts to a set F (X), for every S-scheme X, as well as a map ϕ∗ : F (X) → F (Y ) for every morphism ϕ : Y → X of S-schemes, subject to the following two conditions: a) For every S-scheme X, one has Id∗X = IdF (X) ; b) For every pair of morphisms of S-schemes, ϕ : Y → X and ψ : Z → Y , one has (ϕ ◦ ψ)∗ = ψ ∗ ◦ ϕ∗ . Let X be an S-scheme and let U be a subscheme of Y . For every f ∈ F (X), one also denotes by f |U the element j ∗ (f ) of F (U ) which is the image of f by the map j ∗ : F (X) → F (U ) deduced from the inclusion j : U → X. Definition 1.1.2. — Let S be a scheme, and let F be a presheaf in sets on the category of schemes. One says that F is a sheaf for the Zariski topology, or a Zariski sheaf, if the following condition is satisfied:  Let Y be an S-scheme, let (Yi )i∈I be an open covering of Y , and let (fi ) ∈ F (Yi ) be a family such that fi |Yi ∩Yj = fj |Yi ∩Yj for every i, j ∈ I; then there exists a unique element f ∈ F (Y ) such that f |Yi = fi for every i ∈ I. (1.1.3). — Let S be a scheme. A ring A equipped with a morphism from Spec(A) to S will be called an S-algebra. The category AlgS of S-algebras is defined in the obvious way, and the functor A → Spec(A) identifies its opposite category with the full subcategory of SchS consisting of S-schemes whose underlying scheme is affine. (The terminology “affine S-scheme” might lead to some ambiguity, because the structural morphism of such an S-scheme might not be affine, unless S is separated. We shall therefore use the awkward expression “S-scheme which is affine.”) Every S-scheme admits a covering by open subschemes U which are affine and whose images in S are contained in an affine open subscheme. Consequently, a Zariski sheaf on SchS is determined by its value on schemes of the form Spec(A), for A in AlgS . In particular, we will not make any notational distinction between such a sheaf (e.g., the sheaf associated with a jet scheme) and the associated covariant functor on AlgS . Example 1.1.4. — Let X be an S-scheme, and let hX : Schop S → Sets be the presheaf on SchS associated with X, for which hX (Y ) = HomS (Y, X) for every S-scheme Y , and such that for every morphism of S-schemes f : Y → Y  , the map hX (f ) : HomS (Y  , X) → HomS (Y, X) is equal to u → u ◦ f . Then hX is a Zariski sheaf. Indeed,  let Y be an S-scheme, let (Yi )i∈I be an open covering of Y , and let (fi ) ∈ hX (Yi ) be a family such that fi |Yi ∩Yj = fj |Yi ∩Yj for every i, j ∈ I. For every i, fi is a morphism of S-schemes from Yi to X. By assumption, for every i, j ∈ I, the morphisms fi and fj coincide on Yi ∩ Yj . Since a morphism of schemes is a morphism of locally ringed spaces, hence can be defined locally, there exists a unique morphism f : Y → X whose restriction

§ 1. WEIL RESTRICTION

155

to Yi is equal to fi . This morphism f is the unique element of hX (Y ) such that f |Yi = fi for every i ∈ I. Definition 1.1.5. — Let F be a Zariski sheaf on the category of S-schemes. One says that F is represented by an S-scheme X if it is isomorphic to the functor hX . If F is represented by some S-scheme, one says that it is representable. Let us also recall the Yoneda lemma in this context. Let F be a Zariski sheaf on the category of S-schemes, and let u : hX → F be a morphism of sheaves. Then IdX ∈ hX (X) and f = u(X)(IdX ) is the unique element of F (X) such that for every S-scheme Y and every morphism ϕ : Y → X of S-schemes (hence ϕ ∈ hX (Y )), one has u(Y )(ϕ) = ϕ∗ f . In particular, the functor X → hX , from the category of S-schemes to the category of Zariski sheaves, is fully faithful. In practice, we shall therefore often denote by the same letter a representable Zariski sheaf and a scheme that represents this sheaf. → Sets be Definition 1.1.6. — Let S be a scheme, let F, G : Schop S presheaves in sets on the category SchS , and let Φ : G → F be a morphism of presheaves. One says that Φ is representable if for every S-scheme X and every f ∈ F (X), the fiber product functor G ×F hX is representable. Explicitly, this means that the following property is satisfied: For every S-scheme X and every f ∈ F (X), there exist a scheme Vf , a morphism ϕ : Vf → X, and an isomorphism ψ : hX ×F G → hVf such that hϕ ◦ ψ is the first projection. In particular, if F is representable by a scheme X, then G is representable as well (by the scheme Vf , in the above notation, where f ∈ F (X) is an isomorphism from hX to F ). Definition 1.1.7. — One says that a morphism of presheaves Φ : G → F on SchS is (representable by) an open (resp. a closed) immersion if it is representable and if, with the above notation, the morphism ϕ is an open (resp. a closed) immersion. (1.1.8). — Let S be a scheme, let F : Schop S → Sets be a presheaf on the category of S-schemes, and let G be a subfunctor of F . One says that G is an open (resp. a closed) subfunctor of F if the inclusion from G to F is representable by an open (resp. a closed) immersion. Explicitly, this means the following property: Let X be an S-scheme and let f ∈ F (X); then there exists an open (resp. closed) subscheme V of X such that for every S-scheme Y , a morphism u ∈ HomS (Y, X) factors through V if and only if the element u∗ f of F (Y ) belongs to G(Y ). In particular, if F is representable by a scheme X, then G is representable by an open (resp. closed) subscheme of X.

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If G and G are open (resp. closed) subfunctors of a presheaf F on the category of S-schemes, then G ×F G is an open (resp. closed) subfunctor of F . Definition 1.1.9. — Let S be a scheme, and let F be a Zariski sheaf on the category of S-schemes. One says that a family (Fi )i∈I of open subfunctors of F covers F if the following property is satisfied: Let X be an S-scheme, and let f ∈ F (X); for every i, let Vi be the open subscheme of X representing the functor hX ×F Fi ; then the family (Vi ) is an open covering of X. The following proposition, a local criterion for representability, is a useful tool to establish the representability of some Zariski sheaves. Proposition 1.1.10. — Let S be a scheme, and let F be a Zariski sheaf on the category of S-schemes. Let (Fi )i∈I be a family of representable open subfunctors of F . If this family covers F , then F is representable. Proof. — For every i, let ji : Fi → F be the canonical inclusion, let Ui be an S-scheme, and let ϕi : hUi → Fi be an isomorphism. For every i, j ∈ I, the open subfunctor Fij = Fi ×F Fj of F is representable by an open subscheme Uij of Ui , as well as by an open subscheme Uji of Uj ; let ϕij : Uij → Uji be the resulting isomorphism. One checks readily the equalities ϕii = idUi , ϕij (Uij ∩Uik ) = Uji ∩Ujk , and ϕik = ϕjk ◦ϕij on Uij ∩Uik . Consequently, there exist an S-scheme X, an open covering (Xi ) of X, and isomorphisms fi : Xi → Ui such that Uij = fi (Xi ∩ Xj ) and ϕij ◦ fi = fj on Xi ∩ Xj for every i, j ∈ I. For every i, the composition ji ◦ϕi ◦fi : hXi → F corresponds to an element gi ∈ F (Xi ). The family (gi ) satisfies the gluing property gi |Xi ∩Xj = gj |Xi ∩Xj . Since F is a Zariski sheaf, there exists a unique element g ∈ F (X) such that g|Xi = gi for every i. In turn, this gives a morphism of sheaves g : hX → F . To conclude the proof, it remains to check that g is an isomorphism. Let Y be an S-scheme and let u ∈ F (Y ). For every i, let Yi be the open subscheme of Y which represents hY ×F Fi , and let ui = u|Yi ∈ F (Yi ). Since Fi is representable by Ui , there exists a morphism vi : Yi → Ui which induces ui . The morphisms fi−1 ◦vi : Yi → Xi glue and give rise to a morphism v : Y → X. Viewing v as an element of hX (Y ), one has g(Y )(v) = u and it is the unique such morphism. Corollary 1.1.11. — Let S be a scheme, let F be a Zariski sheaf on the category of S-schemes, and let (Si ) be an open covering of S. Then F is representable if and only if, for every i, the restriction FSi of F to the category of Si -schemes is representable. Proof. — The functor Fi  F ×hS hSi is an open subfunctor of F . Moreover, the family (Fi ) covers F . The corollary thus follows from proposition 1.1.10.

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1.2. The Weil Restriction Functor (1.2.1). — Let S  be an S-scheme, and let X be an S  -scheme. For every S-scheme Y , let us set (1.2.1.1)

RS  /S (X)(Y ) := HomSchS (Y ×S S  , X).

For every morphism of S-schemes f : Y  → Y and every α ∈ HomSchS (Y ×S S  , X), let us set (1.2.1.2)

RS  /S (X)(f ) := (α → α ◦ fS  ).

This defines a presheaf RS  /S (X) : Schop S → Sets on the category of S-schemes, called the Weil restriction of X with respect to S  /S. For every S-scheme Y , for every morphism of S  -schemes g : X  → X and for every β ∈ HomSchS (Y ×S S  , X  ), let us also set (1.2.1.3)

RS  /S (g)(Y ) := (β → g ◦ β).

Then this defines a bifunctor RS  /S (·) : SchS  × Schop S → Sets. Example 1.2.2. — For every S-scheme Y , the first projection Y ×S S  → Y induces a functorial bijection between the sets HomS (Y, S) and HomS  (Y ×S S  , S  ), which are singletons. This shows that the Weil restriction of S  , RS  /S (S  ), is represented by S. Example 1.2.3. — Let X be an S-scheme. The Weil restriction of X with respect to idS : S → S, RS/S (X) is represented by X. Remark 1.2.4. — The Weil restriction of an S-scheme X is not always representable. As an example, let k be an infinite field, let S = Spec(k[T ]), and let S  = Spec(k[T ]/(T ))  Spec(k); let X = A1S  be the affine line over S  . By definition, for every k[T ]-algebra R, one has RS  /S (A1S  )(Spec R) = X(Spec(R ⊗k[T ] (k[T ]/(T )))) = R/(T ). In particular, if R is a k[T ]/(T )-algebra, then RS  /S (A1S  )(R) = X(R); this identifies X with a closed subfunctor of RS  /S (A1S  ). The formula shows that RS  /S (A1S  )(R) is reduced to one element as soon as T is invertible in R, so that the localization map RS  /S (A1S  )(R) → RS  /S (A1S  )(RT ) deduced from the open immersion Spec(RT ) → Spec(R) is rarely injective. Let us explain why this non-separatedness property implies that the functor Y cannot be representable. Let R be the localization of the ring k[T ] at (T ), and let K = k(T ) be its field of fractions. Assume, by contradiction, that the functor RS  /S (A1S  ) is represented by a scheme Y , and let U be an affine open subscheme of Y which meets X. Since R is a local ring, the image of a morphism Spec(R) → Y

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is contained in U as soon as it sends the closed point of Spec(R) to a point of U . This implies that U (R) is infinite. On the other hand, U (K) is reduced to one element, and this contradicts the fact that the map U (R) → U (K) is injective, because U is affine. Proposition 1.2.5. — Let S be a scheme, let S  be an S-scheme, and let X be an S  -scheme. The functor RS  /S (X) is a Zariski sheaf on S. Proof. — This assertion follows from the fact that the functor hX is a sheaf for the Zariski topology. Precisely, let Y be an S-scheme, and let (Yi )i∈I be an open covering of Y ; for every pair (i, j) of elements of I, let Yij = Yi ∩Yj . We need to show that for every family (fi )i∈I , where for every i, j ∈ I, one has fi ∈ RS  /S (X)(Yi ) and fi |Yij = fj |Yij , there exists a unique f ∈ RS  /S (X)(Y ) such that f |Yi = fi for every i ∈ I. Concretely, for every i, j ∈ I, fi is a morphism of S  -schemes from Yi ×S S  to X, and the morphisms fi and fj coincide on Yij ×S S  . The family (Yi ×S S  )i∈I is an open covering of Y ×S S  , and one has Yij ×S S  = (Yi ×S S  ) ∩ (Yj ×S S  ). Since the functor hX associated with the scheme X is a Zariski sheaf, there exists a unique morphism f : Y ×S S  → X whose restriction to Yi ×S S  is equal to fi . This morphism f is the unique element of RS  /S (X)(Y ) such that f |Yi = fi for every i. Proposition 1.2.6. — Let S be a scheme and let S  be an S-scheme. Let X1 , X2 , T be S  -schemes, let f1 : X1 → T and f2 : X2 → T be S  -morphisms, and let p1 and p2 be the two projections from X1 ×T X2 to X1 and X2 , respectively. The canonical morphism of functors (RS  /S (p1 ), RS  /S (p2 )) : RS  /S (X1 ×T X2 ) → RS  /S (X1 )×RS /S (T ) RS  /S (X2 ) is an isomorphism. In other words, the Weil restriction functor respects fiber products. Proof. — Let Y be an S-scheme. Evaluated at Y , the given morphism of functors is the map HomS  (Y ×S S  , X1 ×T X2 ) → HomS  (Y ×S S  , X1 ) ×HomS (Y ×S S  ,T ) HomS  (Y ×S S  , X2 ) given by f → (p1 ◦ f, p2 ◦ f ). This map is a bijection, hence the proposition. Corollary 1.2.7. — Let S be a scheme and let S  be an S-scheme. Let X1 and X2 be S  -schemes, and let p1 and p2 be the two projections from X1 ×S  X2 to X1 and X2 , respectively. The canonical morphism of functors (RS  /S (p1 ), RS  /S (p2 )) : RS  /S (X1 ×T X2 ) → RS  /S (X1 ) × RS  /S (X2 ) is an isomorphism.

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Remark 1.2.8. — Fiber products are a particular case of categorical limits: If X, Y, T are S  -schemes, their fiber product X ×T Y is the limit of the diagram X T Y 

in the category of S -schemes. Since the Weil restriction functor is a right adjoint, proposition 1.2.6 and its corollary are special cases of the general fact that a right adjoint respects categorical limits. Proposition 1.2.9. — Let S be a scheme and let S  be a finite locally free S-scheme. Let j : X  → X be an open (resp. closed) immersion of S  schemes. Then the morphism of functors RS  /S (j) : RS  /S (X  ) → RS  /S (X) is an open (resp. a closed) immersion as well. Proof (cf. (Bosch et al. 1990, Chapter 7, §6, proposition 2)). Let Y be an S-scheme, and let ϕ : hY → RS  /S (X) be a morphism of functors. Observe that ϕ(Y )(idY ) is a morphism of S  -schemes f : Y ×S S  → X. Unfolding the definitions, we need to prove that there exists a unique open (resp. closed) subscheme Y  of Y such that for every scheme Z and every morphism g : Z → Y , the morphism h = f ◦ g ×S IdS  factors through X  if and only if g factors through Y  : h

ZS ¢ Z

gS ¢

g

YS ¢

f

X

Y.

Let us assume that j : X  → X is an open immersion; then f −1 (X  ) is an open subscheme of Y ×S S  . The first projection p1 : Y ×S S  → Y is finite, hence closed, because S  is finite over S. Consequently, Y  = Y p1 (Y ×S S  f −1 (X  )) is open in Y and satisfies the required property. Let us now treat the case where X  → X is a closed immersion. One reduces to the case where S, X, Y , and Z are affine, say S = Spec(R), X = Spec(A), Y = Spec(B), and Z = Spec(C), and that S  = Spec(R ) is the spectrum of an R-algebra R which has a basis (e1 , . . . , en ) as an R-module. Then f corresponds to a morphism of R -algebras f ∗ : A → B ⊗R R , and g corresponds to a morphism of R-algebras g ∗ : B → C. Then h = f ◦(g×S IdS  ) corresponds to the composition h∗ : A → B ⊗R R → C ⊗R R . For a ∈ A, n ∗ ∗ ∗ write B for every i, so that h∗ (a) = n f∗ (a)∗ = i=1 fi (a) ⊗ ei , with fi (a) ∈ n  ∗ i=1 g (fi (a)) ⊗ ei . Let I be the ideal of X in A, and let J = i=1 fi (I)B;  observe that J is an ideal of B. Moreover, h factors through X if and only if I ⊂ Ker(h∗ ), that is, if and only if J ⊂ Ker(g ∗ ), that is, if and only if g

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factors through the closed subscheme Spec(B/J) of Y . This concludes the proof. Corollary 1.2.10. — Let S be a scheme and let S  be a finite and locally free S-scheme. Let X be an S  -scheme such that the functor RS  /S (X) is representable. Let u : X  → X be an immersion (resp. an open immersion, a closed immersion). Then the functor RS  /S (X  ) is representable, and RS  /S (u) is an immersion (resp. an open immersion, a closed immersion). Corollary 1.2.11. — Let S be a scheme, and let S  be a finite and locally free S-scheme. Let X be an S  -scheme such that the functor RS  /S (X) is representable, represented by an S-scheme Y . If X is separated over S  , then Y is separated over S. Proof. — By proposition 1.2.6, the functor RS  /S (X ×S  X) is representable by Y ×S Y . Let i : X → X ×S  X be the diagonal immersion; its image j under the Weil restriction functor satisfies p1 ◦ j = p2 ◦ j = id, hence is the diagonal immersion Y → Y ×S Y . If X is separated over S  , then i is a closed immersion; hence j is a closed immersion as well, so that Y is separated over S. (1.2.12). — Let S be a scheme, let S  be an S-scheme, and let X be an S  scheme. Let T be an S-scheme and let T  = T ×S S  . For every T -scheme Y , the canonical associativity isomorphism Y ×T T  = Y ×T (T ×S S  )  Y ×S S  induces a bijection HomT  (Y ×T T  , X ×S  T  )  HomS  (Y ×S S  , X), which is functorial in Y . This furnishes an isomorphism of functors RT  /T (X ×S  T )  RS  /S (X) ×S hT . In particular, if RS  /S (X) is representable, then RT  /T (X ×S  T ) is representable by the fiber product RS  /S (X) ×S T .

1.3. Representability of a Weil Restriction: The Affine Case (1.3.1). — Let A be a ring, and let A be an A-algebra which is a free Amodule of rank n; let a1 , . . . , an be a basis of A . Let S = Spec(A) and S  = Spec(A ); let X be an affine S  -scheme. In this subsection, we will prove that the Weil restriction of X, RS  /S (X), is representable and describe explicitly a scheme which represents it.

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(1.3.2). — Let (Te )e∈E be a family of indeterminates, and let us assume  that X = AE S  = Spec(A [(Te )e∈E ]) is the corresponding affine space. Set nE = {1, . . . , n} × E. Let B be an A-algebra and let Y = Spec(B). One has canonical bijections HomS  (Y ×S S  , X)  HomA (A [(Te )e∈E ], B ⊗A A )  (B ⊗A A )E .  bi ⊗ ai is a Moreover, the map B n → B ⊗A A given by (b1 , . . . , bn ) → bijection; hence we get a canonical bijection HomS  (Y ×S S  , X)  B nE  HomA (A[T(i,e) ], B)  HomS (Y, AnE S ). This bijection is functorial in Y , which shows that RS  /S (AE S  ) is represented nE by the affine space AS . Observe that nE is finite if E is finite. E (1.3.3). — Let X be a closed subscheme of AE S  . Since RS  /S (AS  ) is reprenE sented by AS , it follows from proposition 1.2.9 that RS  /S (X) is represented by a closed subscheme of AnE S . Let us describe its ideal. Let I be the ideal of X in A [(Te )]. For every f ∈ I, there exists a unique tuple (f1 , . . . , fn ) of polynomials in

A[Ti,e | i ∈ {1, . . . , n}, e ∈ E] such that f (T1,e a1 + . . . + Tn,e an ) = f1 (Ti,e )a1 + . . . + fn (Ti,e )an in A [Ti,e ]. Let J be the ideal of A[(Ti,e )] generated by the polynomials f1 , . . . , fn , where f ranges over a generating set of the ideal I. It follows from the proof of proposition 1.2.9 that RS  /S (X) is the closed subscheme of AnE S defined by J. Proposition 1.3.4. — Let S be a scheme and let S  be a finite and locally free S-scheme. For every affine S  -scheme X, the functor RS  /S (X) is represented by an affine S-scheme. Moreover, if the S  -scheme X is of finite type (resp. of finite presentation), then so is the S-scheme RS  /S (X). Proof. — The preceding arguments establish the lemma when S = Spec(A) and S  = Spec(A ) are affine, and when A is a free S-module. The general case follows since the statements of the proposition are local on S. 1.4. Representability: The General Case Let S be a scheme and let S  be a finite and locally free S-scheme. In this subsection, we establish sufficient conditions for the representability of the Weil restriction of an S  -scheme X with respect to S  /S. Theorem 1.4.1. — Let S be a scheme and let S  be a finite and locally free S-scheme. Let X be an S  -scheme. Any one of the following assumptions implies that the Weil restriction RS  /S (X) is representable:

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(i) For every s ∈ S and every finite set of points P ⊂ X ⊗S κ(s), there is an affine open subscheme of X containing P ; (ii) The structural morphism X → S  is quasi-projective; (iii) The morphism S  → S is a universal homeomorphism. Proof. — Since case (ii) follows from (i), we assume that either hypothesis (i) or (iii) holds. We can reduce to the case where S, and thus S  , are affine. For every affine open subscheme U of X the open subfunctor RS  /S (U ) is representable, as we have seen in the preceding section. By the general representability local criterion of Zariski sheaves (proposition 1.1.10), it thus suffices to prove that these subfunctors, where U ranges over affine open subschemes of X, cover RS  /S (X). Let Y be an S-scheme and let f ∈ RS  /S (X) = HomS  (Y ×S S  , X). Let p and q be the projections from Y ×S S  to Y and S  , respectively. Let y ∈ Y , and let s be the image of y in S. The preimage p−1 (y) of y in Y ×S S  is the finite scheme κ(y) ×S S  ; let P be its image in X under f . By construction, P is a finite subset of X ⊗ κ(s). Moreover, if S  → S is a universal homeomorphism, then Y ×S S  → Y is a homeomorphism; hence P is reduced to a single point. Consequently, there exists an open affine subscheme U of X which contains P : this is the stated assumption in case (i), and follows from the definition of a scheme in case (iii). Then V  = f −1 (U ) is an open subscheme of Y ×S S  which contains p−1 (y). Since S  is finite over S, there exists an open subscheme V of Y such that y ∈ V and p−1 (V ) ⊂ V  . This implies the desired conclusion. Remark 1.4.2. — a) It follows from the proof that RS  /S (X) is covered by affine open subschemes of the form RS  /S (U ), where U is an open affine subscheme of X. If U is of finite type (resp. of finite presentation) over S  , then RS  /S (U ) is of finite type (resp. of finite presentation) over S, by proposition 1.3.4. Consequently, if X is locally of finite type (resp. locally of finite presentation) over S  , then RS  /S is locally of finite type (resp. locally of finite presentation) over S. b) Under assumption (iii), the preceding proof shows that for every affine covering (Ui ) of X, the open subschemes RS  /S (Ui ) form an open covering of X. In particular, if X is quasi-compact, then RS  /S (X) is quasi-compact as well.

§ 2. JET SCHEMES 2.1. Jet Schemes of a Variety (2.1.1). — Let S be a scheme and let X be an S-scheme. Let n ∈ N. For every S-scheme Y , we call a morphism Y ⊗Z Z[t]/(tn−1 ) → X a Y -valued jet of level n on X. We denote by Ln (X/S)(Y ) the set of Y -valued jets of level n on X, i.e.,

§ 2. JET SCHEMES

(2.1.1.1)

163

Ln (X/S)(Y ) := HomSchS (Y ⊗Z Z[t]/(tn+1 ), X).

When Y = Spec(A) is affine, one also speaks of an A-valued jet of level n or of an A-jet. Then (2.1.1.2) Ln (X/S)(A) := Ln (X/S)(Spec(A)) = HomSchS (Spec(A[t]/(tn+1 )), X). For every morphism f : Y  → Y of S-schemes and every γ ∈ Ln (X/S)(Y ), one defines a map Ln (X/S)(f ) : Ln (X/S)(Y ) → Ln (X/S)(Y  ) by Ln (X/S)(f )(γ) = γ ◦ f. These data furnish a presheaf (2.1.1.3)

Ln (X/S) : Schop S → Sets

on the category of S-schemes, called the functor of jets of level n on X. The construction of the functor of jets is functorial in X. Precisely, let g : X  → X be a morphism of S-schemes. The formula (2.1.1.4)

Ln (g)(Y ) := (β → g ◦ β),

where Y is an S-scheme and β ∈ Ln (X  /S)(Y ) is a Y -valued jet of level n, defines a morphism of functors from Ln (X  /S) to Ln (X/S), also denoted by g∗ . When S = Spec(k) is the spectrum of a ring k, we also write Ln (X/k) for the functor Ln (X/S). (2.1.2). — The functor of jets is a particular case of Weil restriction: namely, Ln (X/S) is the Weil restriction of X ⊗Z Z[t]/(tn+1 ) with respect to the morphism S ⊗Z Z[t]/(tn+1 ) → S. Let us also observe that this morphism is finite, is locally free everywhere of rank n + 1, and is a universal homeomorphism. By base change, it suffices to prove this for S = Spec(Z). Then Z[t]/(tn+1 ) is a free Z-module of rank n+1. Moreover, the nilradical of Z[t]/(tn+1 ) being equal to the ideal (t), the ring morphism Z → Z[t]/(tn+1 ) induces an isomorphism on the associated reduced rings. By (ÉGA IV2 , 2.4.3 (vi)), the morphism of schemes Spec(Z[t]/(tn+1 )) → Spec(Z) is thus a universal homeomorphism. Proposition 2.1.3. — Let S be a scheme, let X be an S-scheme, and let n be an integer. a) The functor Ln (X/S) of jets of level n on X is representable. b) If f : Y → X is a closed (resp. open) immersion, then the associated morphism of schemes Ln (f ) : Ln (Y /S) → Ln (X/S) is a closed (resp. open) immersion as well. c) For every open covering (Ui )i∈I of X, the family (Ln (Ui /S)) is an open covering of Ln (X/S). d) Assume that the S-scheme X is affine (resp. is locally of finite type, resp. is locally of finite presentation, resp. is quasi-compact). Then, the S-scheme Ln (X/S) has the same property.

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Proof. — a) By (2.1.2), the representability of the functor of jets of level n follows from theorem 1.4.1, (iii). The property for a morphism to be a closed (resp. open) immersion is stable under base change. Consequently, assertion b) is proposition 1.2.9, while assertion c) had been observed in remark 1.4.2, b). d) If X is affine, resp. quasi-compact, then so is X ⊗Z Z[t]/(tn+1 ). Similarly, if X is (locally) of finite type over S, resp. (locally) of finite presentation over S, then X ⊗Z Z[t]/(tn+1 ) is (locally) of finite type, resp. (locally) of finite presentation over S ⊗Z Z[t]/(tn+1 ). When X is affine, we thus deduce from proposition 1.3.4 that Ln (X/S) is affine and that it is of finite type (resp. of finite presentation) if X is moreover of finite type (resp. of finite presentation). In the general case, it follows from remark 1.4.2 that if X is quasi-compact, resp. (locally) of finite type over S, resp. (locally) of finite presentation over S, then so is Ln (X/S). (2.1.4) Base Change. — Let X be an S-scheme, let T be an S-scheme, and let n be an integer. Let Y be a T -scheme. By base change, every Y -valued arc of level n on X induces an Y -valued arc of level n on X ×S T . This defines a morphism of schemes (2.1.4.1)

Ln (X ×S T /T ) → Ln (X/S) ×S T.

By compatibility of Weil restriction with base change, this morphism is an isomorphism. In the particular case where the morphism T → S is a monomorphism and the structural morphism of X factors (uniquely, necessarily) through T , the projection X ×S T → X is an isomorphism, so that this property implies that the canonical morphism Ln (X/T ) → Ln (X/S) ×S T is an isomorphism. (2.1.5) Fiber Product. — Let X1 and X2 be S-schemes, and let p1 and p2 be the projections from X1 ×S X2 to X1 and X2 , respectively. Let n be an integer. The associated morphisms Ln (p1 ) and Ln (p2 ) define a morphism of schemes (2.1.5.1)

Ln (X1 ×S X2 /S) → Ln (X1 /S) ×S Ln (X2 /S).

By compatibility of Weil restriction with fiber products (proposition 1.2.6), this morphism is an isomorphism. Lemma 2.1.6. — Let S be a scheme and let X be a separated S-scheme. For every integer n, the S-scheme Ln (X/S) is separated. Proof. — Since separatedness is stable under base change, the scheme X ⊗Z Z[t]/(tn+1 ) is separated over Spec(Z[t]/(tn+1 )). Since the Weil restriction functor preserves separatedness, this implies that Ln (X/S) is a separated S-scheme.

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165

2.2. Truncation Morphisms (2.2.1). — Let m, n ∈ N be integers, with m  n. Let X be an S-scheme. Reduction modulo tn+1 induces a morphism of schemes n+1 )) → Spec(Z[t]/(tm+1 )). pm n : Spec(Z[t]/(t

For every S-scheme Y , it induces a map (2.2.1.1)

m (Y ) : Lm (X/S)(Y ) → Ln (X/S)(Y ), θn,X

that associates with every Y -valued jet of level m, γ : Y ⊗Z Z[t]/(tm+1 ) on X, the Y -valued jet of level n on X defined by γ ◦ (IdX × pm n ). This map is called the truncation map from level m to n on Y -valued jets on X. m (Y ) are functorial in Y ; the associated morphism The truncation maps θn,X m of S-schemes, θn,X : Lm (X/S) → Ln (X/S), is also called the truncation morphism from level m to level n. These truncation morphisms satisfy the transition property (2.2.1.2)

p m m = θn,X ◦ θp,X θn,X

for every S-scheme X and every integers m, p, n ∈ N such that m  p  n. n (ϕ) ∈ X(Y ) If ϕ ∈ Ln (X/S)(Y ) is a Y -valued jet of level n, the point θ0,X is called its base point and is also denoted by ϕ(0). m The truncation morphisms θn,X are functorial in X. Namely, for every  morphism g : X → X of S-schemes, and every pair (m, n) of integers such that m  n, one has (2.2.1.3)

m m Ln (g) ◦ θn,X  = θn,X ◦ Lm (g).

(2.2.2). — Let n ∈ N be an integer. The canonical ring morphism from Z to Z[t]/(tn+1 ) induces a morphism of schemes Spec(Z[t]/tn+1 )) → Spec(Z). For every S-schemes X and Y , this gives rise to a map (2.2.2.1)

sn,X (Y ) : X(Y ) → Ln (X/S)(Y ),

which is covariantly functorial in X and contravariantly functorial in Y . The associated morphisms of S-schemes, sn,X : X → Ln (X/S), satisfy (2.2.2.2)

m ◦ sm,X = sn,X θn,X

for every pair (m, n) of integers with m  n. Proposition 2.2.3. — Let S be a scheme, let X be an S-scheme, and let U be an open subscheme of X. Let n be an integer. Then Ln (U/S) identifies n )−1 (U ) of Ln (X/S). with the open subscheme (θ0,X Proof. — Let Y be an S-scheme. Observe that Y is a closed subscheme of Y ⊗Z Z[t]/(tn+1 ) with the same underlying topological space. Consequently the image of a Y -valued jet γ : Y ⊗Z Z[t]/(tn+1 ) → X is contained in U if and n (γ) : Y → X factors through U . This implies the proposition. only if θ0,X Corollary 2.2.4. — If S is an affine scheme, then the truncation morm are affine. phisms θn,X

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Proof. — Let U be an affine open subscheme of X, and let m, n be integers such that m  n. Then Ln (U/S) is an affine open subscheme of Ln (X/S), and Lm (U/S) is an affine open subscheme of Lm (X/S). Moreover, one has m −1 m n m Lm (U/S) = (θ0,X ) (U ) = (θn,X )−1 (θ0,X )−1 (U ) = (θn,X )−1 (Ln (U/S)).

When U ranges over all affine open subschemes of X, the jet schemes Ln (U/S) thus constitute an affine open covering of Ln (X/S). This implies m is affine. that θn,X Corollary 2.2.5. — The morphisms sn,X are closed immersions. n is affine Proof. — We may assume that S is affine. Then the morphism θ0,X and thus separated. Since sn,X is a section of that morphism, the assertion follows from (ÉGA I, 5.4.6).

2.3. Examples Example 2.3.1. — Let X be an S-scheme. The isomorphism Z[t]/(t)  Z induces an isomorphism of functors from L0 (X/S) to the functor hX associated with the scheme X. In other words, the functor L0 (X/S) is representable by X, and the morphism s0,X : X → L0 (X/S) is an isomorphism. Example 2.3.2. — The jet functors Ln (S/S) of S are representable by S; the truncation morphisms are the identity. Example 2.3.3. — The functor of jets of order 1 identifies with the “relative tangent bundle” of X, defined by TX/S = Spec(Sym(Ω1X/S )) (ÉGA IV4 , 16.5.12). Let us explain this explicitly when X and S are affine, say X = Spec(B) and S = Spec(A), for a ring A and an A-algebra B. In this case, TX/S = Spec(Sym(Ω1B/A )), where Ω1B/A is the B-module of relative differentials of B with respect to A. Let R be an A-algebra. A morphism of A-schemes Spec(R) → Spec(Sym(Ω1B/A )) corresponds to the datum (f, D) of a morphism f : B → R of A-algebras and of an A-derivation D : B → R. On the other hand, such a pair (f, D) corresponds to the morphism B → R[t]/(t2 ),

b → f (b) + tD(b)

of A-algebras. This furnishes a bijection Spec(Sym(Ω1X/S ))(Spec(R)) → L1 (X/S)(Spec(R)), functorial in R, hence the claim. Let us moreover explicit the case where B is a A-algebra of finite presentation, say B = A[T1 , . . . , Tn ]/(f1 , . . . , fm ). Then Ω1X/S is the quotient of the

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free B-module with basis dT1 , . . . , dTn by the submodule generated by the differentials df1 , . . . , dfm given by n  ∂fi dTj . dfi = ∂T j j=1

As a consequence, the A-algebra Sym(Ω1X/S ) can be described as the quotient A[T1 , . . . , Tn , U1 , . . . , Un ]/(f1 , . . . , fm , δf1 , . . . , δfm ), where δf1 , . . . , δfm are defined by the formula δfi =

n  ∂fi Uj . ∂T j j=1

Example 2.3.4. — Let k be a ring, let E be a set, let (Te )e∈E be a family of indeterminates, and let AE k = Spec(k[(Te )]) be the corresponding affine k-space. Let n be an integer. For every k-algebra A, the direct sum decomposition of A-modules (2.3.4.1)

A ⊗k (k[t]/(tn+1 )) ∼ = ⊕ni=0 Ati (n+1)E

induces a bijection from Ln (X/k)(A) to Ak . Explicitly, an A-jet on AE k is the datum of a family (xe )e∈E of polynomials of degrees  n; if one n i writes xe = i=0 xi,e t , this bijection associates with (xe )e∈E the family (xi,e )0in . e∈E

This gives an isomorphism of functors from Ln (X/k) to the point functor (n+1)E associated with Ak = Spec(k[(Si,e )0in ]). e∈E

Moreover, if m is an integer such that 0  n  m, the truncation mor(m+1)E (n+1)E m phism θn,X corresponds to the projection Ak → Ak obtained by forgetting the components xi,e , for n < i  m and e ∈ E. By base change, this implies a similar result for every scheme S: the jet (n+1)E , and the truncascheme Ln (AE S /S) identifies with the affine space AS (m+1)E (n+1)E → AS . tion morphisms identify with projections AS (2.3.5). — Let k be a ring, let E be a set, and let X be a closed subscheme of the affine k-space AE k defined by an ideal I. For every integer n, the jet scheme Ln (X/k) is a closed subscheme of the (n+1)E jet scheme Ln (AE . Let us describe its ideal. k /k) = Ak Let (Te )e∈E and (Si,e )0in be families of indeterminates. For every polye∈E

nomial f ∈ k[(Te )e∈E ], there exists a unique family (fm )0mn , where for each m, fm ∈ k[(Si,e )im,e∈E ], such that f ((S0,e + tS1,e + · · · + tn Sn,e )e ) =

n  m=0

fm ((Si,e )0im )tm e∈E

(mod tn+1 ).

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Let In be the ideal of k[(Si,e )0in ] generated by the polynomials fm , for e∈E

f ∈ I and m ∈ {0, . . . , n}. Let us check that the jet scheme Ln (X/k) is the closed subscheme (n+1)E of Ak defined by the ideal In of k[(Si,e )i,e ]. Let A be a k-algebra, and let x be an A-jet of order n on X; as an Ai jet on AE k , it can be written as x = ( in ai,e t )e∈E . Moreover, for every polynomial f ∈ k[(Te )], one has n  f (x) = fm ((ai,e )0im )tm ∈ A[t]/(tn+1 ). m=0

e∈E

Consequently, the A[t]/(tn+1 )-point x of AE k belongs to X(A) if and only if fm ((ai,e )0im ) = 0 for every m ∈ {0, . . . , n} and every polynomial f ∈ I. e∈E

This implies the claim. Observe also that if S is a generating subset of the ideal I, then the ideal In is generated by the polynomials fm , for f ∈ S and 0  m  n. In particular, if E is finite and I is finitely generated, we see explicitly that Ln (X/k) is a finitely presented k-scheme. Example 2.3.6. — Let k be a ring and let X = V (xy) be the union of the coordinates axis in A2k . Let n ∈ N. Let us compute a presentation of the k-variety Ln (X/k). By definition, the equations of the embedding of Ln (X/k) in A2n k are given by the vanishing of the first n coefficients of the polynomial (x0 + x1 t + . . . + xn−1 tn−1 ) · (y0 + y1 t + . . . + yn−1 tn−1 ) in the ring k[t] or, equivalently, by the following system: ⎧ = 0 x0 y0 ⎪ ⎪ ⎨ = 0 x0 y1 + x1 y0 . . . . .. ⎪ ⎪ ⎩ x0 yn−1 + x1 yn−2 + . . . + xn−1 y0 = 0 m : Lm (X/k) → If m ∈ N, with m  n, the truncation morphism θn,X Ln (X/k) corresponds to forgetting the coordinates xi and yi such that n < i  m.

§ 3. THE ARC SCHEME OF A VARIETY In this subsection, we are going to construct the arc scheme associated with a scheme.

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3.1. Arcs on a Variety (3.1.1). — Let S be a scheme and let X be an S-scheme. Let A ∈ AlgS be an S-algebra, that is, a ring endowed with a morphism from Spec(A) to S, cf. §1.1.3. Then A[[t]] is an S-algebra as well, and we call an A-arc on X any A[[t]]-point of X, equivalently any morphism of S-schemes Spec(A[[t]]) → X. For every S-algebra A, the set of A-arcs on X is denoted by (3.1.1.1)

L∞ (X/S)(A) := HomSchS (Spec(A[[t]]), X).

Every morphism of S-algebras f : A → A induces a morphism of Salgebras fˆ: A[[t]]→ A [[t]]; let (3.1.1.2)

L∞ (X/S)(f ) := (ϕ → ϕ ◦ Spec(fˆ)).

These data define a covariant functor L∞ (X/S) : AlgS → Sets, from the category of S-algebras to the category of sets. We call it the functor of arcs on X. If no ambiguity is possible with respect to the base scheme S, we may omit from the notation and simply write L∞ (X). (3.1.2). — The assignment X → L∞ (X/S) is functorial in X: For every morphism of S-schemes f : X → Y and every S-algebra A, the morphism of functors L∞ (f ) : L∞ (X/S) → L∞ (Y /S) associates with an A-arc ϕ ∈ X(A[[t]]) on X the A-arc f ◦ ϕ on Y . (3.1.3) Base Change. — Let S be a scheme, and let X and T be S-schemes. By composition with the given morphism from T to S, every T -ring can be viewed as T -ring. This induces a morphism L∞ (X ×S T ) → L∞ (X/S) ×S T. This morphism is an isomorphism. Let indeed A be an S-ring. Let ϕ ∈ L∞ (X/S)(A) and let g : Spec(A) → T be a S morphism. Then ϕ is an S-morphism from Spec(A[[t]]) to X; viewing A[[t]] as a T -algebra via g, the pair (ϕ, g) defines a T -morphism ψ from Spec(A[[t]]) to X ×S T . This morphism ψ is the unique element of L∞ (X ×S T ) mapping to (ϕ, g). This proves the claim. In particular case where the morphism T → S is a monomorphism and the structural morphism of X factors (uniquely, necessarily) through T , the projection X ×S T → X is an isomorphism, so that this property implies that the canonical morphism L∞ (X/T ) → L∞ (X/S) ×S T is an isomorphism.

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(3.1.4). — Let n be an integer. For every S-algebra A, reduction modulo (tn+1 ) defines a morphism of S-algebras A[[t]]→ A[t]/(tn+1 ). It induces a morphism of functors ∞ θn,X : L∞ (X/S) → Ln (X/S), ∞ which is called the truncation functor of level n. We also write θn,X for θn,X .

For every pair (m, n) of integers with m  n, one has the following transitivity relation ∞ m ∞ ◦ θn,X = θn,X . θm,X

(3.1.4.1) In other words, the diagram

Lm (X/S) ∞ θ m,X

L∞ (X/S)

∞ θ n,X

m θ n,X

Ln (X/S)

commutes. Observe also that these truncation functors are functorial in X. For every morphism g : X → Y of S-schemes and every integer n ∈ N, one has (3.1.4.2)

∞ ∞ Ln (g) ◦ θn,X = θn,Y ◦ Ln (g).

If ϕ ∈ L∞ (X/S)(A) is an A-arc on X, we also write ϕ(0) for its base ∞ (ϕ)(A) ∈ X(A). point, which is the A-point θ0,X Example 3.1.5 (Arcs on curves). — Let k be an algebraically closed field of characteristic zero. Let f ∈ k[x, y] be an irreducible polynomial; let C ⊂ A2k be the integral plane curve defined by f . Assume that f (0, 0) = 0 and that the image of f is irreducible in the power series ring k[[x, y]]; this means that C is analytically irreducible at the origin. Assume moreover that f (0, y) = 0. Then there exist an integer n > 0 and a formal power series y(t) in t · k[[t]] such that f (tn , y(t)) = 0 and such that y(t) is not of the form y((te ) with y((t) ∈ k[[t]] and e > 1. The pair (tn , y(t)) is called a Puiseux expansion of C at (0, 0). Up to a unit in k[[t, Y ]], the polynomial f (x, Y ) is a minimal polynomial of the Puiseux series y(x1/n ) over the field of Laurent series k((x)). Consequently, such a Puiseux expansion completely characterizes the formal germ Spec(k[[x, y]]/(f )) of C at 0. Observe also that the exponent n is equal to the degree of the fraction field of k[[x, y]]/(f ) over the Laurent series field k((x)). The Puiseux expansion (tn , y(t)) is not unique. However, for every other  Puiseux expansion (tn , y  (t)), one has n = n, and there exists an nth root of unity ζ such that y  (t) = y(ζt). Indeed, y(t) and y  (t) must be conjugate over k((t)).

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Each Puiseux expansion (tn , y(t)) defines a point of C(k[[t]]), and thus a k-valued arc on C centered at the origin (0, 0). Conversely, let us show that every k-valued arc on C centered at (0, 0) is of the form (Φ(t)n , y(Φ(t))), for some power series Φ(t) ∈ t·k[[t]]: Every arc on C centered at (0, 0) is obtained by reparameterizing a Puiseux expansion. Let indeed a(t), b(t) be power series in t·k[[t]] such that a(t) = 0 and f (a(t), b(t)) = 0. If N is the order in t of a(t), there exists a power series a1 (t) such that a(t) = a1 (t)N ; then a1 (t) has order 1 and hence can be taken as formal parameter. We may thus assume that a(t) = tN . Then b(t1/N ) is a root of the polynomial f (t, Y ) = 0 in the field k((t1/N )), so that (tn , b(tn/N )) is a Puiseux expansion of C at (0, 0). This is only possible when N is a multiple of n; thus we can take Φ(t) = tN/n . Remark 3.1.6 (Wedges). — Let S be a scheme and let X be an S-scheme. For every integer e  0, one defines the eth wedge functor L (e) (X/S) by the formula (e) L∞ (X/S)(A) = X(A[[t1 , . . . , te ]]), for every S-ring A (“A-valued e-wedges on X”). Equivalently, one can define (0) it by induction on e, by setting L∞ (X/S) = X, and by defining (e) (e−1) L∞ (X/S) = L∞ (L∞ (X/S)/S)

for e > 0. When e = 1, one recovers the functor of arcs. When e = 2, wedges play an important role in the study of the Nash problem; see section 7/2.5 and (Reguera 2006, 5.1). 3.2. Relative Representability Properties Lemma 3.2.1. — Let f : Y → X is a monomorphism of S-schemes. Then the morphism of functors L∞ (f ) is a monomorphism. Proof. — Let A be an S-algebra, and let ϕ, ϕ ∈ L∞ (Y /S) be two Aarcs on Y such that f∗ (ϕ) = f∗ (ϕ ). The arcs ϕ and ϕ are morphisms from Spec(A[[t]]) to Y whose composition with f coincide. Since f is a monomorphism, they coincide, as was to be shown. Lemma 3.2.2. — Let S be a scheme, let X, Y, Z be S-schemes, and let f : X → Z and g : Y → Z be morphisms. The canonical morphism (L∞ (f ), L∞ (g)) : L∞ (X ×Z Y ) → L∞ (X/S) ×L∞ (Z/S) L∞ (Y /Z) is an isomorphism. Proof. — Indeed, for every S-algebra A, the evaluation of this morphism at A identifies with the canonical map (X ×Z Y )(A[[t]]) → X(A[[t]]) ×Y (A[[t]]) Y (A[[t]]),

ϕ → (f ◦ ϕ, g ◦ ϕ)

which is a bijection, by definition of the fiber product X ×Z Y .

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Proposition 3.2.3. — Let S be a scheme, let X be an S-scheme, let U be an open subscheme of X, and let j : U → X be the inclusion. The canonical morphism L∞ (U/S) → L∞ (X/S) ×X U ∞ induced by the morphisms L∞ (j) and θ0,U is an isomorphism. In particular, the morphism L∞ (j) is representable by an open immersion. Corollary 3.2.4. — Let (Ui ) be an open covering of X. Then the family of open subfunctors L∞ (Ui /S) covers L∞ (X/S). Proof. — Indeed, let V be a scheme, and let ϕ : hV → L∞ (X/S) be a morphism of functors. Let us write ϕ0 = θ0,X (ϕ); this is a morphism of schemes from V to X. For every i, let Vi be the inverse image of L∞ (Ui /S) in V ; it is an open subscheme of V . Moreover, Vi = ϕ−1 0 (Ui ). Consequently, the family (Vi ) is an open covering of V . Proposition 3.2.5. — Let X be an S-scheme, and let Y be a closed subscheme of X; let j : Y → X be the inclusion. The morphism L∞ (j) : L∞ (Y /S) → L∞ (X/S) is representable by a closed immersion. Proof. — Let V be an S-scheme, and let f : hV → L∞ (X/S) be a morphism of functors. We need to prove that the fiber product functor L∞ (Y /S) ×L∞ (X/S) hV is representable by a scheme and that the canonical morphism to V is a closed immersion. By corollary 3.2.4, we may assume that X is affine, say X = Spec(R). We may also assume that V is affine, say V = Spec(A), where A is an S-ring, so that the morphism f corresponds to an A-arc ϕ : Spec(A[[t]]) → X. Let ϕ∗ : R → A[[t]] be the corresponding morphism of rings; define of maps ϕ∗n : R → A, for n ∈ N, by  ∗ a family ∗ n ϕn (r)t . Let J be the ideal of R defining Y in X; the formula ϕ (r) = then ϕ factors through Y if and only if ϕ∗ (J) = 0. Let I be the ideal of A generated by the images of J under the maps ϕ∗n . Then, the fiber product L∞ (Y /S) ×L∞ (X/S) hV is representable by the closed subscheme Spec(A/I) of V = Spec(A). 3.3. Representability of the Functor of Arcs (3.3.1). — Let S be a scheme and let X be an S-scheme. With respect to m the truncation morphisms θn,X , the jet schemes Ln (X/S) form a projective system; this projective system admits a limit in the category of functors over ∞ define a morphism AlgS . By equation (3.1.4.1), the truncation functors θn,X of functors L∞ (X/S) → lim Ln (X/S), ←− n

in the category of functors over AlgS . This morphism will be the key to the representability of the functor of arcs.

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173

We begin with an elementary, but important, case. Proposition 3.3.2. — Let S be a scheme and let X be an S-scheme which is affine. Then the canonical morphisms L∞ (X/S) → lim Ln (X/S) → Spec( lim O(Ln (X/S))) ←− −→ n

n∈N

are isomorphisms. In particular, the arc functor L∞ (X/S) of X is representable by an S-scheme which is affine. Proof. — Let R = OX (X); this is an S-algebra. Let A be an S-algebra, and let us consider the canonical diagram L∞ (X/S )(A)

lim Ln (X/S )(A) n

Hom S (R, A [[t]])

lim Hom S (R, A [[t]]/(tn +1 )) n

lim Hom S (O(Ln (X/S )), A) n

Hom S (lim O(Ln (X/S )), A). n

It is commutative. Moreover, all arrows besides the top-left horizontal one are bijective, so that this arrow is bijective too. Consequently, the morphism of functors L∞ (X/S) → limn Ln (X/S) on AlgS is an isomorphism, and ←− the functor L∞ (X/S) is representable by the spectrum of the S-algebra limn O(Ln (X/S)). −→ Corollary 3.3.3. — Let S be an affine scheme and let f : X → Y be a morphism of S-schemes. If f is affine, then the morphism of functors L∞ (f ) is representable by an affine morphism. Proof. — Let U be an affine open subscheme of Y . By assumption, f −1 (U ) is affine, so that L∞ (U/S) and L∞ (f −1 (U/S)) are representable by affine schemes, and L∞ (f ) induces a morphism between them. Since L∞ (Y /S) is covered by the open subfunctors of the form L∞ (U/S), where U ranges over all affine open subschemes of Y , this implies the corollary. Example 3.3.4. — Keep the notation of (2.3.5). Let k be a ring and let X be an affine k-scheme. Let (Te )e∈E be a family of indeterminates, and let AE k = Spec(k[(Te )]) be the corresponding affine space. Let X be a closed subscheme of AE . By proposition 3.3.2, the functor L∞ (X/k) is representable by an affine k-scheme. Let us describe it explicitly. Let S = (Si,e )(i,e)∈N×E be a family of indeterminates, and let AkN×E be the corresponding affine space. For every k-algebra A, an A-arc ϕ ∈ E a family (ϕe ) ∈ (A[[t]])e . For every e, let us write ϕe = L ∞ (Ak )(A) is i N×E . Associating with ϕ this i∈N xi,e (ϕ)t , for some family (xi,e ) ∈ A family (xi,e ) defines an isomorphism of functors from L∞ (AE k ) to (the functor given by) AkN×E . In particular, L∞ (A1k )  AN k = Spec(Z[S0 , S1 , . . . ]).

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Let I be the ideal of X in k[(Te )e∈E ]. Let A be a k-algebra. As above, an A-arc ϕ ∈ X(A[[t]]) corresponds to a family of power series (ϕe ) ∈ (A[[t]])E such that f ((ϕe )) = 0 in A[[t]] for every f ∈ I. For every polynomial f ∈ k[(Te )e∈E ], let f0 , . . . , fn , . . . be the polynomials in the indeterminates (Si,e )i,e given by the relation  f ((S0,e + tS1,e + · · · + tn Sn,e + . . .)e ) = fi ((Sj,e )j∈{0,...,i},e∈E )ti . i0

It follows from this definition that an arc ϕ ∈ X(A[[t]] corresponds to a point x = (xi,e ) ∈ AkN×E (A) such that fi (x) for every i ∈ N and every f ∈ I (or every f in a generating family of I). Consequently, the functor L∞ (X/k) is represented by the closed subscheme of AkN×E whose ideal is generated by the polynomials fi , for f in a given generating family of I and i ∈ N. Example 3.3.5. — Let k be a field. Let X = V (xy) be the union of the coordinate axis in A2k . Let us compute a presentation of the k-scheme L∞ (X/K). By definition, the equations of the embedding of L∞ (X/k) in 2 (AN k ) are given by the vanishing of all the coefficients of the power series (x0 + x1 t + . . . + xn−1 tn−1 + . . .) · (y0 + y1 t + . . . + yn−1 tn−1 + . . .) in the ring k[[t]] or, equivalently, by the following infinite ⎧ x0 y0 = ⎪ ⎪ ⎪ ⎪ = ⎨ x0 y1 + x1 y0 ... ... ⎪ ⎪ x0 yn−1 + x1 yn−2 + . . . + xn−1 y0 = ⎪ ⎪ ⎩ ... ...

system: 0 0 0

∞ : L∞ (X/k) → Ln (X/k) is given by forgetting The truncation morphism θn,X the coordinates xi and yi , for i > n.

(3.3.6). — The key to the representability property of the arc functor is the following theorem of Bhatt (2016). Theorem (Bhatt). — Let A be a ring and let I be an ideal such that A is I-adically complete. For every scheme X, the canonical map from X(A) to lim X(A/I n ) is bijective. ←− Corollary 3.3.7. — Let S be a scheme and let X be an S-scheme. a) The canonical morphism L∞ (X/S) → lim Ln (X/S) ←− is an isomorphism of functors. b) The functor L∞ (X/S) of arcs on X is representable by an S-scheme. ∞ are affine. c) If S is affine, then the morphisms θn,X d) If X is quasi-compact (resp. quasi-separated over S, resp. separated over S), then so is L∞ (X/S).

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Proof. — a) This is a direct consequence of Bhatt’s theorem, applied to power series rings A[[t]] and the ideal I = (t). b) We may assume that S is affine. Recall then from corollary 2.2.4 that m are affine, for every pair (m, n) of integers the truncation morphisms θn,X such that m  n. By (ÉGA IV3 , proposition 8.2.3), the limit limn Ln (X/S) ←− of the projective system (Ln (X/S))n∈N of S-schemes is representable by an S-scheme. By a), this shows that the functor L∞ (X/S) is representable. c) It follows from the explicit construction of the scheme representing lim Ln (X/S) that the canonical morphisms θn,X : L∞ (X/S) → Ln (X/S) ←− are affine, for all integers n  0. Finally, assertion d) follows from c).

(3.3.8). — The proof of theorem 3.3.6 in Bhatt (2016) relies on techniques of derived algebraic geometry due to Lurie, which go far beyond the scope of this book. Consequently, we shall only give some brief indications on this proof. Moreover, the important consequence for motivic integration does not really lie in the representability of the functor L∞ (X/S) but rather in that of the projective limit lim Ln (X/S). We have seen in the proof of corollary 3.3.7 ←− how that representability follows from the fact that the truncation morphisms m are affine. θn,X We first detail a few important particular cases: a) Let us first assume that X is affine, isomorphic to the spectrum of a ring R. Then one has natural identifications X(A/I n )  Hom(R, A/I n ), for every integer n ∈ N, as well as X(A)  Hom(R, A). By assumption, the canonical map A → lim A/I n is an isomorphism of rings. Consequently, the ←− map X(A) → lim X(A/I n ) is bijective. ←− b) Let us now assume that X = Pm R is a projective space over a ring R, and let π : Pm R → Spec(R) be the projection. Let (ϕn ) ∈ lim X(A/I n ) be a compatible family of morphisms of schemes. ←− By the affine case, applied to Spec(R), there exists a unique morphism ψ : Spec(A) → Spec(R) such that ψ ≡ π ◦ ϕn modulo I n , for every n ∈ N. For every integer n ∈ N∗ , the morphism ϕn then corresponds to an A/I n module Pn , locally free of rank 1, together with a system (un,0 , . . . , un,m ) of m + 1 elements which generate it. Let un : (A/I n )m+1 → Pn be the corresponding surjection. By assumption, the modules Pn form a projective system, and the morphisms un are compatible. Fix a section v1 : P1 → (A/I)m+1 of u1 , so that v1 ◦ u1 is a projector of (A/I)m+1 . For every integer n  2, one may choose a section un of vn which is compatible with vn−1 modulo I n−1 . This gives a compatible family (vn ◦ un ) of projectors, which converges to an element w ∈ Mm+1 (A) such that w2 = w. The image of this projector w is a projective submodule of Am+1 whose reduction modulo I n identifies with Pn , for every integer n  1, hence a morphism ϕ : Spec(A) → Pm R, such that ϕ ≡ ϕn modulo I n for every integer n  1, and it is the unique morphism satisfying that property.

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c) If the theorem holds for a scheme X, let us show that it holds for every closed subscheme Y of X and for every open subscheme U of X. Injectivity is obvious. In the case of a closed subscheme Y , surjectivity follows from the remark that a morphism ϕ : Spec(A) → X factors through Y if and only  if it factors through Y modulo I n for every n (because n I n = 0); in the case of an open subscheme U , we note that an element a ∈ A is invertible if and only if it is invertible modulo I, so that a morphism ϕ as above factors through U if and only if it factors through U modulo I. Together with b), this implies that theorem 3.3.6 holds for schemes which are quasi-projective over an affine scheme. d) Let us now return to the general case of theorem 3.3.6, and let us prove that the map λ : X(A) → lim X(A/I n ) is injective. As explained in Bhatt ←− (2016, 4.6), a trick due to Gabber allows us to assume that X is quasicompact and quasi-separated. Let f, g : Spec(A) → X be two morphisms of schemes such that f ≡ g (mod I n ) for every n ∈ N. Let Z be the fiber product Spec(A) ×X×X X with respect to the morphism (f, g) : Spec(A) → X × X and the diagonal immersion δ : X → X × X. The first projection p1 : Z → Spec(A) is then an immersion; in particular, the scheme Z is quasi-affine. By assumption, for every n ∈ N, the pair (f, g) factors through the diagonal modulo I n . By the quasi-affine case, we obtain an element in lim Z(A/I n )  Z(A), hence a ←− morphism w from Spec(A) to Z such that the composition p1 ◦ w : Spec(A) → Z → Spec(A) is the identity morphism modulo I n , for every n ∈ N. Necessarily, the ring morphism (p1 ◦ w)∗ : A → A is equal to IdA , so that p1 ◦ w = IdSpec(A) . Then p1 is an epimorphism, hence an isomorphism, so that f = g. e) The proof that the map λ is surjective is more involved, and we only sketch the arguments, referring to Bhatt (2016) for details. We start from a compatible family (fn ), where fn ∈ X(A/I n ), and we need to prove that there exists f ∈ X(A) such that f ≡ fn (mod I n ). Recall that Dperf (X) denotes the full subcategory of the quasi-coherent derived category of X consisting of perfect complexes. For every n, the morphism fn induces a functor fn∗ : Dperf (X) → Dperf (Spec(A/I n )). Moreover, the functors (fn∗ ) are compatible, and passing to the limit, they furnish a functor ϕ from Dperf (X) to the category lim Dperf (Spec(A/I n )). One proves (Bhatt 2016, Lemma 4.2) ←− that the latter category identifies with Dperf (Spec(A)). One can then check that H 0 (ϕ) is a cocontinuous symmetric monoidal functor from the category of quasi-coherent sheaves on X to the category of quasi-coherent sheaves on Spec(A), i.e., to the category of A-modules. By a theorem of Brandenburg and Chirvasitu (2014) (which extends an earlier theorem of Lurie), this functor ϕ corresponds to a unique morphism u : Spec(A) → X which then induces the given family (fn ) of morphisms. Remark 3.3.9. — Arc schemes have natural descriptions in terms of formal schemes. Let S be a scheme. For every scheme X, let us denote by X the

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t-adic completion of X ⊗Z Z[t]. In particular, if X is affine, say X = Spec(A), then X = Spf(A[[t]]). If X is an S-scheme, then X is an S-formal scheme, where S is the t-adic completion of S ⊗Z Z[t]. Let X, Y be S-schemes. For every S-scheme Y , one then has canonical bijections L∞ (X/S)(Y )  lim HomS (Y ⊗Z Z[t]/(tn+1 ), X) ←− n

 lim HomS⊗Z Z[t]/(tn+1 ) (Y ⊗Z Z[t]/(tn+1 ), X ⊗Z Z[t]/(tn+1 )) ← − n  HomS (Y, X), where we have indicated by HomS and HomS the sets of morphisms in the category of S-schemes and of formal S-schemes, respectively. When A is a noetherian S-ring and X is a separated S-variety, it follows from Grothendieck existence theorem (ÉGA III2 , corollaire 5.1.8) that the canonical map from HomS (Spec(A[[t]]), X) to HomS (Spf(A[[t]]), X) is a bijection. In this case, one recovers the bijective character of the canonical map L∞ (X/S)(A) → limn Ln (X/S)(A), without recourse to Bhatt’s theorem. ←− Remark 3.3.10. — Let S be a scheme, and let X be an S-scheme which is (locally) of finite type. The description of L∞ (X/S) as a projective limit of S-schemes which are (locally) of finite type, with affine transition morphisms, shows that L∞ (X/S) is (locally) countably generated over S. It is not of finite type in general (see example 3.3.4, as well as corollary 3.7.11). Remark 3.3.11. — Let k be a field and let X be a k-variety. Let ξ be a closed point of L∞ (X/k) and let K be its residue field. Observe that L∞ (X/k) is locally of countable type, so that the field extension K/k is a countably generated algebra, hence [K : k]  ℵ0 . When moreover k is uncountable, this implies that K is algebraic over k. Let indeed a ∈ K. If a were transcendental over k, then the k-algebra K would contain the subfield k(a), which is isomorphic to the k-algebra k(T ) of rational functions. By decomposition in simple terms, this k-algebra has dimension  Card(k) over k, hence dimk (K)  Card(k), hence Card(k)  ℵ0 . In particular, if k is algebraically closed and uncountable (e.g., if k = C), then every closed point of L∞ (X) is a k-point (Ishii 2004, proposition 2.10). However, if k is countable, every countably generated field extension of k is countably generated as a k-vector space, hence appears as the residue field of L∞ (A1 ) = Spec(k[(Tn )n∈N ]) at some closed point (Watanabe, Yoshida); see also Ishii (2004, proposition 2.11). 3.4. Base Point and Generic Point of an Arc (3.4.1). — Let X be a topological space and let x, y be points of X. Recall that one says that x is a specialization of y and that y is a generization of x, if x belongs to the closure of {y} in X. In particular, a closed subset is stable

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under specialization, and an open subset is stable under generization. It also follows from this definition that a subset U of X is dense if every point of X has a generization in U . Let f : X → Y be a continuous map of topological spaces; let x, y ∈ X. If x is a specialization of y, then f (x) is a specialization of f (y). Indeed, the inverse image of {f (y)} is closed in X and contains y, hence contains x; consequently, f (x) belongs to {f (y)}. (3.4.2). — Let S be a scheme and let X be an S-scheme. A point ξ of L∞ (X/S) is called an arc of X. Let K be an extension of the residue field of ξ. As for any scheme, the point ξ gives rise to a K-point of L∞ (X/S), i.e., a K-arc ϕξ : Spec(K[[t]]) → X on X. ∞ (ξ), The image of the closed point of Spec(K[[t]]) under ϕξ is equal to θ0,X hence does not depend on the choice of the extension K; it is called the base point of the arc ξ and is naturally denoted by ϕξ (0). The image ηξ of the generic point of Spec(K[[t]]) under ϕξ does not depend on the choice of K either and is called the generic point of the arc ξ. By lemma 3.4.3 below, the closure of ηξ in X is the smallest closed subscheme Y of X such that ξ ∈ L∞ (Y /S). Lemma 3.4.3. — Let S be a scheme and let X be an S-scheme. Let ξ ∈ L∞ (X/S) be an arc on X; let b be its base point and x be its generic point. a) One has b ∈ {x}: the base point of an arc is a specialization of its generic point. b) Let Z be a closed subscheme of X. The point ξ belongs to L∞ (Z/S) if and only if η belongs to Z. c) Let U be an open subscheme of X. The point ξ belongs to L∞ (U ) if and only if b belongs to U . In fact, assertion c) is equivalent to the property that L∞ (U/S) identifies ∞ −1 ) (U ) of L∞ (X/S). with the open subscheme (θ0,X Proof. — a) Let F be the residue field of ξ and ϕ : Spec(F [[t]]) → X be the corresponding morphism. In Spec(F [[t]]), the closed point is a specialization of its generic point, and these points map to the base point b and to the generic point x of ξ, respectively. Assertion a) thus follows from the continuity of ϕξ . b) If ξ belongs to L∞ (Z), then b and η belong to Z. Conversely, assume that η ∈ Z. Since Z is closed, this implies that b ∈ Z as well. Since the ring F [[t]] is an integral domain, the morphism ϕξ factors through Z; hence ξ ∈ L∞ (Z). c) If ξ belongs to L∞ (U ), then b and η belong to U . Conversely, assume that b ∈ U . Since U is open, this implies that η ∈ U as well. Since U is open, the morphism ϕξ factors through U and ξ ∈ L∞ (U ). Lemma 3.4.4. — Let S be a scheme and let X be an S-scheme. The closed immersion Xred → X induces a homeomorphism L∞ (Xred /S) → L∞ (X/S).

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Proof. — The functor L∞ (·/S) respects closed immersions, so that the morphism L∞ (Xred /S) → L∞ (X/S) is a closed immersion. In particular, it is injective and closed. To prove that it induces a homeomorphism of the underlying topological spaces, it thus suffices to prove that it is surjective. Let ξ ∈ L∞ (X/S), let F be its residue field, and let ϕξ : Spec(F [[t]]) → X be the corresponding F -arc on X. Since F [[t]] is an integral domain, the morphism ϕξ factors through Xred . In particular, ξ belongs to L∞ (Xred ), as was to be shown. Example 3.4.5. — Let k be a field. Let k alg be an algebraic closure of k. Let X be an integral k-curve and let x be a closed point of X. Let us denote by R the local ring OX,x of X at x. Let m be the maximal ideal of  be the m-adic completion of R. Let us assume that m is generated R. Let R by t1 , . . . , tn . It is easy to see that the datum of the ti induces a surjective continuous ring morphism of local k-algebras (3.4.5.1)

 k[[T1 , . . . , Tn ]]→ R

with kernel I. A geometrical parameterization of R at m with respect to the presentation (t1 , . . . , tn ) of m is the datum of a continuous morphism of local k-algebras γ : k[[T1 , . . . , Tn ]]→ k alg [[T ]]. such that I ⊂ Ker(γ). By proposition 3.2.3, we easily observe that γ ∈ L∞ (X/k)(k alg ). Let PR be the set of geometrical parameterizations of R (at m with respect to (t1 , . . . , tn )). We say that a geometrical parameterization γ  ∈ PR is induced by γ ∈ PR if there exists a power series σ ∈ T K[[T ]] such that γ  = γ(σ(T )). Two geometrical parameterizations γ1 andγ2 of R are said equivalent if γ1 is induced by γ2 and γ2 by γ1 . A geometrical parameterization is primitive if there exist no power series σ ∈ T 2 K[[T ]] and no geometrical parameterization γ˜ ∈ PR such that γ = γ˜ (σ(T )). Let us denote by R the integral closure of R in its fraction field. Let us remark that the ring R ⊗k k alg is a semi-local ring. A geometrical branch of R is a maximal ideal of the ring R ⊗k k alg , or equivalently a maximal ideal  ⊗k k alg . For every geometrical branch m , we deduce from the m -adic of R  ⊗k k alg a unique continuous morphism of k-algebras completion of R γm : k[[T1 , . . . , Tn ]]→ k alg [[T ]] which extends the morphism p defined by formula (3.4.5.1). It is not hard to verify, thanks to the definition of the integral closure, that γm ∈ PR is primitive and that the map m → [γm ] defines a bijection from the set of geometrical branches to that of classes of equivalent primitive parameterizations.

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3.5. Constant Arcs (3.5.1). — For every ring A, the natural morphism of rings A[[t]]→ A admits a section, which maps an element a ∈ A to the “constant power series” a + 0 · t + . . . . This induces a morphism of S-schemes s∞,X : X → L∞ (X/S), which, for every S-algebra A, maps an A-point f ∈ X(A) to a “constant Aarc” s∞,X (f ). This morphism also corresponds to the family (sn,X ), where sn,X : X → Ln (X) is the constant section of Ln (X). ∞ ◦ s∞,X = sn,X ; in particular, identifying L0 (X/S) with X, One has θn,X ∞ one has θ0,X ◦ s∞,X = IdX . The arcs contained in s∞,X (X) are called the constant arcs of X. Otherwise, we say that the considered arc is nonconstant. Proposition 3.5.2. — The morphism s∞,X is a closed immersion. ∞ Proof. — We may assume that S is affine. The morphism θ0,X is affine, hence separated, and s∞,X is a section of that morphism. Consequently, the claim follows from (ÉGA I, 5.4.6).

Corollary 3.5.3. — Let k be a field. Let K be a field extension of k. If K is algebraic over k, then the closed immersion s∞ : Spec(K) → L∞ (Spec(K/k)) is bijective. Consequently, the canonical morphism ∞ (θ0,Spec(K) )red : L (Spec(K/k))red → Spec(K/k)

is an isomorphism. Proof. — Let us assume that K is algebraic over k. Let L be a field extension of k, and let ϕ : K → L[[t]] be a morphism of k-algebras. It follows from lemma 3.5.4 below that the image of ϕ is contained in L. Consequently, the ∞ induces a bijection from L∞ (X) to the image of the morphism θ0,Spec(K) closed immersion s∞,X . This implies the claim. Lemma 3.5.4. — Let k be a field, let K be an algebraic extension of k, and let A be a k-algebra. Then every morphism of k-algebras, ϕ : K → A[[t]], factors through A. Proof. — The map ϕ0 : K → given by u → ϕ(u)(0) is a morphism of rings. A ∞ Let u ∈ K; write ϕ(u) = n=0 an tn , and let us prove that ϕ(u) is constant, that is, ϕ(u) = ϕ0 (u) = a0 . First assume that u is separable over k. Let P ∈ k[T ] be the minimal polynomial of u, so that P  (u) is invertible in K. Let us notice that P (a0 ) = P (ϕ0 (u)) = ϕ0 (P (u)) = 0 and P  (a0 ) = ϕ0 (P  (u)) is invertible in A as well. Let us prove by induction that an = 0 for every integer n  1; indeed, if a1 = · · · = an−1 = 0, then ϕ(u) = a0 + an tn (mod tn+1 ); hence the Taylor formula (at order 1) shows that 0 = P (ϕ(u)) = P (a0 ) + P (a0 )an tn (mod tn+1 ); this implies that an = 0. Consequently, an = 0 for every n  1 and ϕ(u) = ϕ0 (u) is constant.

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Otherwise, let p be the characteristic of the field k and let q be the smallest q is separable. By the first case, ϕ(uq ) is constant. power of p such that u q q Since ϕ(u ) = ϕ(u) = n0 aqn tnq , this implies that an = 0 for every n  1, as was to be shown. Remark 3.5.5. — Let k be an imperfect field of characteristic p > 0, let a be in k k p , and let K = k[T ]/(T p − a). Then the space of arcs of Spec(K) is given by L∞ (Spec(K)/k) = Spec(k[(Ti )i∈N ]/(T0p − a, (Tip )i1 )). In particular, it is not reduced, so that L∞ (Spec(K)/k) is not isomorphic to Spec(K). Lemma 3.5.6. — Let S be a scheme and let X be an S-scheme. If the S-scheme L∞ (X/S) is reduced, then the S-scheme X is reduced. ∞ Proof. — Since L∞ (X/S) is reduced, the morphism θ0,X : L∞ (X/S) → X factors through Xred . The factorization s∞,X

∞ θ0,X

X −−−→ L∞ (X/S) −−−→ X, of idX then implies that idX factors through Xred as well. Consequently, X = Xred and X is reduced. Remark 3.5.7. — It is an interesting question to find geometric conditions on X equivalent to the reducedness of the S-scheme L∞ (X/S). Sebag (2011) proves that if X is an integral plane curve over a field k of characteristic 0, then L∞ (X/k) is reduced if and only if X is smooth. We also refer to (Kpognon and Sebag 2017) for related algorithmic questions. More generally, Sebag (2017) shows that if X is an integral variety of arbitrary dimension L∞ (X/k) is reduced, then the OX -module Ω1X is torsion-free. Conversely, it follows from Mustaţă (2001, Propositions 1.7, 4.12) that if X is an integral local complete intersection k-variety with rational singularities, then L∞ (X/k) is reduced. 3.6. Renormalization of Arcs (3.6.1). — Let S be a scheme and let us denote by S be the formal scheme  Z Z[[t]]. Let ϕ : S → S be an adic morphism of formal schemes, ı.e., it S⊗ is given by a compatible family (ϕn ) of morphisms ϕn : S ⊗Z Z[t]/(tn+1 ) → S ⊗Z Z[t]/(tn+1 ). The important case is the one where S is an affine scheme, say S = Spec(A). Then S = Spf(A[[t]]), and the morphism ϕ is determined by the  power series ϕ∗ (t). Let us pose f = ϕ∗ (t) = n0 an tn ∈ A[[t]]. Then, a0 , the constant term of f , is nilpotent. Conversely, for any power series f ∈ A[[t]] whose constant term is nilpotent, there exists a unique adic endomorphism ϕ of S such that ϕ∗ (t) = f . Moreover, the morphism ϕ of formal schemes extends naturally to a morphism of schemes ϕ˜ : Spec(A[[t]]) → Spec(A[[t]]).

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In any case, for every S-ring A, the morphism ϕ induces, by base change to A, a continuous endomorphism ϕA of Spec(A[[t]]). (3.6.2). — Let X be an S-variety. For every S-algebra A and every A-valued arc γ ∈ L∞ (X)(A), viewed as a morphism γ : Spec(A[[t]]) → X, we consider the A-valued arc γ ◦ ϕA : Spec(A[[t]]) → X. It is also denoted γ(fA∗ (t)). Proposition 3.6.3. — Let S be a scheme and let ϕ be an adic endomorphism of the formal scheme S; let X be an S-variety. There exists a unique morphism of S-schemes ϕ∗X : L∞ (X/S) → L∞ (X/S) such that, for every S-algebra A and every A-valued arc γ ∈ X(A[[t]]), one has ϕ∗X (γ) = γ ◦ ϕA . Proof. — Obviously, the maps X(A[[t]]) → X(A[[t]]) given by γ → γ ◦ ϕA are functorial in A. Consequently, they induce a morphism of schemes ϕ∗X as stated. Remark 3.6.4. — a) The construction shows that one has (IdS )∗X = IdL∞ (X/S) . Similarly, if ψ is another adic endomorphism of S, then one has ∗ (ψ ◦ ϕ)∗X = ϕ∗X ◦ ψX . In particular, if ϕ is an automorphism, then ϕ∗X is an automorphism as well, for every S-scheme X. b) Let f : X → Y be a morphism of S-schemes. The natural diagram L∞ (X/S ) L ∞ (f )

L∞ (Y /S)

ϕ∗X ϕ∗Y

L∞ (X/S) L ∞ (f )

L∞ (Y /S)

is commutative. c) Similarly, the morphisms ϕ∗X are compatible with the formation of fiber products. d) For every open subscheme U of X, L∞ (U/S) is an open subscheme of L∞ (X/S) and (ϕ∗X )−1 (L∞ (U/S)) = L∞ (U/S). If, moreover, S is affine, this implies that the morphism ϕ∗X is affine: If U is affine, then L∞ (U/S) is affine as well, and these affine open subschemes of L∞ (X/S) cover L∞ (X/S). Corollary 3.6.5. — Let S be a scheme and let X be an S-scheme. There exists a unique action of the multiplicative monoid scheme A1S on L∞ (X/S), A1S ×S L∞ (X/S) → L∞ (X/S) such that, for every S-algebra A, every element a ∈ A, and every A-valued arc γ ∈ X(A[[t]]), the image of (a, γ) is the arc γ(at). Proposition 3.6.6. — With the notation of proposition 3.6.3, let us assume that S = Spec(A) is affine and that in the power series ϕ∗ (t) ∈ A[[t]], the nonzero coefficient of smallest degree is invertible in A. Then for every Svariety X, the morphism ϕ∗ : L∞ (X/S) → L∞ (X/S) is a closed immersion.

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Proof. — We may assume that X is a closed subscheme of the affine space AdA . Since L∞ (X/S) is a closed subscheme of L∞ (AdA ), we may even assume that X = AdA . Since ϕ∗ is compatible with fiber products, we then reduce to the case  d = 1,n so that L∞ (X/R) = Spec(A[(xn )n∈N ]). Let us write ϕ∗ (t) = n>0 cn t ∈ A[[t]] as the power series inducing ϕ, and let m be For the smallest integer such thatncm = 0; by assumption, cmis invertible. Fn (x)tn , where, every power series γ = xn t , one can write γ(f (t)) = for every integer n, Fn is a linear form in x0 , . . . , xn . By construction, ϕ∗ is associated with the morphism of rings F˜ : R[(xn )] → R[(xn )] which maps xn to Fn (x), for every integer n. One has F0 = x0 , F1 = · · · = Fm−1 = 0, and Fm = x1 cm . More generally, for every integer n  1, the polynomial Fmn − cnm xn is a linear form in x1 , . . . , xn−1 . Since cm is invertible in A, the morphism of rings F˜ is surjective. Consequently, ϕ∗ is a closed immersion, as was to be shown. 3.7. Differential Properties of Jets and Arc Schemes Proposition 3.7.1. — Let S be a scheme, let X, Y be S-schemes, and let f : Y → X be a morphism of S-schemes. Assume that f is formally étale (resp. formally unramified, resp. formally smooth). a) Let m and n be integers such that m  n. The canonical morphism Lm (Y /S) → Lm (X/S) ×Ln (X/S) Ln (Y /S), m , is an isomorphism (resp. is a induced by the morphisms Lm (f ) and θn,Y monomorphism, resp. is surjective). b) Let n be an integer. The canonical morphism

L∞ (Y /S) → L∞ (X/S) ×Ln (X/S) Ln (Y /S), ∞ , is an isomorphism (resp. is a induced by the morphisms L∞ (f ) and θn,Y monomorphism, resp. is surjective).

Proof. — We do all proofs at the same time; in case b), set m = ∞, read A[[t]]/(tm+1 ) as A[[t]] and “A-jet of level m” as “A-arc.” Denote the indicated morphism by λ. Let A be an S-ring, let ϕ ∈ Lm (X)(A), and let ψ  ∈ Ln (Y )(A) be such m (ϕ) = f∗ (ψ  ); the preimage Λ of (ϕ, ψ  ) by λ is the set of elements that θn,X m ψ ∈ Lm (Y )(A) such that f∗ (ψ) = ϕ and θn,Y (ψ) = ψ  . Let us prove that Card(Λ)  1 if f is formally unramified, Card(Λ) = 1 if f is formally étale, and Card(Λ)  1 if f is formally smooth. The jet of level n, ψ  , is a morphism from Spec(A[t]/(tn+1 )) to Y , and the jet of level m, ϕ, is a morphism from Spec(A[[t]]/(tm+1 )) to X which coincides m (ϕ) = f ◦ ψ  . with the jet f ◦ ψ  at level n; we thus have θn,X

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Let us first assume that f is formally unramified. Observe that for every integer p such that n < p  m, the ideal (tn+1 ) of A[t]/(tp+1 ) is nilpotent. By the definition of a formally unramified morphism, there exists at most one p m (ϕ) and such that θn,Y (ψ) = morphism ψ : A[t]/(tp+1 ) → Y which lifts θp,X  ψ . When m is finite, one may take p = m and one concludes that Card(Λ)  m (ψ), 1. When m = ∞, this shows that for every p, the set of truncations θp,Y for ψ ∈ Λ, has at most one element. Necessarily, Card(Λ)  1 in this case too. Let us now assume that f is formally smooth. Set ψn = ψ  . One then sees by induction that for every integer p such that n < p  m, there exists, for every integer p such that n < p  m, an A-arc of level p ψp : Spec(A[t]/(tp+1 )) → Y such that f∗ (ψp ) coincides with ϕ modulo tp+1 , p (ψp ) = ψp−1 . If m is finite, then set ψ = ψm . If m is infinite, then and θn,Y the family (ψp ) defines an A-point of limp Lp (Y /S), hence, by corollary 3.3.7, ←− an A-arc ψ of Y . In both cases, ψ is an A-jet of level m such that f∗ (ψ) = ϕ m (ψ) = ψ  , implying that Λ is nonempty, as desired. and θn,Y Finally, one concludes that Λ has exactly one element if f is formally étale. From what precedes, we obtain that the indicated morphism λ is a monomorphism when f is formally unramified, and an isomorphism if f is formally étale. When f is formally smooth, we take for A a field, and we obtain that λ is surjective. Remark 3.7.2. — Under the assumption that f is formally smooth, the proof of proposition 3.7.1 established the indicated morphisms have a section after base change to any affine scheme, a much stronger property than mere surjectivity. Corollary 3.7.3. — Let S be a scheme, let X, Y be S-schemes, and let f : Y → X be a morphism of S-schemes. Assume that f is étale (resp. formally étale). a) For every m ∈ N, the morphism Lm (f ) is étale (resp. formally étale). b) The morphism L∞ (f ) is formally étale. Proof. — Let us apply proposition 3.7.1 with n = 0. We see that for every m ∈ N ∪ {∞}, the morphism Lm (f ) : Lm (Y ) → Lm (X) is deduced from the m morphism f by base change by the morphism θ0,Y . By (ÉGA IV4 , proposition 17.1.3), this implies that Lm (f ) is formally étale; if f is étale, it follows similarly from (ÉGA IV4 , proposition 17.3.3) that L∞ (f ) is étale. Proposition 3.7.4. — Let S be a scheme and let f : Y → X be a morphism of S-schemes. Assume that f is smooth (resp. formally smooth). a) For every integer m ∈ N, the morphism Lm (f ) is smooth (resp. formally smooth). b) The morphism L∞ (f ) is formally smooth.

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Proof. — We prove at the same time that Lm (f ) and L∞ (f ) are formally smooth. In the latter case, set m = ∞ and, for every ring A, read “jet of level m” as “arc” and A[[t]]/(tm+1 ) as A[[t]]. Let A be an S-ring and let J be a nilpotent ideal of A, and let B = A/J. Let ϕ be an A-valued jet of level m on X, and let ψ be an A-valued jet of level m on Y , such that f∗ (ψ) coincides with ϕ modulo J. We thus have S-morphisms ϕ : Spec(A[[t]]/(tm+1 ) → X and ψ : Spec(B[[t]]/(tm+1 )) → Y such that f ◦ ψ = ϕ ◦ j, where j : Spec(B[[t]]/(tm+1 )) → Spec(A[[t]]/(tm+1 )) is the canonical closed immersion. The ideal of j ∗ is generated by J, hence is nilpotent. Since f is formally smooth, there exists an S-morphism ψ  : Spec(A[[t]]/(tm+1 )) → Y which coincides with ψ modulo J and such that f ◦ ψ  = ψ. This proves that the morphism Lm (f ) is formally smooth. Let us assume that f is smooth and that m < ∞. By definition, the morphism f is locally of finite presentation and formally smooth. By what precedes, Lm (f ) is formally smooth. It is also locally of finite presentation (proposition 2.1.3), so that the morphism Lm (f ) is smooth. Proposition 3.7.5. — Let S be a scheme, and let X be a smooth S-scheme, of pure relative dimension d. m a) For every pair (m, n) of integers such that m  n, the morphism θn,X d(m−n)

is a locally trivial fibration for the Zariski topology, with fiber AS . ∞ is a locally trivial fibration for b) For every integer n, the morphism θn,X the Zariski topology, with fiber Ad∞ S . In particular, these morphisms are surjective and open. In this statement, we denote by Ad∞ the dth power of the infinite dimenS sional affine space over S, A∞ S = Spec(OS ([T0 , T2 , . . . ])). Proof. — We treat both cases at the same time. In case b), we set m = ∞ as well as d(m − n) = d∞ for every integer n ∈ N. Let f : X → S be the structural morphism of X. For every point x of X, there exist an open neighborhood U of x in X and an étale morphism p : U → AdS of S-schemes. Then Lm (U/S) is an open subscheme m reof Lm (X/S), Ln (U/S) is an open subscheme of Ln (X/S), and θn,X m stricts to the truncation morphism θn,U/S : Lm (U/S) → Ln (U/S). This m morphism θn,U factors as a composition q

λ

Lm (U/S) − → Lm (AdS /S) ×Ln (Ad /S) Ln (U/S) − → Ln (U/S) S

where, according to proposition 3.7.1, λ is an isomorphism and q is the second projection. As computed in examples 2.3.4 and 3.3.4, for every k ∈ N ∪ {∞}, the scheme Lk (AdS /S) of jets of level k of the affine space can be identified with Adk S , so that the truncation morphismes are projections. In particular, d(m−n) m dm the morphism θn,A  AS ×S d is isomorphic to the projection AS S

d(m−n)

dn m Adn . By what precedes, θn,U is induced from S → AS with fiber AS this trivial fibration by base change under the morphism Ln (p). This shows

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m that, over the open subscheme Ln (U/S) of Ln (X/S), the morphism θn,X is d(m−n)

a trivial fibration with fiber AS . By proposition 2.1.3, c), those open subschemes Ln (U/S) form an open covering of Ln (X/S). This concludes the proof of the proposition. Corollary 3.7.6. — Let k be a field, let X be a smooth integral k-scheme of dimension d, let Y be an integral subscheme of X, and let e = dim(Y ). n )−1 (Y ) of Ln (X/k) is intea) For every integer n, the k-subscheme (θ0,X gral of dimension nd + e; it is smooth if Y is smooth. ∞ −1 b) The k-subscheme (θ0,X ) (Y ) of L∞ (X/k) is integral; it is formally smooth if Y is smooth. n : Ln (X/k) → X is smooth Proof. — By proposition 3.7.5, the morphism θ0,X (resp. formally smooth in case b)) and surjective, and its fibers, being isomorphic to Adn (resp. Ad∞ ), are irreducible. By base change, the same holds for its restriction to Y . By (ÉGA Isv , proposition 0.2.1.14), the scheme n )−1 (Y ) is thus irreducible. If Y is smooth, then the structural morphism (θ0,X n of (θ0,X )−1 (Y ) is then smooth (resp. formally smooth).

Corollary 3.7.7. — Let k be a field and let X be a smooth integral k-scheme of dimension d. a) For every integer n, the k-scheme Ln (X/k) is integral and smooth, of dimension (n + 1)d. b) The k-scheme L∞ (X/k) is integral and formally smooth. Proof. — This is the particular case Y = X of corollary 3.7.6. Corollary 3.7.8. — Let X be a smooth k-variety. For every point x ∈ X such that dimx (X)  1, there exists a nonconstant arc in L∞ (X)(κ(x)) whose base point is equal to x. Proof. — Let d = dimx (X). The arcs with base point x correspond with the ∞ above x. By proposition 3.7.5, this space is isomorphic to Ad∞ fiber of θ0,X κ(x) , of which the constant arc at x is a κ(x)-rational point. Since d∞ > 0, there exist nonconstant arcs as claimed. Example 3.7.9. — Let X ⊂ A2R = Spec(R[T1 , T2 ]) be the real affine curve with equation T22 = T12 (T1 − 1). Every arc γ ∈ L∞ (X/R)(R) based at the origin (0, 0) is constant. Indeed, such an arc corresponds with a pair  (x,n y) ∈ xn t and R[[t]] such that y 2 = x2 (x − 1) and x(0) = y(0) = 0. Write x = y = yn tn . Assume that the power series y is nonconstant, and let m be its order at 0. Then the order of x at 0 is equal to m as well, and one has 2 2 = −x2m . Since ym = 0, this implies 0 < ym = −x2m  0, a contradiction. ym Proposition 3.7.10. — Let S be a scheme and let X be an S-scheme. The following assertions are equivalent: (i) X is formally unramified over S; ∞ : L∞ (X/S) → X is an isomorphism. (ii) The morphism θ0,X

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n (iii) For every integer n  0, the morphism θ0,X : Ln (X/S) → X is an isomorphism; n : Ln (X/S) → X is an (iv) There exists an integer n  1 such that θ0,X isomorphism; If X is noetherian, these assertions are also equivalent to the following: (v) The arc scheme L∞ (X/S) is noetherian. n Proof. — Let n ∈ N ∪ {∞}. The formula IdX = θ0,X ◦ sn,X implies that n n θ0,X is an epimorphism. Consequently, in order to show that θ0,X is an isomorphism, it suffices to prove that it is a monomorphism. Let us assume that X is formally unramified over S. Applying proposition 3.7.1 to the morphism X → S and using that L∞ (S/S) = S, we ∞ ∞ is a monomorphism. Consequently, θ0,X is an see that the morphism θ0,X isomorphism. This establishes the implication (i)⇒(ii). m is an isomorphism and let Let m ∈ N ∪ {∞}. Let us assume that θ0,X m n m m n ∈ N be such that n < m. The relation θ0,X = θ0,X ◦ θn,X implies that θn,X is a monomorphism. Consequently, since n m m m m m (sn,X ◦ θ0,X ) ◦ θn,X = sn,X ◦ θ0,X = θn,X ◦ sm,X ◦ θ0,X = θn,X , n n one has sn,X ◦ θ0,X = IdLn (X) . In particular, θ0,X is a monomorphism, hence an isomorphism. This proves in particular the implication (ii)⇒(iii), and the implication (iii)⇒(iv) is obvious. 1 is an Let us finally assume (iv). By what precedes, the morphism θ0,X 1 isomorphism. Since L1 (X/S) = Spec(Sym(ΩX/S )) (example 2.3.3), one has

Ω1X/S = 0. By (ÉGA IV4 , proposition 17.2.1), X is then formally unramified over S. Let us now assume that the scheme X is noetherian. The implication (iv)⇒(v) is then clear. Let us establish the implication (v)⇒(iv). The question is local on X, so we may assume that S and X are affine. Let n ∈ N∗ ; associating with an A-arc ϕ on X, the arc ϕ(tn ) defines a morphism (3.7.10.1)

∞ Fn,X : L∞ (X/S) → L∞ (X/S),

which is a closed immersion. By assumption, the scheme L∞ (X/S) is noethe∞ rian. So the decreasing family (Fn!,X (L∞ (X/S)))n1 of closed subschemes is ultimately constant. Their intersection, consisting of constant arcs, is the image of the morphism s∞,X . Thus, there exists an integer N such that for every arc ϕ, the arc ϕ(tN ) on X is constant; hence ϕ is constant. We con∞ are isomorphisms and inverse one clude that the morphisms s∞,X and θ0,X of the other. Corollary 3.7.11. — Let k be a field and let X be a k-variety. Then the following assertions are equivalent: (i) The k-scheme X is étale;

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n (ii) For every n ∈ N ∪ {∞}, the morphism θ0,X : Ln (X/k) → X is an isomorphism of k-schemes. (iii) There exists an element n ∈ N∗ ∪ {∞} such that the morphism n θ0,X : Ln (X/k) → X is an isomorphism of k-schemes; (iv) The k-scheme L∞ (X/k) is of finite type.

§ 4. TOPOLOGICAL PROPERTIES OF ARC SCHEMES In this section, we begin the study of topological properties of arc schemes. 4.1. Connected Components of Arc Schemes Proposition 4.1.1. — Let S be a scheme and let X be a connected Sscheme. Then the arc scheme L∞ (X/S) is connected. Proof. — We consider the reparameterization morphism H : L∞ (X/S) ×S A1S → L∞ (X/S), (γ(t), λ) → γ(λt). The image of H(·, 0) coincides with the image of the canonical section sX,∞ : X → L∞ (X/S), hence is connected. Now let γ be any point in L∞ (X/S) and denote by k its residue field. The image of the morphism H(γ, ·) : A1k → L∞ (X/S) is a connected subset of L∞ (X/S) that contains γ = H(γ, 1) and H(γ, 0), hence intersects the image of sX,∞ . It follows that L∞ (X/S) is connected. Corollary 4.1.2. — Let S be a scheme and let X be an S-scheme. We endow every connected component of X with its reduced subscheme structure. Then the map C → L∞ (C/S) defines a bijection between the set of connected components of X and the set of connected components of the arc scheme. Proof. — Every point of L∞ (X/S) is contained in L∞ (C/S) for a unique connected component C of X, because Spec k[[t]] is reduced and connected for every field k. Moreover, for every connected subspace A of L∞ (X/S), the ∞ (A) is connected and thus contained in a unique connected truncation θ0,X component C of X; it then follows that A is contained in L∞ (C/S). Hence, the subsets L∞ (C/S) are the connected components of L∞ (X/S).

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4.2. Irreducible Components of Arc Schemes Lemma 4.2.1. — Let k be a field. Let X be a k-scheme, let (Xi )i∈I be the family of its irreducible components. Then, we have the following properties: a) For every i, L∞ (Xi ) is a closed subscheme of L∞ (X). b) We have the decomposition L∞ (X) =



L∞ (Xi )

i∈I

and L∞ (Xi ) ⊂ L∞ (Xj ), if i = j. c) Assume moreover that X is smooth over k. L∞ (Xi ) are the irreducible components of L∞ (X).

Then the k-schemes

Proof. — a) Since Xi is a closed subscheme of X, this follows from the fact that the arc scheme functor respects closed immersions. b) Let ξ ∈ L∞ (X), let η be its generic point, and let i ∈ I be such that η ∈ Xi . Since Xi is closed in X, we deduce from lemma 3.4.3 that ξ ∈ L∞ (Xi ). Consequently, L∞ (X) =



L∞ (Xi ).

i∈I

Let i, j ∈ I be such that L∞ (Xi ) ⊂ L∞ (Xj ). Since every point of X is ∞ is surjective. the base point of a constant arc, the truncation morphism θ0,X i ∞ By applying the morphism θ0,X to this inclusion, we obtain that Xi ⊂ Xj ; hence Xi = Xj by definition of an irreducible component. c) Since X is smooth over k, its irreducible components are open; hence Xi is smooth over k, for every i ∈ I. Applying corollary 3.7.7 to Xi , it follows that L∞ (Xi ) is integral, for every i ∈ I. Consequently, the closed subschemes L∞ (Xi ) are the irreducible components of L∞ (X). Lemma 4.2.2 (Ishii and Kollár 2003). — Let k be a field and let X be a k-variety. Every arc ξ in X possesses a generization whose base point is the generic point of ξ. Proof. — Let F be the residue field of ξ, and let ϕ : Spec(F [[t]]) → X be the corresponding morphism. Let us consider the morphism of k-schemes: (4.2.2.1)

θ : Spec(F [[u, v]]) → Spec(F [[t]]),

associated with the morphism of rings from F [[t]] to F [[u, v]] given by f → f (u + v). Then, the morphism, ϕ ◦ θ : Spec(F [[u, v]]) → X,

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is an F [[u, v]]-point of X, equivalently, a F [[u]]-point γ of L∞ (X); we sum up this situation in the diagram Spec(F [[u, v]])

Spec(F [[u]])

θ

Spec(F [[t]]) ϕ

γ

L∞ (X)

X

which makes apparent that γ is an arc on L∞ (X). The base point γ(0) ∈ L∞ (X) of γ is obtained by setting u = 0, hence, is equal to ϕ, by the very definition of θ. On the other hand, the generic point of γ is the arc ψ : Spec(F ((u))) → L∞ (X) on X. By lemma 3.4.3, ψ is a generization of ϕ. This concludes the proof. Corollary 4.2.3. — Let k be a field and let X be a k-variety. Let U be an open subscheme of X and let Z be its complement. Then the k-scheme L∞ (U ) is dense in L∞ (X) L∞ (Z). Proof. — Let C be an irreducible component of L∞ (X) which is not contained in L∞ (Z). Let ξ ∈ C be its generic point, let F be its residue field, and let ϕ : Spec(F [[t]]) → X be the corresponding morphism. Let x ∈ X be the generic point of the arc ξ. Since ξ is the generic point of C and C ⊂ L∞ (Z), one has ξ ∈ L∞ (Z). Consequently, one has x ∈ Z, that is, x ∈ U . By lemma 4.2.2, there exists a generization η of ξ which is an arc whose base point equals x, so that η ∈ L∞ (U ). Consequently, the closure of L∞ (U ) contains the arc ξ; hence it contains C. This concludes the proof. Corollary 4.2.4. — Let k be a field and let X be a k-variety. The set of irreducible components of L∞ (X) is finite. Proof. — We argue by induction on the dimension of X. Let K be the perfect closure of k; one has L∞ (XK ) = L∞ (X)K ; moreover, the canonical map L∞ (X)K → L∞ (X) is a homeomorphism (ÉGA IV2 , proposition 2.4.5); we may thus assume that k is perfect. We may also assume that X is reduced. Then, the smooth locus Xsm is a dense open subscheme of X. We denote by Xsing the complement of Xsm in X, endowed with its induced reduced structure (1) . By corollary 4.2.3, L∞ (Xsm ) is dense in L∞ (X) L∞ (Xsing ). Consequently, for every irreducible component C of L∞ (X), either C is the closure of an irreducible component of L∞ (Xsm ), or C is an irreducible component of L∞ (Xsing ). By induction, the set of irreducible components of L∞ (Xsing ) is finite. Moreover, it follows from lemma 4.2.1 that the set of irreducible components of L∞ (Xsm ) is finite. This concludes the proof of the corollary.

(1) We will later endow X sing with a more natural schematic structure; however, in this chapter, it is sufficient to consider its reduced structure.

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4.3. Kolchin’s Irreducibility Theorem Lemma 4.3.1. — Let k be a field. Let X be an irreducible k-curve. Then the associated arc scheme L∞ (X) is irreducible. Proof. — By lemma 3.4.4, we may assume that curve X is reduced. By (3.1.3) and (ÉGA IV2 , 2.4.5), we also may assume that the field k is perfect up to making a base change to its perfect closure. Let h : X  → X be the normalization of X. This is a proper and birational morphism. Since k is perfect, the curve X  is k-smooth. We claim that the induced morphism L∞ (h) is surjective. Let K be a field extension of k and let x ∈ L∞ (X)(K). If x is constant (see subsection 3.5), it has a preimage in L∞ (X  ), because h is surjective. If x is not constant, the valuative criterion of properness implies that x has a preimage in L∞ (X  ) (see proposition 4.4.2 below). Let us assume that X is irreducible. Since X  is smooth, we deduce from corollary 3.7.7 that the k-scheme L∞ (X  ) is irreducible. Since L∞ (h) is surjective, we conclude that the k-scheme L∞ (X) is irreducible. The rest of this subsection is devoted to generalize this statement in higher dimensions. Proposition 4.3.2 (Ishii and Kollár 2003). — Let k be a field of characteristic 0 and let X be a reduced k-variety. Every point of L∞ (X) whose base point belongs to Xsing has a generization which does not belong to L∞ (Xsing ) and whose base point belongs to Xsing . Proof. — Let ξ be a point of L∞ (X), let F be an extension of its residue field, and let ϕ : Spec(F [[t]]) → X be the corresponding morphism. Let Y be the closure of the generic point of ξ, that is, the smallest closed subscheme of X which contains the image of ϕ. If Y ⊂ Xsing , we may take ψ = ϕ; hence let us assume that Y ⊂ Xsing . By lemma 4.2.2, there exists a generization ψ of ϕ in L∞ (Y ) whose base point is the generic point of ξ. Up to replacing the arc ϕ by ψ, we assume that ϕ(0) is the generic point of Y . We also assume, as we may, that F is algebraically closed. Since char(k) = 0 and X is reduced, the singular locus Xsing of X is nowhere dense in X, hence dim(Y )  dim(X) − 1. If dim(Y ) = dim(X) − 1, we set Z = X. Otherwise, there exists an integral hypersurface Z1 of X which contains Y and such that X is smooth at the generic point of Z1 . We repeat this process until we reach a closed integral subscheme Z of X such that dim(Y ) = dim(Z)−1. We thus have Y ⊂ Z ⊂ X, dim(Z) = dim(Y )+1, and Z ⊂ Xsing . Let p : Z  → Z be the normalization of Z, and let Y  ⊂ Z  be the inverse image of Y , endowed with its reduced scheme structure. The morphism p is finite and surjective. Since F is algebraically closed, there exists a point b ∈ Y  (F ) such that p(b ) = ϕ(0); observe that b is a generic point of Y  .

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Since char(k) = 0, the morphism pY : Y  → Y , deduced from p by restriction, is generically étale. Consequently, it is étale in a neighborhood of b , so that there exists a unique arc ϕ : Spec(F [[t]]) → Y  such that p ◦ ϕ = ϕ and ϕ (0) = b (proposition 3.7.1). By construction, the point b is a codimension 1 point of Z  ; since Z  is normal and char(k) = 0, b is a smooth point of Z  , so that there exists an integral open subscheme U  of Z  which is smooth over k and such that ∞ −1 (U  ∩ Y  ) of L∞ (U  ) ϕ ∈ L∞ (U  ). By corollary 3.7.6, the subspace (θ0,U )  is irreducible. Consequently, the arc ϕ is a specialization of an arc ψ  such that ψ  (η) is the generic point of U  and such that ψ  (0) belongs to Y  . Then, the arc ϕ = p ◦ ϕ is a specialization of the arc ψ = p ◦ ϕ ; the point ψ(η), being equal to the generic point of Z, is a smooth point of X; and the base point ψ(0) belongs to p(Y  ) = Y ⊂ Xsing . Corollary 4.3.3. — Let k be a field of characteristic 0. Let X be a kvariety and let U be a dense open subscheme of X. Then L∞ (U ) is dense in L∞ (X). Proof. — Let Z = X U . Let ϕ ∈ L∞ (X), and let us prove that the arc ϕ has a generization which belongs to L∞ (U ). Applying successively proposition 4.3.2 and lemma 4.2.2, the arc ϕ has a generization ψ whose base point ψ(0) does not belong to Xsing ; hence ψ ∈ L∞ (Xsm ). More precisely, there exists a unique irreducible component V of Xsm which contains ψ(0), and ψ ∈ L∞ (V ). By proposition 3.7.7, L∞ (V ) is irreducible. Since U is dense and V is open in X, the intersection U ∩ V is nonempty so that proposition also implies that L∞ (U ∩ V ) is a nonempty subset of L∞ (V ). Consequently, L∞ (U ∩ V ) is dense in L∞ (V ); hence the arc ϕ has a generization in L∞ (U ), as claimed. Since ϕ is arbitrary, this proves that L∞ (U ) is dense, as claimed. Theorem 4.3.4 (Kolchin). — Let k be a field of characteristic zero and let ∞ (C) X be a k-variety. For every irreducible component C of L∞ (X), Y = θ0,X ∞ −1 is the unique irreducible component of X such that C = (θ0,X ) (Y ). In particular, X is irreducible if and only if L∞ (X) is irreducible. Proof. — By lemma 3.4.4, we may assume that X is reduced. Let us first assume that X is irreducible, and let us prove that L∞ (X) is irreducible. Since char(k) = 0, its smooth locus Xsm is a dense open subscheme of X, hence is irreducible, so that lemma 3.7.7 asserts that L∞ (Xsm ) is irreducible. Moreover, L∞ (Xsm ) is dense in L∞ (X), by corollary 4.3.3, so that L∞ (X) is irreducible. Let us now treat the general case. For every irreducible component Y of X, the closed subset L∞ (Y ) of L∞ (X) is then irreducible, and it follows from lemma 4.2.1 that these subsets are the (pairwise distinct) irreducible compo∞ (L∞ (Y )) = Y . nents of L∞ (X). Considering constant arcs, one also has θ0,X This concludes the proof.

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193

Remark 4.3.5. — This result is classically referred to as the Kolchin irreducibility theorem and is an analogue for arc schemes of a theorem in differential algebra (Kolchin 1973, IV.17, proposition 10). Theorem 4.3.4, as well as Kolchin’s statement, do not hold true in positive characteristic. Let, for example, k be a field of characteristic p > 0. Let us consider the polynomial f = T1p + T2p T3 ∈ k[T1 , T2 , T3 ], and let us denote by X the hypersurface in A3k = Spec(k[T1 , T2 , T3 ]) defined by f . As a polynomial in T3 , the polynomial f has degree 1, and its coefficients are coprime in k[T1 , T2 ]; this implies that f is irreducible. In particular, the k-variety X is irreducible. Its reduced singular locus is the closed subset Xsing = V (T1 , T2 ). One has Xsing  A1k ; in particular, L∞ (Xsing ) is a closed irreducible subset of L∞ (X). Since Xsm is smooth and irreducible, its arc space L∞ (Xsm ) is an irreducible open subset of L∞ (X). Its closure is thus an irreducible closed subset of L∞ (X). Let us prove that these two irreducible sets are the irreducible components of L∞ (X). The arc space L∞ (X) is a closed subscheme of L∞ (A3k ) = Spec(k[(Si,n )]), where the indices (i, n) run over {1, 2, 3} × N. Let U be the trace on L∞ (X) of the open subset D(S3,1 ) of L∞ (A3k ). For every extension K of k, its K3 points are the arcs (x(t), y(t), z(t)) ∈ K[[t]] such that x(t)p + y(t)p z(t) = 0 and the derivative z  (t) of z does not vanish at 0. In particular, U contains the point associated to the arc given by (0, 0, t), so that U = ∅. Let us prove that U ⊂ L∞ (Xsing ). Let indeed K be an extension of k, and let ξ = (x(t), y(t), z(t)) ∈ L∞ (X)(K) be an arc such that z  (0) = 0. Let us prove that x(t) = y(t) = 0. Arguing by contradiction, we assume that y(t) = 0 and consider its order d at 0; define a ∈ K × by y(t) = atd + . . . . Derivating the equation x(t)p + y(t)p z(t) = 0, we obtain y(t)p z  (t) = 0; since z  (0) = 0, this implies y(0) = 0, hence d  1 and x(t)p = −y(t)p z(p) = −ap tpd − ap z  (0)tpd+1 + · · · , a contradiction since all the exponents that appear in the expansion of x(t)p are multiples of p. Consequently, y(t) = 0; hence x(t)p = 0, so that x(t) = 0. In other words, ξ ∈ L∞ (Xsing )(K), as was to be shown. Consequently, L∞ (Xsing ) is an irreducible closed subset of L∞ (X) of nonempty interior, hence is an irreducible component of L∞ (X). By corollary 4.2.3, any other irreducible component is contained in the closure of L∞ (Xsm ). We thus have shown that L∞ (Xsing ) and L∞ (Xsm ) are the two irreducible components of L∞ (X). In particular, L∞ (X) is not irreducible. It directly follows from theorem 4.3.4 and lemma 4.2.1 the following statement:

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Corollary 4.3.6. — Let k be a field of characteristic zero and let X be a k-variety. Let (Xi )i∈I be the family of the irreducible components of X, and then the family (L∞ (Xi ))i∈I is that of the irreducible components of L∞ (X). Remark 4.3.7. — If X is a singular variety over a field of characteristic zero, the jet schemes Ln (X) can be reducible even if X is irreducible. Actually, the irreducibility of the jet schemes Ln (X) is related to the singularities of X. Let k be a field of characteristic zero, and let X be a normal integral separated k-variety which is a locally complete intersection. Then Mustaţă (2001) has shown that X has rational singularities if and only if the k-variety Ln (X) is irreducible for every n  0. We will discuss this statement, together with related results, in theorem 7/2.4.11. 4.4. Application of the Valuative Criterion (4.4.1). — Let f : Y → X be a proper morphism of k-schemes. Let x ∈ L∞ (X) be an arc on X and let F be its residue field. By adjunction, the point x corresponds to a morphism of k-schemes ϕx : Spec(F [[t]]) → X. Let us denote by ηF = Spec(F ((t))) the generic point of Spec(F [[t]]), and let us assume that there exists a morphism of k-schemes Spec(F ((t))) → Y which makes the following diagram of morphisms of k-schemes to commute: (4.4.1.1)

Y

Spec(F ((t))) ηF

Spec(F [[t]])

f ϕx

X.

Then, the valuative criterion of properness asserts that there exists a unique morphism of k-schemes ψx : Spec(F [[t]]) → Y which makes the following diagram of morphisms of k-schemes to commute: (4.4.1.2)

Y

Spec(F ((t))) ηF

Spec(F [[t]])

ψx ϕx

f

X.

Proposition 4.4.2. — Let k be a field. Let f : Y → X be a proper and birational morphism of k-schemes. Let U be the largest open subscheme of X over which f is an isomorphism, let Z = X U , and let E = f −1 (Z). The morphism L∞ (f ) induces a bijection L∞ (Y )

L∞ (E) → L∞ (X)

L∞ (Z).

Proof. — Let x ∈ L∞ (X) L∞ (Z). Let F be the residue field of x, and let ϕx : Spec(F [[t]]) → X be the corresponding morphism. Since x ∈ L∞ (Z), one has ϕx (ηF ) ∈ U . Since f is an isomorphism over U , there exists a unique

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morphism ψx : Spec(F ((t))) → Y such that f ◦ψx over ηF . We thus are in the situation of diagram (4.4.1.1). Since f is proper, the valuative criterion of properness implies the existence of a unique morphism ψx : Spec(F [[t]]) → Y coinciding with ψx on ηF and such that f ◦ ψx = ϕx (diagram (4.4.1.2)). The morphism ψx corresponds to an F -valued arc y ∈ L∞ (Y ) such that f (y) = x. It is not contained in L∞ (E). Moreover, it is the unique arc satisfying this property. That concludes the proof. 4.5. Irreducible Components of Constructible Subsets in Arc Spaces (4.5.1). — We have recalled in §2/1.3 the general definition of a constructible subset of a scheme. In the case of arc schemes, they have a simple presentation. In fact, the following lemma shows that constructible subsets of arc spaces coincide with the so-called cylinders in motivic integration! Lemma 4.5.2. — Let k be a field and let X be a k-variety. a) For every integer n ∈ N and every constructible subset D of Ln (X), ∞ )−1 (D) of L∞ (X) is constructible; it is closed (resp. open) the subset (θn,X if D is closed (resp. open). b) For every constructible subset C of L∞ (X), there exist an integer n ∈ ∞ )−1 (Cn ). N and a constructible subset Cn of Ln (X) such that C = (θn,X Moreover, if C is closed (resp. open), then Cn can be taken to be closed (resp. open). Proof. — By assumption, X is noetherian, hence quasi-compact and quasiseparated. Moreover (corollary 3.3.7), one has L∞ (X) = limn Ln (X), and ←− n+1 are affine. The lemma is thus a particular the truncation morphisms θn,X case of (ÉGA IV3 , théorème 8.3.11). Proposition 4.5.3. — Let X be a k-variety. Let us assume that resolution of singularities holds for k-varieties of dimension  dim(X). Then the set of irreducible components of any constructible subset of L∞ (X) is finite. Proof. — We may assume that X is reduced and connected. Let d = dim(X); we prove the proposition by induction on d. Let C be a constructible subset of L∞ (X). Let n ∈ N, and let S be a con∞ )−1 (S) (lemma 4.5.2). Since structible subset of Ln (X) such that C = (θn,X Ln (X) is a noetherian scheme, S can be written as a finite disjoint union of integral subschemes Si of Ln (X), and C is the union of the constructible ∞ sets (θn,X )−1 (Si ). Consequently, we may assume that S is itself an integral subscheme of Ln (X). If d = 0, then X is a point; precisely, there exists a finite extension K ∞ is of k such that X = Spec(K). By corollary 3.5.3, the morphism θ0,X then a homeomorphism. Consequently, C is a point; hence the assertion of proposition 4.5.3 is clear in this case.

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Assume that d  1. Let f : Y → X be a resolution of singularities of X. Then we have the following commutative diagram of morphisms of k-schemes: L∞ (Y )

L ∞ (f )

∞ θ n,Y

L∞ (X) ∞ θ n,X

Ln (Y )

L n (f )

θ 0n,Y

Ln (X) θ 0n,X

Y

f

X.

Let D = L∞ (f )−1 (C) and T = Ln (f )−1 (S); endow T with its unique reduced structure of Ln (Y ). By the commutativity of the above diagram, we have D = L∞ (f )−1 (C)

∞ −1 ) (S) = L∞ (f )−1 (θn,X

∞ −1 Ln (f )−1 (S) ) = (θn,Y ∞ −1 ) (T ). = (θn,Y

By lemma 4.4.2, we also know that the morphism of k-schemes L∞ (f ) induces a continuous bijection (4.5.3.1)

L∞ (Y )

L∞ (f )−1 (L∞ (Xsing )) → L∞ (X)

L∞ (Xsing ).

By restriction, L∞ (f ) thus induces a continuous bijection D

L∞ (f )−1 (L∞ (Xsing )) → C

L∞ (Xsing ).

Since the k-variety Y is smooth over k, it follows from proposition 3.7.5 that ∞ the morphism of k-scheme θn,Y is surjective and open and that its fibers are integral. Since T is a subscheme of the noetherian scheme Ln (Y ), it has a finite number of irreducible components. We then deduce from (ÉGA Isv , ∞ −1 ) (T ) → T , Chapitre 0, proposition 2.1.14), applied to the morphism (θn,Y ∞ −1 that the k-scheme (θn,Y ) (T ) has a finite number of irreducible components too. Consequently, C L∞ (Xsing ) has a finite number of irreducible components. On the other hand, we observe that ∞ ∞ )−1 (S) ∩ L∞ (Xsing ) = (θn,X )−1 (S ∩ Xsing ). C ∩ L∞ (Xsing ) = (θn,X sing

Since dim(Xsing ) < d, the induction hypothesis shows that C ∩ L∞ (Xsing ) has finitely many irreducible components and this concludes the proof of the proposition.

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§ 5. THE THEOREM OF GRINBERG–KAZHDAN–DRINFELD Let k be a field and let X be a k-variety. The main result of this section is theorem 5.1.1, due to Drinfeld (2002), which gives a “finite dimensional” model of the formal completion of the arc space L (X) at a given arc. It generalizes to an arbitrary field a previous result of Grinberg and Kazhdan (2000), valid over the complex numbers. 5.1. Formal Completion of the Space of Arcs Theorem 5.1.1. — Let k be a field and let X be a k-variety. Let γ ∈ X(k[[t]]) be a k-arc on X which does not factor through the singular locus Xsing of X. Let n be the dimension of X at the generic point of γ. Then there exist  an affine k-scheme of finite type Y and a point y ∈ Y (k) such that L (X)γ n   is isomorphic to Yy ×k D∞ , where D∞ = Spf(k[[(Tm )m∈N ]]). Every formal k-scheme Yy which satisfies the former isomorphism is called  a finite dimensional model of the formal neighborhood L (X)γ . There exist  infinitely many finite dimensional models of L (X) ; remark 5.5.5 clarifies γ

the link between all these models. Remark 5.1.2. — A consequence of theorem 5.1.1 is that the nilradical of the complete local ring O L (X) γ is nilpotent. Based on this remark, Bourqui and Sebag (2017b) have observed that the condition that the arc is nondegenerate cannot be omitted from theorem 5.1.1. Let indeed k be a field of characteristic zero such that −1 is not a square in k, and let X be the plane curve with equation X 2 + Y 2 = 0; let γ be the constant arc at the origin.  In that case, they show that the nilideal of the ring of functions of L (X)γ contains element of arbitrary large nilpotence order. We refer to Bourqui and Sebag (2017b) for details. Remark 5.1.3. — If the variety X is assumed to be reduced, under the assumptions of theorem 5.1.1, Bourqui and Sebag (2017c) prove that the unique formal branch of X containing the arc γ is a formal disk if and only if  the formal neighborhood L (X)γ is a formal disk. This result suggests that the finite dimensional models introduced in the statement of theorem 5.1.1 could be a “geometric measure” for the singularities of X. (5.1.4). — To prove theorem 5.1.1, we first give a description of the for mal completion L (X)γ using A-valued arcs, where A runs over a suitable category of test-rings over k. The rest of this section and the next two ones are thus devoted to technical complements. In particular, we recall the Weierstrass preparation theorem for power series and explain how the Jacobi

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criterion allows to reduce the proof of the theorem to the case where X is a complete intersection. In section 5.4, we will finally prove theorem 5.1.1 and give some examples. (5.1.5). — Let k be a field. A topological k-algebra A is said to be admissible (ÉGA Isv , 0, 7.1.2) if it is separated and complete and if it possesses an ideal of definition, namely, an open ideal I of A whose powers converge to 0. We write Admk for the category of admissible k-algebras. Let LocAlgk be the full subcategory of Admk consisting of completions of local k-algebras with residue field isomorphic to k. An object of LocAlgk  = lim A/mn , where A is a local algebra with is thus a k-algebra of the form A ←−  is a local k-algebra, its maximal maximal ideal m such that k = A/m. Then A   ideal is given by m = Ker(A → A/m), and it is separated and complete for  → A/mn ))n . the topology defined by the family of ideals (Ker(A  may not be complete Note however that, unless A is noetherian, the ring A  is the algebra  for the m-adic topology. This is, for example, the case when A k[[(Tm )m∈N ]] that appears in the statement of theorem 5.1.1. Observe that this algebra is the m-adic completion of the k-algebra A = k[(Tm )]m with  and the m-adic  = m · A,  respect to its maximal ideal m = (Tm ). One has m  topology of A is not complete; see Stacks Project, tag 05JA. We define the category Testk of test-rings over k to be the full subcategory of LocAlgk whose objects are local k-algebras with residue field equal to k and whose maximal ideal is nilpotent. (Note that such an algebra (A, m) is canonically isomorphic to its m-adic completion.) Let us denote by ι : Testk → LocAlgk the inclusion functor. Lemma 5.1.6. — Let (R, m) be a local k-algebra with residue field k and  induces a let A be a test-ring over k. The completion morphism j : R → R  bijection from HomLocAlgk (R, A) to the set of local homomorphisms R → A. Proof. — Let ϕ : R → A be a local homomorphism. By assumption, ϕ(m) is contained in the maximal ideal of A, and A is complete, so that ϕ induces a  → A in the category LocAlgk . One has ϕ morphism ϕ : R  ◦ j = ϕ.  Conversely, let ψ : R → A be a morphism in LocAlgk such that ψ ◦ j = ϕ; let us prove that ψ = ϕ.  Let mA be the maximal ideal of A, and let p be an  for every n, let xn be the image of x integer such that mpA = (0). Let x ∈ R; n n  ; hence in R/m . One has x − j(xn ) ∈ m ψ(x) − ϕ(xn ) = ψ(x) − ψ ◦ j(xn ) ∈ mnA ,  implies that so that ψ(x) = ϕ(xn ) for n  p. Applying this relation to ϕ ψ = ϕ,  as was to be shown.  the category of contravariant (5.1.7). — If C is a category, we denote by C functors from C to the category Sets of sets.

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Let A be an object of LocAlgk . As usual, we denote by hA the functor  k . For A, A in LocAlgk , the HomLocAlgk (A, ·); it is an object of LocAlg natural maps (hA , hA ) (5.1.7.1) HomLocAlg (A, A ) ∼ = Hom k

 LocAlg k



are functorial in A, A ; the Yoneda lemma asserts that they are bijective. Let hA := hA ◦ ι be the restriction of this functor hA to the subcategory Testk . When A varies, these functors give rise to a functor (5.1.7.2)

k , h : LocAlgk → Test

A → hA

from the category LocAlgk to the category of set-valued functors on the category of test-rings. Lemma 5.1.8. — Let k be a field. With the notation of (5.1.7), the functor h is fully faithful. Proof. — By the Yoneda lemma (equation (5.1.7.1)), we have to prove that for all objects A, A of LocAlgk , the natural (and functorial) map (5.1.8.1)

HomLocAlg  (hA , hA ) → HomLocAlg  (hA ◦ ι, hA ◦ ι) k

k

is bijective. Let us first prove that it is injective. Let ϕ1 , ϕ2 : hA → hA be morphism of functors such that ϕ1 ◦ ι = ϕ2 ◦ ι. Let B be an object of LocAlgk ; let us choose a local k-algebra (R, m) with residue field k and an isomorphism  = lim R/mn . For every integer n, the k-algebra R/mn is a test-ring. BR ←− By assumption, one has an equality ϕ1 (R/mn ) = ϕ2 (R/mn ) : HomLocAlgk (A, R/mn ) → HomLocAlgk (A , R/mn ). On the other hand, for i ∈ {1, 2}, the map ϕi (B) : HomLocAlgk (A, B) → HomLocAlgk (A , B) identifies with the projective limit of the maps ϕi (R/mn ) under the isomorphism B  lim R/mn . Consequently, ϕ1 (B) = ϕ2 (B), which ←− proves the injectivity of the map (5.1.8.1). For the surjectivity, let ϕ ∈ HomLocAlg  (hA ◦ ι, hA ◦ ι), and let us prove k the existence of a morphism of functors ϕ˜ : hA → hA which lifts ϕ. Again, let B be an object of LocAlgk and let (R, m) be a local k-algebra with residue field k with an isomorphism B  limm R/mn . For every integer n, R/mn is a ←− test-ring, and we can consider the map ϕ(R/mn )) : HomLocAlgk (A, R/mn ) → HomLocAlgk (A , R/mn ). The family (ϕ(R/mn )) is a projective system; passing to the limit and using the isomorphism B  lim R/mn , we get a map ←− ϕ(B) ˜ : HomLocAlgk (A, B) → HomLocAlgk (A , B). One then checks that, when B varies, the family of maps (ϕ(B)) ˜ is a morphism of functors ϕ˜ : hA → hA . By construction, it lifts ϕ. Remark 5.1.9. — Let A, A be two objects in the category LocAlgk . Explicitly, lemma 5.1.8 asserts that a morphism of functors ϕ : hA → hA is induced by a unique morphism f : A → A of admissible local k-algebras.

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As a consequence, the morphism ϕ is an isomorphism if and only if f is an isomorphism. We thus sum up the content of this lemma by saying that a local k-algebra A in LocAlgk is determined by its associated functor hA . (5.1.10). — Let k be a field, let X be a k-variety, and let γ ∈ X(k[[t]]) be a k-arc on X. Considering γ as a k-point of L (X), we consider the formal completion L (X)γ of L (X) at γ; by definition it is the formal spectrum of the mγ -adic completion O L (X),γ of the local ring OL (X),γ of L (X) at γ. Let us give an explicit description, in terms of arcs, of the associated functor on the category Testk . By construction, the algebra O L (X),γ is an object of LocAlgk . Consequently, determining the formal completion L (X)γ is equivalent to de scribing, functorially, the sets HomLocAlgk (OL (X),γ , A), for all test-rings A over k. By lemma 5.1.6, this even amounts to describing functorially the sets Hom(OL (X),γ , A) of local homomorphisms of k-algebras from OL (X),γ to A, that is, of diagrams (5.1.10.1)

OL (X ),γ

ϕA

A

k. Since the k-algebra A is local, the diagram (5.1.10.1) also corresponds to a commutative diagram of morphisms of k-schemes: (5.1.10.2)

Spec(A[[T ]])

γA

X.

γ

Spec(k[[T ]]) 5.2. Weierstrass Theorems for Power Series (5.2.1). — Let A be a local ring and let m be its maximal ideal; let us assume that A is complete for the m-adic topology. Let T = (T1 , . . . , Tn ) be a finite family of indeterminates. For m ∈ Nn , we write T m = T1m1 . . . Tnmn and |m| = m1 + · · · + mn .  Let g ∈ A[[T ]]; write g = m∈Nn gm T m ∈ A[[T ]]. Let d ∈ N; one says that g is regular of degree d with respect to Tn if gm ∈ m for every m ∈ Nn such that |m| < d and if the coefficient of Tnd is invertible in A. Proposition 5.2.2 (Weierstrass division theorem) Let f, g ∈ A[[T ]] be formal power series with coefficients in A, and let d ∈ N. If g is regular of degree d with respect to Tn , then there exists a unique pair (q, r) with q ∈ A[[T ]] and r ∈ A[[T1 , . . . , Tn−1 ]][Tn ] such that f = gq + r and r is a polynomial of degree < d in Tn .

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201

Proof. — The ring A[[T1 , . . . , Tn−1 ]] is local and complete for the topology defined by its maximal ideal m + (T1 , . . . , Tn−1 ). We thus may assume that S be the linear n = 1; we then write T for T1 . Let ]], and let  M = A[[T an+d T n . The assertion is endomorphism of M defined by S( an T n ) = equivalent to the existence of a unique q ∈ M such that S(gq) = S(f ). Let us prove that q → S(gq) is invertible. Since g is regular of degree d, there exist a polynomial g1 ∈ A[T ] of degree < d, all of whose coefficients belong to m, and an invertible power series g2 ∈ A[[T ]] such that g = g1 + T d g2 . For every q ∈ M , one has S(gq) = S(g1 q) + g2 q = g2 (q − R(q)), where R(q) = −g2−1 S(g1 q). It suffices to prove that IdM − R is an invertible endomorphism of M . By construction, the image of R iscontained in mM . Since M is complete for the m-adic topology, the series m0 Rm defines an endomorphism S  of M such that S  ◦ (IdM − R) = (IdM − R) ◦ S  = IdM . This shows that IdM − R is invertible and the proposition follows. Corollary 5.2.3. — Let f, g ∈ A[[T ]] and let I be an ideal of A such that f g ∈ I[[T ]]. If g ∈ A[[T ]] is regular of degree d with respect to Tn , then f ∈ I[[T ]]. For I = (0), this statement shows in particular that a power series in A[[T ]] which is regular with respect to Tn is not a zero divisor in A[[T ]]. Proof. — Let h = f g. The element g¯ = g mod I is regular of degree d with ¯ = g¯ · f¯ is the Weierstrass division of h ¯ by g¯. respect to Tn in (A/I)[[T ]], and h ¯ ¯ Since h = 0, this division is given by 0 = g¯ · 0; hence f = 0, due to the uniqueness of the decomposition, so that f ∈ I[[T ]]. Proposition 5.2.4 (Weierstrass preparation theorem) Let f ∈ A[[T ]] be a formal power series. If f ≡ 0 (mod m), then there exists an automorphism σ of A[[T ]] which is continuous for the T -adic topology and such that σ(f ) is regular of degree d with respect to Tn . Proof. — Let μ be the smallest element of Nm , for the lexicographic order, such that fμ ∈ m. For i ∈ {1, . . . , n}, let ei = (μi+1 + 1) . . . (μn + 1). We observe that en = 1; from the relation ei = (μi+1 + 1)ei+1 , we deduce by decreasing induction that   ej μj = ei+1 − ej μj = 1 ei − j>i

j>i+1

for every i ∈ {1, . . . , n}. For any a, b ∈ Nm , let us write a · b = a1 b1 + · · · + an bn . Then there exists e a unique continuous automorphism σ of A[[T ]] such that σ(Tj ) = Tj + Tnj for j ∈ {1, . . . , n − 1} and σ(Tn ) = Tn . Let d = e · μ, and let us prove that σ(f ) is regular of degree d with respect to Tn . We can expand σ(f ) explicitly as follows:  fm (T1 + Tne1 )m1 . . . (Tn−1 + Tnen−1 )mn−1 Tnmn σ(f ) = m∈Nn

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=





m∈Nn

p∈Nn 0pj mj pn =0

 fm

 m1 mn−1 pn−1 e·(m−p) Tn . ... T1p1 . . . Tn−1 p1 pn−1

 m, we shall For each m ∈ Nn and each p ∈ Nn such that pn = 0 and fm ∈ show that the degree of the corresponding monomial is > d, except in the case where m = μ and p = 0. This degree is given by p

n−1 Tne·(m−p) ) = e · m + (e − 1) · p; deg(T1p1 . . . Tn−1

it is at least equal to e · m, and equals e · m if and only if p = 0. Since fm ∈ m, one has m  μ. If m = μ, we obtain a monomial of degre > d, unless p = 0 in which case we get fm Tnd . Let us thus assume that m > μ. The degree of the corresponding monomial is  e · m; hence it suffices to prove that e · m > d. By definition of the lexicographic order, there exists an integer i ∈ {1, . . . , n} such that m1 = μ1 , . . . , mi−1 = μi−1 , and mi > μi . Then, n n   e · m − d = e · (m − μ) = ei (mi − μi ) + ej (mj − μj )  ei − ej μj = 1, j=i+1

j=i+1

which proves that the corresponding monomial has degree > d. This concludes the proof. 5.3. Reduction to the Complete Intersection Case In this section, we use the Jacobi criterion of smoothness to reduce the proof of theorem 5.1.1 to the case where X is a complete intersection in an affine space. Such reductions were classically used in the first papers on motivic integration, notably Denef and Loeser (1999). (5.3.1). — Let k be a field, let X be a k-variety, and let γ ∈ X(k[[t]]) = L (X)(k) be a k-arc on X. Let U be an open neighborhood of γ(0) in X. The canonical inclusion i identifies L (U ) with an open subscheme of L (X) (X)γ of containing γ. In particular, it induces an isomorphism L (U )γ → L formal completions. (5.3.2). — Let r be an integer and let f1 , . . . , fr ∈ k[T1 , . . . , Tn ]. Recall that the Jacobian matrix J(f1 , . . . , fr ) is the matrix of size r × n whose entries are the partial derivatives ∂fi /∂Tj , for i ∈ {1, . . . , r} and j ∈ {1, . . . , n}. A minor of size r of this matrix is a determinant of the form det(∂fi /∂Tji ) for a family (j1 , . . . , jr ) of elements of {1, . . . , r}. Lemma 5.3.3. — Let I be an ideal of k[T1 , . . . , Tn ] and let X = V (I). Let x be a point of X and let r be an integer. Then X is smooth over k of relative codimension r at the point x if and only if there exist f1 , . . . , fr ∈ I,

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203

a minor g of size r of the Jacobian matrix J(f1 , . . . , fr ), and an element h ∈ ((f1 , . . . , fr ) : I) such that g(x)h(x) = 0. Proof. — By the Jacobian criterion (Bosch et al. (1990), prop. 2.2/7), the morphism p is smooth of relative codimension r at x if and only if it can be defined, in a neighborhood of x, by r equations f1 , . . . , fr whose Jacobian matrix has rank r at x. Since k[T1 , . . . , Tn ] is noetherian, the ideal I is finitely generated, and the condition that the ideal (f1 , . . . , fr ) coincides with I in a neighborhood of x means exactly that IOX,x = (f1 , . . . , fr )OX,x . Since X is affine, this is equivalent to the existence of h ∈ A such that h(x) = 0 and hI ⊂ (f1 , . . . , fr ). This establishes the lemma. Corollary 5.3.4 (Jacobian Criterion). — The nonsmooth locus Xsing of X/k is the closed subscheme of X whose ideal is generated by all products of the form gh, for an integer r  0, elements f1 , . . . , fr ∈ I, a minor g of size r of the Jacobian matrix J(f1 , . . . , fr ), and an element h ∈ ((f1 , . . . , fr ) : I). Proposition 5.3.5. — Let k be a field, and let X be a closed subscheme of Ank defined by an ideal I of k[T1 , . . . , Tn ]; let Xsing be the nonsmooth locus of the canonical morphism X → Spec(k). Let γ be a k-arc on X and let x = γ(0) be its base point. Let us assume that γ ∈ L∞ (Xsing ). Then there exist an integer  0 and polynomials f1 , . . . , fr ∈ I defining a closed subscheme Y = V (f1 , . . . , fr ) of Ank such that the following properties hold: a) There exists a minor of size r of the Jacobian matrix J(f1 , . . . , fr ) which does not vanish at γ; b) There exists a unique irreducible component of X containing the arc γ, and its dimension is equal to n − r. c) The canonical closed immersion j : X → Y induces an isomorphism ∼ (X)γ − → L (Y )j(γ) . j∗ : L

Proof. — Since γ is not contained in the singular locus of X, there exist an integer r, elements f1 , . . . , fr ∈ I, a minor of rank r of the Jacobian matrix J(f1 , . . . , fr ), and h ∈ ((f1 , . . . , fr ) : I) which do not vanish at γ. Let Y = V (f1 , . . . , fr ), and let j : X → Y be the canonical closed immersion. Since X is smooth of dimension n − r at the generic point of γ, there exists a unique irreducible component of X containing γ, and it has dimension n − r. To prove that j induces an isomorphism of formal completions, we need to prove that for every test-ring (A, mA ) over k, the associated map (Y )j(γ) (A) is bijective. Injectivity is clear, since j is a L (X)γ (A) → L  (Y )j(γ) (A); by the monomorphism; let us thus show surjectivity. Let θA ∈ L  description of the formal completion of L (Y ) at j(γ), θ corresponds to an arc θA ∈ Y (k[[t]]) which is congruent to j(γ) modulo mA . Let f ∈ I and let us prove that f (θA ) = 0. By assumption, one has hf ∈ (f1 , . . . , fr ); hence h(θA )f (θA ) = 0. For, modulo mA , one has h(θA ) ≡ h(γ) = 0. By corollary 5.2.3 of the Weierstrass division theorem, this implies that f (θA ) = 0. Consequently, the arc θA is contained in X. This concludes the proof.

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5.4. Proof of the Theorem of Grinberg–Kazhdan–Drinfeld We now prove theorem 5.1.1. (5.4.1). — By §5.3.1 and proposition 5.3.5, we may assume that X is a complete intersection in an affine space. We assume that X is defined = Spec(k[X1 , . . . , Xn , Y1 , . . . , Yr ]) by a family f = (f1 , . . . , fr ) of in An+r k r polynomials such that the determinant of the matrix D = (∂fi /∂Yj )1i,jr does not vanish at γ. The dimension of the unique irreducible component of X containing γ is thus equal to n. Let C be the adjugate matrix of D and J = det(C); one thus has CD = DC = JIr . The arc γ is viewed as a pair (x0 , y0 ) where x0 ∈ (k[[t]])n and y0 ∈ (k[[t]])r ; we may also assume that γ(0) = 0. Let d = ordt (J(x0 , y0 )); by assumption, this is a nonnegative integer, and the power series u0 = t−d J(x0 , y0 ) is invertible. (5.4.2). — To prove theorem 5.1.1, we will first use the equations of X to analyze precisely the set of A-arcs that reduce to γ, for every test-ring A. This will allow us to define an affine k-variety Z, a k-point z ∈ Z, and a functorial family of maps X(A) → Z(A) × mN , for every test-ring (A, m), such that γ → (z, 0). We will finally prove that these maps are bijective. (5.4.3). — Let (A, m) be a test-ring. Let us consider an A-arc reducing to γ modulo m; it corresponds to a pair (x, y), where x ∈ (A[[t]])n and y ∈ (A[[t]])n , which is congruent to (x0 , y0 ) modulo m and such that f (x, y) = 0 in Ar . Since the power series J(x, y) ∈ A[[t]] is congruent to J(x0 , y0 ) modulo m, it is regular of degree d with respect to t. Let J(x, y) = q·u be the Weierstrass factorization of J(x, y), where q ∈ A[t] is a monic polynomial congruent to td modulo m and u ∈ A[[t]] is an invertible power series. Let x = gq 2 + x ˜ and y = hq + y˜ be the Weierstrass divisions of the entries of x and y by the regular power series q 2 and q, respectively. One has g ∈ (A[[t]])n and h ∈ (A[[t]])r , while the entries of x ˜ and y˜ are elements of A[t] of degrees < 2d and < d, respectively. Since y ≡ y˜ (mod q) and f (x, y) = 0, we first have (5.4.3.1)

f (˜ x, y˜) ≡ 0

(mod q).

More precisely, the first-order Taylor formula writes 0 = f (x, y) = f (x, y˜ + hq) = f (x, y˜) + DY f (x, y˜) · hq

(mod q 2 )

= f (x, y˜) + qDY f (x, y˜) · h (mod q 2 ), so that C(x, y˜) · f (x, y˜) ≡ −qJ(x, y˜)h ≡ 0

(mod q 2 ).

Since x ≡ x ˜ (mod q 2 ), we then have (5.4.3.2)

C(˜ x, y˜) · f (˜ x, y˜) ≡ 0 (mod q 2 ).

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Since x ≡ x ˜ (mod q), y ≡ y˜ (mod q) and J(x, y) is divisible by q, we also have (5.4.3.3)

J(˜ x, y˜) ≡ 0 (mod q).

We associate with the A-arc (x, y) the triple ϕA (x, y) = (q, x ˜, y˜). Since q is monic, it is identified with the vector coefficients of q − td , an element ˜ and y˜ are identified with their vectors of Ad ; similarly, the polynomials x of coefficients, so that the triple (q, x ˜, y˜) is viewed as an element of Ad × 2d n d r (1+2n+r)d (A ) × (A ) = A . It satisfies the relations (5.4.3.1), (5.4.3.2), (1+2n+r)d and (5.4.3.3). These equations define a closed subscheme Z of Ak . The point associated with the arc (x0 , y0 ) is given by the triple z = ˜0 , y˜0 ), where x ˜0 and y˜0 are the remainders of the WeierϕA (x, y) = (td , x strass divisions of x0 and y0 by t2d and td , respectively. When A varies among all the test-rings over k, the maps (ϕA ) define a  z . morphism ϕ of formal schemes from L (X)γ to Z Let g0 ∈ k[[t]] be the image of g modulo m. Set ψA (x, y) = g − g0 ; this is an element of (m[[t]])n , which we see as an element of (mN )n . The family of  (X)γ to Dn∞ , where maps (ψA ) defines a morphism of formal schemes from L D∞ = Spf(k[[(Tn )n∈N ]]) is the “infinite dimensional” formal disk. (5.4.4). — At this point, we have defined a morphism of formal k-schemes  z ×  Dn∞ . (ϕ, ψ) : L (X)γ → Z We now prove that this morphism is an isomorphism. Proposition 5.4.5. — Let (A, m) be a test-ring, let (q, x ˜, y˜) ∈ Z(A) be a triple which reduces to z modulo m, and let g ∈ (A[[t]])n be a family of power ˜ + gq 2 . There exists a unique series congruent to g0 modulo m; set x = x r element y ∈ (A[[t]]) such that f (x, y) = 0, y ≡ y0 (mod m), and y ≡ y˜ (mod q). Lemma 5.4.6. — Let us keep the notation of proposition 5.4.5. For every integer a  1, there exists an element y ∈ (A[[t]])r , unique modulo (qma A[[t]])r , such that f (x, y) ≡ 0 (mod ma ), y ≡ y0 (mod m) and y ≡ y˜ (mod q). Proof. — We prove the lemma by induction on a. Let us first treat the case a = 1. Let y  ∈ (A[[t]])r be any lift of y0 modulo m, and let y  = hq + r be its Weierstrass division by q. By reduction modulo m, we obtain the Weierstrass division of y0 by td , so that r ≡ y˜0 ≡ 0 (mod m). The element y = y  − r satisfies the requested condition. Uniqueness follows from corollary 5.2.3. Let us now assume that the claim holds for a  1, and let us prove it for a + 1. We start from y  ∈ (A[[t]])r satisfying the conditions at level a, and we seek for y ∈ (A[[t]])r which satisfies them at level a + 1; let w = y  − y. By the uniqueness assertion at level a, one must have w = y  − y ∈ (qma A[[t]])r ; conversely, for any such w, y = y  − w is congruent to y0 modulo m and to y˜ modulo q. Since J(˜ x, y˜) ≡ 0 (mod q), J(x, y  ) is a multiple of q as well, so

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that there exists a unique power series u ∈ A[[t]] such that J(x, y  ) = qu. ¯, so that u ≡ u0 (mod m); in Modulo m, one has J(x0 , y0 ) ≡ J(x, y  ) ≡ td u particular, u is invertible in A[[t]]. Since y  ≡ y˜ (mod q), one has f (x, y  ) ≡ 0 x, y˜) (mod q). Consequently, C(x, y  ) · f (x, y  ) ≡ (mod q), and C(x, y  ) ≡ C(˜ 2 0 (mod q ). There thus exists a unique element w ∈ (A[[t]])r such that J(x, y  )w = C(x, y  ) · f (x, y  ), and one has w ∈ (qA[[t]])r . Then, J(x, y  )DY f (x, y  ) · w = DY f (x, y  ) · J(x, y  )w = DY f (x, y  ) · C(x, y  ) · f (x, y  ) = J(x, y  ) · f (x, y  ), so that DY f (x, y  ) · w = f (x, y  ), since J(x, y  ) is regular in A[[t]]. Since a  1, one has 2a  a + 1; hence f (x, y  − w) ≡ f (x, y  ) − DY f (x, y  ) · w ≡ 0 (mod ma+1 ). We thus have proved that the element y = y  − w ∈ (A[[t]])r satisfies the conclusion of the claim. It remains to show its uniqueness. If y and y  are two such elements, then w = y  − y is an element of (qma A[[t]])r such that DY f (x, y) · w ≡ 0 (mod ma+1 ). This implies J(x, y)w ∈ (ma+1 [[t]])r . By Weierstrass division in (A/ma+1 )[[t]], it then follows that w ∈ (ma+1 [[t]])r ; hence w ∈ (qma+1 [[t]])r , by corollary 5.2.3. Proof of prop. 5.4.5. — For every integer a  1, let ya ∈ (A[[t]])r be any element such that f (x, ya ) = 0 (mod ma ), ya ≡ y0 (mod m) and ya ≡ y˜ (mod q). By the previous lemma, the sequence (ya ) converges, when a → +∞, to an element y ∈ (A[[t]])r such that f (x, y) = 0, y ≡ y0 (mod m), and y ≡ y˜ (mod q). Moreover, any element y  ∈ A([[t]])r satisfying these relations is congruent to ya modulo ma , for every a  1, hence is equal to y. In summary, we have associated with the k-arc γ a k-scheme Z of finite type (based on equations (5.4.3.1), (5.4.3.2), and (5.4.3.3)) and defined a  z ×  Dn∞ . We have then proved that morphism of formal schemes L (X)γ → Z this morphism is an isomorphism by showing that it induces a bijection on A-points, for every test-ring A. This concludes the proof of theorem 5.1.1. Example 5.4.7. — Let f ∈ k[x1 , . . . , xn ] be a polynomial such that f (0) = 0. Let X be the hypersurface of An+2 = Spec(k[x1 , . . . , xn , xn+1 , y]) with equation xn+1 y + f (x1 , . . . , xn ) = 0. The nonsmooth locus of X is defined by the equations f (x1 , . . . , xn ) =

∂f ∂f (x1 , . . . , xn ) = · · · = (x1 , . . . , xn ) = xn+1 = y = 0. ∂x1 ∂xn

§ 5. THE THEOREM OF GRINBERG–KAZHDAN–DRINFELD

207

Let γ 0 ∈ L (x) be the k-arc given by γ10 (t) = · · · = γn0 (t) = 0,

0 γn+1 (t) = t,

y 0 (t) = 0.

Let A be a test-ring over k, and let m be its maximal ideal. Let γ be an A-arc on X such that γ ≡ γ 0 (mod m); this means γ1 , . . . , γn , y ∈ m[[t]],

γn+1 − t ∈ m[[t]].

The derivative of γn+1 (t) is equal to 1 (mod m), hence is invertible at t = 0. Since γn+1 (0) ∈ m, Hensel’s lemma implies that the equation γn+1 (a) = 0 has a unique solution a ∈ m. Let γn+1 (t) = (t − a)u(t) + γn+1 (a) be the Weierstrass division of γn+1 by the regular power series t − a. Modulo m, we obtain t ≡ γn+1 (t) ≡ tu (mod m), so that u(t) ≡ 1 (mod m). Moreover, f (γ1 (a), . . . , γn (a)) = 0. Conversely, let a ∈ m, let γ1 (t), . . . , γn (t) ∈ m[[t]] be such that ˜(t) ∈ 1 + m[[t]]. The power series f (x1 (a), . . . , xn (a)) = 0, and let u f (γ1 (t), . . . , γn (t)) belongs to m[[t]] and vanishes at t = a, hence can be uniquely written (t − a)ϕ(t), for some power series ϕ ∈ m[[t]]. Since u ˜(t) is invertible in A[[t]], it follows that there exists a unique power series y(t) ∈ m[[t]] such that (t − a)˜ u(t)y(t) + f (γ1 (t), . . . , γn (t)) = 0. The map t → t − a induces an isomorphism of A[[t]] onto itself which respects [[t]]. We thus can rewrite the preceding discussion as follows: every arc γ ∈ L (X)(A) is of the form (x1 (t − a), . . . , xn (t − a), (t − a)(1 + v˜(t − a))), y(t − a)) for a unique family (x1 , . . . , xn , v˜) of power series in m[[t]] such that f (x1 (0), . . . , xn (0)) = 0, and y(t) is equal to the quotient ˜(t)). f (x1 (t), . . . , xn (t))/t(1 + u Let Y ⊂ An be the hypersurface defined by f , and let y = 0 be the origin  (X)γ is of An ; one has y ∈ Y . The arguments that precede show that L isomorphic to the product of Yy by the formal spectrum of the ring of power series with coefficients in k and variables xm,j (for 1  m  n and j  1) and u ˜j (for j  0). Remark 5.4.8. — Let R = k[[t]]. As we shall see in the next chapter, the notion of an arc scheme L (X), which parameterizes R-points of a kscheme X, generalizes to that of a Greenberg scheme Gr(X) of a formal R-scheme X, parameterizing its R-points. As shown in Bourqui and Sebag (2017a), the arguments used in the proof of theorem 5.1.1 extend naturally to γ  S ×  Dn∞ , that more general situation and furnish an isomorphism Gr(X) where S is an affine noetherian adic formal k-scheme of finite type. 5.5. Gabber’s Cancellation Theorem and Consequences The aim of this appendix is to present theorem 5.5.2, a cancellation theorem in the context of formal geometry (in arbitrary dimension) due to O. Gabber. It can be used, in particular, to reinforce significantly the meaning of theorem 5.1.1; see Bourqui and Sebag (2017a).

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(5.5.1). — Let us fix some notation used in this section. One says that a complete local noetherian k-algebra A with residue field k-isomorphic to k is cancellable if there exists a complete local noetherian k-algebra B whose residue field is k-isomorphic to k such that A is isomorphic to B[[T ]]. A straightforward dimension argument shows that for every such k-algebra A, there exist N ∈ N and a noncancellable k-algebra B such that A is isomorphic to B[[T1 , . . . , TN ]] (as admissible k-algebras). Theorem 5.5.2 (Gabber). — Let k be a field. Let A and B be two complete local noetherian k-algebra with residue field k-isomorphic to k. Let I and J be (possibly infinite) sets. Assume that the admissible k-algebras A[[(Ti )i∈I ]] and B[[(Uj )j∈J ]] are isomorphic. Then, up to exchanging A and B, there exists a finite subset I  ⊂ I such that A[[(Ti )i∈I  ]] and B are isomorphic (as admissible k-algebras). In particular, if both A and B are noncancellable, then they are isomorphic. A proof of this theorem can be found in Bourqui and Sebag (2017a, §7), which essentially follows a private communication of Gabber to the authors of that paper. A slightly weaker statement of this theorem appears in Bouthier et al. (2016). Theorem 5.5.2 also generalizes an older version valid in case the sets I, J are finite; see Hamann (1975, theorem 4). Corollary 5.5.3. — For every complete local noetherian k-algebra A whose residue field is k-isomorphic to k, there exist in fact a unique noncancellable k-algebra B and a unique integer N  0 such that A is k-isomorphic to B[[T1 , . . . , TN ]]. We denote this k-algebra B by Amin . Proof. — Let A and B be two complete local noetherian k-algebra with residue field k-isomorphic to k. Let mA (resp. mB ) be the maximal ideal of A (resp. B). Clearly, we may assume that AandB are noncancellable. We set A := A[[(Ti )i∈I ]] and B  := B[[(Uj )i∈J ]], and let ϕ : A → B  be an isomorphism. We have a natural injective morphism ιA : A → A admitting a retraction ρA given by Ti → 0. We define analogously ιB and ρB . Let us note that, if mA (resp. mB  ) is the maximal ideal of A (resp.  B ), the ideal m2A (resp. m2B  ) is not closed for this topology and its closure m2A (resp. m2B  ) coincides with the kernel of the projection A[[(Ti )i∈I ]]→ A[(Ti )i∈I ]/mA , (Ti )i∈I 2 (resp. B[[(Uj )j∈J ]]→ B[(Uj )j∈J ]/mB , (Uj )j∈J 2 ). Identifying, via ιA , mA /m2A with a subvector space of mA /m2A , we have a decomposition  k ti (5.5.3.1) mA /m2A ∼ = mA /m2A ⊕ i∈I

(where we denote by ti the class of Ti modulo m2A ) and a similar decomposition for B  .

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209

By lemma 5.5.4 below, we obtain that the composition ϕ

ρB ιA A ∼ h : A→ = B → B ∼ mB /m2 . By this and a straightforward induces an isomorphism mA /m2A = B induction, we infer that, for every integer n ∈ N, the morphism h induces a surjection of finite dimensional k-vector spaces :

→ mnB /mn+1 hn : mnA /mn+1 A B . n+1 n Thus, one has dim(mnB /mn+1 B )  dim(mA /mA ). Exchanging the roles of A in B, we get the opposite inequalities, hence the equality of the dimensions for all n. Thus, the morphism hn is an isomorphism for every integer n ∈ N; hence, the morphism h is an isomorphism.

Lemma 5.5.4. — Identifying, via ϕ, mA /m2A and mB  /m2B  , the images of mA /m2A by ιA and mB /m2B by ιB coincide. Proof. — Assume that there exists an element f ∈ mA such that the class of the element ϕ(f ) in mB  /m2B  does not belong to mB /m2B . Thus, there exists j0 ∈ J such that ∂Uj0 ϕ(f ) is invertible, thanks to decomposition (5.5.3.1). Now, let us define the morphism ψ : B  → B  [[T ]] by ⎧ = IdB ⎨ ψ|B ψ(Uj ) = Uj for every element j = j0 ⎩ ψ(Uj0 ) = Uj0 + T. In particular, we observe that the formula ∂T ψ = ψ∂Uj0 holds true. Let us denote by evA : A[[T ]]→A the evaluation morphism given by T → 0. Composing with ρA ◦ ϕ−1 : B  → A, we get the following morphism ϕ

ψ ιA A ∼ μ : A→ = B  → B  [[T ]]→ A[[T ]]

which has the following properties: ) evA ◦ μ = IdA ∂T (μ(f )) ∈ (A[[T ]])× . Let us denote by evA/f A : (A/f A)[[T ]]→A/f A the evaluation morphism given by T → 0. Then, composing the morphism μ with A[[T ]]→ (A/f A)[[T ]], one gets a new morphism μ : A → (A/f A)[[T ]] such that the morphism evA/f A ◦ μ coincides with the quotient morphism A → A/f A, and μ (f ) = T u(T ) with u(T ) ∈ (A/f A)[[T ]] satisfying u(0) ∈ (A/f A)× . Let us show that μ is an isomorphism, which will contradict the fact that A is noncancellable; hence, it will prove the lemma. In order to do so, we consider the f -adic filtration on A and the T -adic filtration on (A/f A)[[T ]]. By (ÉGA I, corollaire 7.3.6), A and (A/f A)[[T ]] are separated and complete for the topologies induced by these filtrations. Hence, since μ respects these filtrations, we are reduced to show that μ induces an isomorphism on the

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level of the homogeneous parts of the associated graded rings. But, for every integer n ∈ N, we may identify canonically T n (A/f A)[[T ]]/T n+1 (A/f A)[[T ]] with A/f A. Then, via this identification, the morphism induced by μ on the homogeneous parts of degree n reads as f n A/f n+1 A −→ A/f A [f n a] −→ u(0)n [a] whose inverse is defined by [b] → [f n u(0)−n b], since u(0) is invertible. Remark 5.5.5. — Keep the notation of the previous section. If the k˜ the element algebra A happens to be the completion of a local k-algebra A, f ∈ mA in the proof of lemma 5.5.4 may be assumed to belong to mA˜ , since every element of f + m2A will have the required properties. Since one has  ˜ A˜ ∼ A/f = A/f A, it shows that if A and B are assumed to be completions of local k-algebras essentially of finite type, the last assertion in the statement of theorem 5.5.2 still holds true under the weaker hypothesis that A and B are noncancellable as elements which are completions of local k-algebras essentially of finite type. (5.5.6). — Let us underline an important consequence of theorem 5.5.2, which implies a kind of “uniqueness” in the choice of the formal k-scheme Yˆy in the statement of theorem 5.1.1. Corollary 5.5.7. — Let X be a k-variety. Keep the notation of theorem 5.1.1. If (Y, y), (Y  , y  ) are two affine pointed k-varieties which realize the isomorphism of theorem 5.1.1. Then, the complete local noetherian k-algebras (OY,y )min , (OY  ,y )min are isomorphic (as admissible local k-algebras). In particular, there exist two integers m, m ∈ N and an isomorphism of formal  k k[[T1 , . . . , Tm ]]∼  k k[[T1 , . . . , Tm ]]. k-schemes O* = O* Y,y ⊗ Y,y ⊗

CHAPTER 4 GREENBERG SCHEMES

(0.0.1). — Let R be a complete discrete valuation ring, let m be its maximal ideal, and let k be its residue field. When R = k[[t]] and X is a k-scheme, we defined in chapter 3 the schemes of jets Ln (X/k) and the scheme of arcs L∞ (X/k) on X whose k-points are in canonical bijection with X(R/mn+1 ) and X(R), respectively. If we set X = X ⊗k R, these sets are in natural bijection with X (R/mn+1 ) and X (R). In this chapter, we will generalize the construction of the jet and arc schemes starting from an arbitrary R-scheme. This framework was first put forward by Looijenga (2002) (when R = k[[t]]), and fully developed by Sebag (2004b). Following Greenberg (1961), we will construct for any R-scheme X the Greenberg schemes Grn (X) and Gr∞ (X) whose k-points are in canonical bijection with X(R/mn+1 ) and X(R), respectively. If R has mixed characteristic, this construction requires the assumption that k is perfect. The original papers of Greenberg are difficult to read, because they use a mixture of Weil’s language of algebraic geometry and Grothendieck’s theory of schemes. We have made an effort to present the construction in a modern form, and we extend it to formal schemes over R. (0.0.2). — In order to construct these Greenberg schemes, it is crucial to have a precise understanding of the structure of complete discrete valuation rings. This is rather straightforward in equal characteristic, but more subtle in the mixed characteristic case, where the ring of Witt vectors W (k) will play an essential role. We will recall the general theory of complete discrete valuation rings in section 1, starting with the definition and basic properties of the Witt ring in section 1.1. In this section, proofs are sometimes only sketched, but we give references to literature. When R = k[[t]] or R = W (k), the ring Rn = R/mn+1 is naturally identified with the set of k-points of a ring scheme, respectively, denoted Arcn © Springer Science+Business Media, LLC, part of Springer Nature 2018 A. Chambert-Loir et al., Motivic Integration, Progress in Mathematics 325, https://doi.org/10.1007/978-1-4939-7887-8_4

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or Wn+1 , which is isomorphic as a scheme to the affine space An+1 of dik mension n + 1. In the absolutely ramified case, the construction of such a ring scheme is more complicated and is the subject of section 2. We will then define the Greenberg schemes in section 3. When R = k[[t]] and X is defined over k, we recover the arc spaces of chapter 3. The structure of the Greenberg schemes will be studied in detail in chapter 5.

§ 1. COMPLETE DISCRETE VALUATION RINGS In this section, we collect some preliminary material on complete discrete valuation rings. The most important results are structure theorems that describe the shape of such rings. In equal characteristic, they are all isomorphic to rings of formal power series over a field. In mixed characteristic, their structure is more complicated; the basic building blocks are the rings of Witt vectors. 1.1. Witt Vectors Throughout this section, p is a fixed prime number. (1.1.1). — We briefly recall the construction and basic properties of the ring of Witt vectors. We refer to chapter IX of Bourbaki (2006) or to chapter II of Serre (1968) for a more detailed account. We consider for every integer n  0 the polynomial n  n−i n n−1 pi Xip = X0p + pX1p + . . . + pn X n wn = i=0

in Z[X0 , . . . , Xn ]. These polynomials are called the p-typical Witt polynomials. Theorem 1.1.2 (Witt). — For every polynomial Φ in Z[X, Y ], there exists a unique sequence (ϕ0 , ϕ1 , . . .) of elements in Z[X0 , Y0 , X1 , Y1 , . . .] such that wn (ϕ0 , . . . , ϕn ) = Φ(wn (X0 , . . . , Xn ), wn (Y0 , . . . , Yn )) for every integer n  0. Moreover, ϕn is a polynomial in the variables X0 , Y0 , . . . , Xn , Yn . Sketch of proof. — We can write every variable Xn as a polynomial in the Witt polynomials w0 , . . . , wn , with coefficients in Z[1/p]. Consequently, for any ring A in which p is invertible, the map w : AN → AN given by w(a0 , a1 , . . . ) = (w0 (a0 ), w1 (a0 , a1 ), . . . ) is a bijection. Applying this remark to the ring A = Z[1/p][X0 , Y0 , X1 , Y1 , . . . ],

§ 1. COMPLETE DISCRETE VALUATION RINGS

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one has (ϕ0 , ϕ1 , . . . ) = w−1 (Φ(w0 (X0 ), w0 (Y0 )), Φ(w1 (X0 , X1 ), w1 (Y0 , Y1 )), . . .). This furnishes existence and uniqueness of such polynomials ϕ0 , ϕ1 , . . ., with coefficients in Z[1/p]. The subtle point is that the polynomials so obtained have coefficients in Z; for a proof, we refer to Bourbaki (2006, chapter IX, §1, no 3) or to Serre (1968, Chapter II, §6, theorem 5). (1.1.3). — We apply Theorem 1.1.2 in two special cases. If Φ is the polynomial X + Y , then we denote by Sn the corresponding polynomial ϕn , for every n. Similarly, if Φ is the polynomial XY , then we denote by Pn the polynomial ϕn . Example 1.1.4. — We have S0 = X0 + Y0 , P0 = X0 Y0 ,

X0p + Y0p − (X0 + Y0 )p , p P1 = X0p Y1 + Y0p X1 + pX1 Y1 .

S1 = X1 + Y1 +

The expressions become quite complicated for higher values of n. To get an idea of their complexity, the reader is invited to compute the polynomials S2 and P2 by himself. (1.1.5). — Let A be a ring. For every integer n > 0, we denote by Wn (A) the set An endowed with the operations (a0 , . . . , an−1 ) + (b0 , . . . , bn−1 ) = (S0 (a0 , b0 ), . . . , Sn−1 (a0 , b0 , . . . , an−1 , bn−1 )), (a0 , . . . , an−1 ) · (b0 , . . . , bn−1 ) = (P0 (a0 , b0 ), . . . , Pn−1 (a0 , b0 , . . . , an−1 , bn−1 )).

Similarly, we denote by W (A) the set AN endowed with the operations (a0 , a1 , . . .) + (b0 , b1 , . . .) = (S0 (a0 , b0 ), S1 (a0 , b0 , a1 , b1 ), . . .), (a0 , a1 , . . .) · (b0 , b1 , . . .) = (P0 (a0 , b0 ), P1 (a0 , b0 , a1 , b1 ), . . .). Theorem 1.1.6. — Let A be ring. a) For every ring A and every integer n > 0, the addition + and multiplication · define a ring structure on Wn (A), resp. on W (A). The neutral element for the addition is (0, . . . , 0), resp. (0, 0, . . .), and the neutral element for the multiplication is (1, 0, . . . , 0), resp. (1, 0, 0, . . .). b) For every pair (m, n) of integers such that m  n  1, the truncation maps Wm (A) → Wn (A), W (A) → Wn (A),

(a0 , . . . , am−1 ) → (a0 , . . . , an−1 ) (a0 , a1 , . . .) → (a0 , . . . , an−1 )

are morphisms of rings. The corresponding morphism W (A) → lim Wn (A) ←− n1

is an isomorphism. c) For every morphism of rings ϕ : A → B, the canonical maps Wn (A) → Wn (B) and W (A) → W (B) are morphisms of rings.

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Proof. — We first observe that for every morphism of rings A → B, the maps W (A) → W (B) and Wn (A) → Wn (B) obtained by functoriality are compatible with the laws + and ·, because they are defined by polynomials in the coordinates with Z-coefficients. Since the polynomials S0 , . . . , Sn−1 only depend on the variables Xi and Yi for i < n, the map W (A) → Wn (A) is compatible with the laws + and · as well. Also, by definition of the polynomials Sn and Pn , the map w : W (A) → AN ,

(a0 , a1 , . . .) → (w0 (a0 ), w1 (a0 , a1 ), . . .)

respects addition and multiplication. We have observed that w is bijective when p is invertible in A. In this case, the two laws + and · endow W (A) with the unique ring structure for which the map w is an isomorphism of rings; its neutral elements are as stated in the theorem. Applying Theorem 1.1.2 to the polynomial Φ(X, Y ) = −X, we also see that we can express the coordinates of the additive inverse of (a0 , a1 , . . .) in W (A) as polynomials in the coordinates a0 , a1 , . . . with coefficients in Z. Assume that A is a subring of a ring B in which p is invertible. By what precedes, W (B) is a ring, and W (A) is a subset of W (B) which is closed under addition, multiplication, and additive inversion and contains the neutral elements for addition and multiplication. It follows that W (A) is a subring of W (B). In the general case, let us write the ring A as the quotient of a polynomial ring B = Z[(Ti )i∈I ] with integer coefficients; since B is a subring of the ring Q[(Ti )i∈I ], we know that W (B) is a ring. Observe that the canonical map W (B) → W (A) deduced from the projection B → A is surjective and is compatible with the laws + and · on W (B) and W (A). Consequently, these laws endow W (A) with the structure of a ring, the neutral elements being as in the statement of the theorem. A similar argument applies to the rings Wn (A). Definition 1.1.7. — For every ring A and every integer n > 0, we call Wn (A) the ring of p-typical Witt vectors of length n with coefficients in A, and we call W (A) the ring of p-typical Witt vectors with coefficients in A. Remark 1.1.8. — Although it is not indicated in the notation, the rings Wn (A) and W (A) depend on the choice of the prime number p. By their very construction, the p-typical Witt polynomials induce morphisms of rings Wn (A) → An , for every n ∈ N, and W (A) → AN . If p is invertible in A, the proof of Theorem 1.1.6 shows that these morphisms are isomorphisms, so that the structure of Wn (A) and AN is of interest mainly in the case where p is not invertible in A. In the sequel, we will mostly use the Witt vectors in the case where A is an Fp -algebra. Then the p-typical Witt vectors will simply be called Witt vectors, and the choice of the prime number p will be tacitly made.

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215

Example 1.1.9. — It follows from Example 1.1.4 that the map A → W1 (A),

a → (a)

is an isomorphism of rings. Remark 1.1.10. — Let Φ ∈ Z[X] be a polynomial, and let (ϕn )n∈N be the corresponding sequence of polynomials furnished by Theorem 1.1.2. Let A be a ring. For every a = (am )m 0, the kernel of the truncation morphism W (A) → Wm (A) is the ideal generated by pm . This is false if A = Ap .

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1.2. Complete Discrete Valuation Rings and Their Extensions (1.2.1). — Let R be a discrete valuation ring. Let K be its quotient field, let m be its unique maximal ideal, and let k = R/m be its residue field. The ideal m is principal, and every generator of m is called a uniformizer in R. The valuation of R associates to any nonzero element a ∈ R the largest integer n ∈ N such that a ∈ mn ; in particular, units are elements of valuation 0, and uniformizers are elements of valuation 1. We will denote the valuation of a by ord(a). We say that R has equal characteristic if the characteristics of the fields k and K coincide; otherwise, we say that R has mixed characteristic. If p denotes the characteristic of the field k, we also say that R has equal characteristic p or mixed characteristic (0, p), accordingly. Assume that R has mixed characteristic (0, p). Then p = 0 in R, and the valuation of p, the largest integer e  0 such that p ∈ me , is called the absolute ramification index of R. Since the residue characteristic of R is p, one has e  1. We say that R is absolutely unramified if e = 1 and absolutely ramified otherwise. Observe that R is absolutely unramified if and only if p is a uniformizer in R. of 0 for a The family of ideals (mn )n0 is a basis of open neighborhoods  ring topology on R, called the m-adic topology. Since n0 mn = 0, this topology is separated. Moreover, R is complete if and only if the natural morphism  = lim R/mn R→R ←− n→∞

is an isomorphism.  is a discrete valuation ring, called the (separated) completion The ring R  of  = lim m/mn , and the image in R of R. Its maximal ideal is the ideal m ←−  any uniformizer of R is a uniformizer of R. For every integer n, the canonical  m   n is an isomorphism. In particular, the ring R morphism from R/mn to R/ is a complete discrete valuation ring. Example 1.2.2. — Here are two basic examples of complete discrete valuation rings to keep in mind. a) Let k be a field and let R be the ring k[[t]] of formal power series with coefficients in k. Then R is a complete discrete valuation ring of equal characteristic with residue field k, and t is a uniformizer in R. b) Let p be a prime number and let k be a perfect field of characteristic p. Then the ring R = W (k) of Witt vectors with coefficients in k is a complete discrete valuation ring of mixed characteristic (0, p); its residue field is k and p is a uniformizer in R. It is thus absolutely unramified. These properties are proven in Bourbaki (2006, Chapter IX, §1, no 8, prop. 8); let us recall the argument. It follows from (1.1.14) that the ideal generated by p in W (k) coincides with the image V (W (k)) of the Verschiebung map. This is also the kernel of the truncation morphism W (k) → W1 (k) = k.

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Consequently, the ideal (p) is maximal. More generally, the kernel of the surjective ring morphism W (k) → Wn (k) is the ideal generated by pn for every n  0. Since W (k) is the projective limit of the rings Wn (k), it is separated and complete with respect to the p-adic topology. In particular, the ideal (p) is the unique maximal ideal of W (k). It is also clear from (1.1.14) that p is not a zero-divisor in W (k). Now it is easy to see that we can write every nonzero element of W (k) in a unique way as the product of a power of p with a unit. It follows that W (k) is a discrete valuation ring. (1.2.3). — Let R be a complete discrete valuation ring with fraction field K and residue field k. An extension of R is a complete discrete valuation ring R endowed with an injective local morphism R → R . Such an extension is finite if the morphism R → R is a finite ring morphism, that is to say, if R is a finitely generated R-module. In that case, R is a free R-module of finite rank, the fraction field K  of R is a finite extension of K, and R is the integral closure of R in K  . Conversely, for every finite extension L of K, the integral closure of R in L is a complete discrete valuation ring and a finite extension of R (Serre 1968, II, §3, prop. 3). The degree [R : R] of a finite extension R → R is the degree [K  : K] of the corresponding extension of fraction fields. It is equal to the rank of R as a free module over R. Let R be an extension of R. We denote by m the maximal ideal of R and by k  its residue field. Then the morphism R → R induces a field extension k → k  . The ramification index of R over R is the largest integer e > 0 such that mR is contained in (m )e . If e = 1, then every uniformizer in R is also a uniformizer in R . The extension R → R is called unramified if e = 1 and k  is a separable field extension of k. The extension is called totally ramified if k → k  is an isomorphism. If R → R is finite, then k  is a finite extension of k. In that case, one has (1.2.3.1)

[R : R] = e · [k  : k];

see Serre (1968, II, §3, cor. 1). For every finite extension R → R such that k  is separable over k, there exists a unique subextension R → R0 such that R → R0 is unramified and R0 → R is totally ramified; see Serre (1968, III, §5, corollary 3). We say that a finite extension L of K is unramified, resp. totally ramified, if this property holds for the integral closure R of R in L. The ramification index of L over K is defined to be the ramification index of R over R. (1.2.4). — For every finite separable extension k  of k, there exists a finite unramified extension R → R such that k  is the residue field of R and such that R → R induces the given extension k → k  . The extension R is unique up to unique R-isomorphism (Serre 1968, III, §5, theorem 2). If R = k[[t]], then R  k  [[t]]. If k is perfect and R = W (k), then R  W (k  ).

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Lemma 1.2.5. — Let us assume that the characteristic of K is a prime number p. Let K  be a finite extension of K such that (K  )p ⊂ K, and let R be the integral closure of R in K  . For every finite unramified extension S of R, the ring S ⊗R R is an unramified extension of R . Proof. — We may assume that R = k[[t]]. Let kS be the residue field of S; it is a separable extension of k by assumption. Moreover, t is a uniformizer in S, so that we may assume that S = kS [[t]]. Let then k  be the residue field of R and u be a uniformizer of R . Since K  is purely inseparable over K, k  is purely inseparable over k. Since the extension kS /k is separable, the tensor product kS = kS ⊗k k  is a field and is a separable extension of kS . The lemma follows, because S ⊗R R  kS [[u]]. (1.2.6). — The totally ramified extensions of R are described as follows. Let R be a finite totally ramified extension of R, of degree e > 0, and choose a uniformizer π  in R . There exists a unique morphism of R-algebras f : R[T ] → R that sends T to π  . This morphism f is surjective. It is proved in Serre (1968, I, §6, prop. 18) that the kernel of f is generated by an Eisenstein polynomial of degree e, that is, a polynomial P ∈ R[T ] of the form P = T e + a1 T e−1 + · · · + ae−1 T + ae where a1 , . . . , ae ∈ m and ae ∈ / m2 . Conversely, for every Eisenstein polynomial Q in R[T ] of degree e > 0, the R-algebra R[T ]/(Q) is a finite totally ramified extension of R of degree e, and the class of T is a uniformizer; see Serre (1968, I, §6, prop. 17). (1.2.7). — A finite extension K  of K is called tame if its ramification index is not divisible by the characteristic p of k and if the corresponding extension of residue fields k → k  is separable. Let us assume that k is separably closed. In this case, the tame extensions of K can be described explicitly. Let K s be a separable closure of K and fix a uniformizer π ∈ K. First observe that the degree of a tame extension of K is prime to p, since it is equal to the ramification index. Conversely, let d be a positive integer which is not divisible by p, and let K (d) be the subfield of K s obtained by adjoining a d-th root π 1/d of the uniformizer π. Since k is separably closed and d is prime to p, it follows from the Hensel lemma (see 1/1.3.2) that K contains all d-th roots of unity, so that the extension K (d) /K is Galois; the classical Kummer map σ → σ(π 1/d )/π 1/d identifies its Galois group with the group μd (K) of d-th roots of unity in K. It follows from Cassels and Fröhlich (1986, I.8, prop. 1) that K (d) is the unique extension of K of degree d contained in K s . The union of all the finite tame extensions of K in K s is a subfield of s K , called the tame closure of K in K s and denoted by K t . This is an infinite Galois extension, and its Galois group Gal(K t /K) is isomorphic to

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221

the procyclic group μ(p) (k) =

lim μd (k) ←−

d>0, pd

where the indices d are ordered by divisibility and the transition morphisms in the projective system are given by μde (k) → μd (k),

x → xe .

1.3. The Structure of Complete Discrete Valuation Rings Let R be a complete discrete valuation ring with residue field k. (1.3.1). — Assume that R has equal characteristic. A field of representatives for R is a subfield κ of R such that the restriction of the projection morphism R → k defines an isomorphism κ → k. By Theorem 1 of Bourbaki (2006, chapter IX, §3, no 3), a field of representatives always exists. The choice of a field of representatives determines a ring morphism k → R whose composition with the reduction morphism R → R/m = k is the identity on k. If we also choose a uniformizer π in R, then we can define a morphism of k-algebras k[[t]]→ R,

t → π

which is an isomorphism by Theorem 2 of Bourbaki (2006, chapter IX, §3, no 3). Thus every complete discrete valuation ring of equal characteristic is (non-canonically) isomorphic to a ring of formal power series in one indeterminate. If k is a perfect field of characteristic p > 0, then R has a unique field of representatives, namely, the subring consisting of all the elements that have a pn -th root for every n > 0. (1.3.2). — Assume that R has mixed characteristic and that k is perfect. Then we can also give an explicit description of R in terms of the ring of Witt vectors W (k). By proposition 8 in Serre (1968, II, §5), there exists a unique multiplicative section τ : k → R of the reduction morphism R → k. The image τ (a) of an element a ∈ k is called the Teichmüller representative of a, and τ is called the Teichmüller map. Example 1.3.3. — a) Let us assume that k is a finite field. Let q = Card(k) and let P be the polynomial T q − T . Every element a ∈ k is a root of P , and P  (a) = −1. It thus follows from Hensel’s lemma that there exists a unique root a ˜ of P in R whose residue class in k is equal to a. Consequently, one has τ (a) = a ˜. b) The map τ : k → W (k) from (1.1.13.1) is the Teichmüller map of W (k). Theorem 1.3.4. — Let k0 be a perfect field of characteristic p > 0. Let R be a complete discrete valuation ring of mixed characteristic, let k be its residue field, and let ϕ : k0 → k be a morphism of fields. There exists a unique

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injective local homomorphism W (k0 ) → R which induces the morphism ϕ on residue fields. It is given explicitly by  −n τ (ϕ(apn p))pn . (a0 , a1 , . . .) → n0

a) The extension W (k0 ) → R is finite if and only if k is a finite extension of k0 . b) Assume that k is a finite extension of k0 . Then, the ramification index of the extension W (k0 ) → R is equal to the absolute ramification index of R. c) If k0 → k is an isomorphism and R is absolutely unramified, then W (k0 ) → R is an isomorphism. Proof. — This follows from Proposition 10 (and its proof) in Serre (1968, chapter II) and the results in section 1.2. In particular, W (k) is the unique absolutely unramified complete discrete valuation ring of mixed characteristic with residue field k. Example 1.3.5. — There exists a unique isomorphism W (Fp ) → Zp . It is given explicitly by (a0 , a1 , . . .) →



τ (an )pn

n0

where τ (an ) is the Teichmüller representative of an in Zp . (1.3.6). — Let R be a complete discrete valuation ring of mixed characteristic, let k be its residue field, and assume that k is perfect. We denote by e the absolute ramification index of R. By Theorem 1.3.4, there exists a unique local homomorphism W (k) → R that induces the identity map on the residue fields. Thus R is a W (k)-algebra in a natural way; it is a finite totally ramified extension of W (k) of degree e. More precisely, let π be a uniformizer in R. By section 1.2.4, there exists an Eisenstein polynomial P ∈ W (k)[T ], of degree e, such that P (π) = 0, inducing an isomorphism of W (k)-algebras W (k)[T ]/(P ) → R,

T → π.

Conversely, for every Eisenstein polynomial Q in W (k)[T ] of degree e > 0, the ring W (k)[T ]/(Q) is a complete discrete valuation ring whose residue field is k and such that the class of T is a uniformizer; its absolute ramification index is equal to e. In this way, we obtain a precise description of all complete discrete valuation rings of mixed characteristic with perfect residue field. If k  is a finite extension of k, then this description implies that W (k  ) ⊗W (k) R is a finite unramified extension of R with residue field k  . We have seen in (1.2.4) that such an extension is unique up to unique R-isomorphism. The following lemma describes the truncations R/mn+1 as W (k)-modules; it will lead to the definition of the ring schemes Rn in section 2.

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Lemma 1.3.7. — Let R be a complete discrete valuation ring of mixed characteristic, let π be a uniformizer in R, and let k be the residue field of R. We assume that k is perfect and we denote by p its characteristic. Let e be the absolute ramification index of R. Let n  0 be an integer; define integers q and r by n + 1 = qe + r, where 0  r < e; let also = min(e − 1, n). Then the canonical morphism of W (k)-modules ϕ : W (k)+1 → R/mn+1 ,

(x0 , . . . , x ) → x0 + x1 π + · · · + x π 

is surjective; its kernel is given by Ker(ϕ) = pq+1 W (k)r ⊕ pq W (k)+1−r . Proof. — One has R = W (k)[π]. Since the minimal polynomial of π in W (k)[T ] has degree e, the ring R is generated by (1, π, . . . , π e−1 ) as a W (k)-module. Since = e − 1 or π +1 ∈ mn+1 , we see that the ring R/mn+1 is generated by (1, π, . . . , π  ) as a W (k)-module. Consequently, the canonical morphism of W (k)-modules ϕ : W (k)+1 → R/mn+1 ,

(x0 , . . . , x ) → x0 + x1 π + · · · + x π 

is surjective. Let u, i be nonnegative integers such that 0  i < e. Since n + 1 = qe + r and 0  r < e, we observe that pu π i ∈ mn+1 if and only if either u  q + 1, or u = q and i  r. In particular, the kernel of ϕ contains pq+1 W (k)r ⊕ pq W (k)+1−r , so that ϕ induces a surjective homomorphism ψ : Wq+1 (k)r ⊕ Wq (k)+1−r → R/mn+1 . It remains to prove that ψ is an isomorphism. To that aim, it suffices to show that the length of its source, which is given by (q + 1)r + q( + 1 − r) = q( + 1) + r, is equal to n + 1, the length of its target. If = e − 1, this follows from the definition of the integers q and r. Otherwise, = n < e − 1, so that n + 1 < e, in which case q = 0 and r = n + 1, and the desired equality holds as well. (1.3.8). — With the hypotheses and notation of Lemma 1.3.7, the canonical commutative diagram (1.3.8.1)

W (k)r ⊕ W (k)+1 −r

(x 0 ,...,x  )

Wq +1 (k)r ⊕ Wq (k)+1 −r



xi π i

R

R/mn +1 .

will be called the standard presentation of the ring R/mn+1 .

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Observe that when n + 1 < e, then q = 0, r = n + 1, and = n, so that the standard presentation reduces to a diagram of W (k)-modules: (1.3.8.2)

(x0 ,...,xn )

W (k)n+1

W1 (k)n+1

xi π i

R

R/mn+1 .

(1.3.9). — Let us compare the standard presentation of the ring R/mn+2 with that of R/mn+1 . As in the statement of Lemma 1.3.7, let = min(e − 1, n) and write n + 1 = qe + r. Similarly, let  = min(e − 1, n + 1) and write n + 2 = q  e + r . If 0  r < e − 1, one has q  = q and r = r + 1; otherwise, r = e − 1, q  = q + 1 and r = 0. If n  e − 1, then  = e − 1 = , so that the two standard presentations fit in the commutative diagram W (k)e

R

Wq+1 (k)r+1 Å Wq (k)e−r−1

R/mn+2

Wq+1 (k)r Å Wq (k)e−r

R/mn+1 ,

where the vertical map in the lower left corner is induced by the truncation map Wq+1 (k) → Wq (k) on the (r + 1)-th summand and the identity map on all other summands. On the other hand, if n  e−2, then  = n+1 = +1. Then the standard presentations give rise to the following diagram W (k)n+2

R

W1 (k)n+2



R/mn+2

W1 (k)n+1



R/mn+1 ,

where the vertical map in the lower left corner forgets the last summand.

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225

§ 2. THE RING SCHEMES Rn Let R be a complete discrete valuation ring, let m be its maximal ideal, and let k be its residue field. If R has mixed characteristic, we assume moreover that k is perfect. For every integer n  0, we put Rn = R/mn+1 . The aim of this section is to construct a projective system (Rn )n0 of ring schemes over k such that Rn (k) = Rn possessing good functorial properties with respect to extensions of k.

2.1. Construction: The Equal Characteristic Case (2.1.1). — We assume that R has equal characteristic. We choose a section of the surjection R → k, endowing the ring R with the structure of a kalgebra. Our constructions will depend on this choice. (2.1.2). — For every integer n  0, we consider the functor Rn : A → A ⊗k Rn from the category of k-algebras to the category of rings. Let t ∈ R be a uniformizer. It gives rise to an isomorphism Rn  k[t]/t(n+1) , hence to functorial isomorphisms of rings Rn (A)  A[t]/(tn+1 ). We can canonically identify A[t]/(tn+1 ) with Ln (A1k )(A), the set of A-valued jets of level n in the affine line. Thus the functor Rn is represented by . Observe that the isomorphism of functors the scheme Ln (A1k )  An+1 k Rn  Ln (A1k ) depends on the choice of t. The ring structure on the functor Rn gives rise to addition and multiplication morphisms on the scheme Ln (A1k ) that turn it into a ring scheme. They are given explicitly by the morphisms of k-algebras + : Spec(k[X0 , . . . , Xn , Y0 , . . . , Yn ]) → Spec(k[X0 , . . . , Xn ]), Xi → Xi + Yi

for all i,

· : Spec(k[X0 , . . . , Xn , Y0 , . . . , Yn ]) → Spec(k[X0 , . . . , Xn ]), Xi →

i 

Xj Yi−j

for all i.

j=0

(2.1.3). — For every pair (m, n) of integers such that m  n  0, reduction modulo mn+1 defines a truncation morphism Rm → Rn . It induces a truncation morphism of k-ring schemes Rm → Rn . Let t ∈ R be an uniformizer. Under the above identification of Rn with Ln (A1k ), this truncation morphism m 1 1 is identified with the truncation morphism θn,A 1 : Lm (Ak ) → Ln (Ak ). k

(2.1.4). — The limit of the projective system (Rn )n0 of functors,  k R := lim (A ⊗k Rn ). A → A⊗ ←− n0

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is a k-ring scheme R∞ . Every uniformizer t ∈ R gives rise to functorial isomorphisms R∞ (A)  A[[t]], hence to an isomorphism of set-valued functors R∞  L∞ (A1k )  AN k , the arc scheme of the affine line over k.

2.2. Construction: The Mixed Characteristic Case (2.2.1). — Let us now assume that R has mixed characteristic (0, p) and that the residue field k is perfect. Then R has a canonical structure of a W (k)-algebra (Theorem 1.3.4), and R is a free W (k)-module of finite rank. Let e = [R : W (k)] be the absolute ramification index of R. (2.2.2). — Assume that R is absolutely unramified, so that e = 1 and R  W (k). In this case, we put Rn = Wn+1 ⊗Z k for every integer n  0 (mind the shift in indexation) and R∞ = W∞ ⊗Z k. (2.2.3). — When e is arbitrary, the most obvious attempt to generalize the construction of Rn leads to the following definitions. First consider the functor (2.2.3.1)

"n : Alg → Rings, R k

A → W (A) ⊗W (k) Rn

from the category of k-algebras to the category of rings. Alternatively, for every integer m such that me  n + 1 (so that pm Rn = 0), define a functor  R n,m by (2.2.3.2)

 R n,m : Algk → Rings,

A → Wm (A) ⊗W (k) Rn .

The natural truncation morphisms W (A) → Wt (A) → Wm (A) for t  m give rise to morphisms of functors "n → R + Ψn,t : R n,t

and

+  Ψtn,m : R n,t → Rn,m .

One has Ψn,m = Ψtn,m ◦ Ψn,t . "n and Example 2.2.4. — It is not true in general that these functors R  R n,m are representable by schemes. Recall that for every scheme X and every field extension L → L , the natural map X(L) → X(L ) is injective. We will construct such an extension "2 (L) → R "2 (L ) and R + +  so that the morphisms R 2,2 (L) → R2,2 (L ) are not injective. We assume that k has characteristic 2 and let R = W (k)[π]/(π 2 − 2).

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227

The ring R is a complete discrete valuation ring with residue field k and absolute ramification index e = 2 (see section 1.3.6). We have π 3 = 2π so that "2 (A) = W (A) ⊗W (k) R2 = W (A)[π]/(π 2 − 2, 2π) R and 2 + R 2,2 (A) = W2 (A) ⊗W (k) R2 = W2 (A)[π]/(π − 2, 2π) for every k-algebra A. Let L be the field k(t) and denote by L its perfect closure. Let ξ = (0, t, 0, . . . ) ∈ W (L), let ξ2 be its image (0, t) in W2 (L), and let ξ  and ξ2 be their images in W (L ) and W2 (L ). By the explicit formulas given in section 1.1.14, the element ξ does not belong to the ideal 2 · W (L) of W (L), and ξ2 does not belong to 2 · W2 (L), because t is not a square in L. However, their images ξ  and ξ2 belong to 2 · W (L ) and 2 · W2 (L ), respectively. It follows that the map "2 (L) = W (L)[π]/(π 2 − 2, 2π) → W (L )[π]/(π 2 − 2, 2π) = R "2 (L ) R sends the nonzero element ξπ to ξ  π = 0. In particular, this map is not injective. Similarly, the map 2  2 + +  R 2,2 (L) = W2 (L)[π]/(π − 2, 2π) → W2 (L )[π]/(π − 2, 2π) = R2,2 (L )

maps ξ2 π to ξ2 π = 0, so is not injective. It is also worth pointing out that the morphisms Ψ2,2 (L) and Ψ32,2 (L) are not injective: if we set η = (0, 0, t, 0, . . . ) ∈ W (L), then the image of πη in + + R 2,3 (L) is nonzero, whereas its image in R2,2 (L) vanishes. (2.2.5). — Let π be a uniformizer in R. Let n  0 be an integer. Let = min(e − 1, n), and define integers q and r by n + 1 = qe + r, where 0  r < e. We set n0 = · · · = nr−1 = q + 1 and nr = · · · = n = q. By Lemma 1.3.7, the choice of π determines a standard presentation of Rn of the form ∼

Wn0 (k) ⊕ · · · ⊕ Wn (k) − → Rn ,

(x0 , . . . , x ) → x0 + x1 π + · · · + x π  .

The operations of addition and multiplication on Rn can be expressed by polynomials with k-coefficients in the coordinates of the Witt rings Wni (k). These polynomials define a ring scheme structure on r RnGr = Wn0 ×k · · · ×k Wn = Wq+1 ×k Wq+1−r ,

which is different from the product ring structure, in general. It is the unique ring scheme structure such that, for every k-algebra A, the map of W (A)modules Φn (A) : W (A) ⊗W (k) Rn → RnGr (A) = Wn0 (A) ⊕ · · · ⊕ Wn (A) that sends π i to (1, 0, . . . , 0) ∈ Wni (A) for i = 0, . . . , , is a ring morphism. The underlying scheme of RnGr is isomorphic to the affine space An+1 , and the k standard presentation of Rn can be interpreted as an isomorphism RnGr (k) → Rn .

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"n → The morphisms Φn (A) define a morphism of ring functors Φn : R Similarly, for every integer m  n0 and every k-algebra A, we have a morphism Wm (A) → RnGr (A), and these define a morphism of ring functors Gr  Φn,m : R n,m → Rn . RnGr .

Lemma 2.2.6. — Let A be a k-algebra. Let m, t be integers such that t  m  n0 . a) The morphisms Φn (A), Φn,m (A), Ψtn,m (A), and Ψn,m (A) are surjective. b) If Ap = A, these morphisms are isomorphisms. Proof. — Point (a) is clear from the construction of these morphisms. So let us now assume that Ap = A. In this case, pm generates the kernel of the canonical morphisms W (A) → Wm (A) and Wt (A) → Wm (A). Since pm vanishes in Rn because m  n0 , it follows at once that all the morphisms in the statement are isomorphisms. (2.2.7). — Let k be a field and let F be a presheaf on the category of affine k-schemes, equivalently, a covariant functor F : Algk → Sets from the category of k-algebras to the category of sets. Let A be a k-algebra and let B be an A-algebra. Let i : A → B be the natural morphism. There are two natural morphisms j1 , j2 : B → B ⊗A B, respectively, given by b → b ⊗ 1 and b → 1 ⊗ b, and one has j1 ◦ i = j2 ◦ i. The functor F then induces a diagram F (B Ä A B) F (j1 )

F (A)

F (i)

F (B) F (j2 )

F (B Ä A B)

One says that F is a fpqc-sheaf if for every k-algebra A and every faithfully flat A-algebra B, the map F (i) is injective and its image is the equalizer of the two maps F (j1 ) and F (j2 ). Every presheaf F has an fpqc-sheafification: there exists a morphism ϕ : F → F  from F to an fpqc-sheaf F  such that, for every fpqc-sheaf G and every morphism of functors f : F → G, there exists a unique morphism of functors f  : F  → G such that f = f  ◦ ϕ. A fundamental theorem of Grothendieck asserts that the functor associated with a k-scheme is an fpqc-sheaf. Definition 2.2.8. — For every integer n  0, we define Rn to be the "n with respect to the fpqc-topology on the category Alg . sheafification of R k Similarly, for every integer n  0 and every integer m  (n + 1)/e, we define  Rn,m as the sheafification of R n,m with respect to the fpqc-topology.

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(2.2.9). — By functoriality of the construction of the fpqc-sheaf associated with a presheaf on the opposite category of k-algebras, the canonical mort "n → R +   phisms Ψn,m : R n,m and Ψn,m : Rn,t → Rn,m , for t  m, give rise to morphisms of fpqc-sheaves Rn → Rn,m and Rn,t → Rn,m , which we will still Gr denote by Ψn,m and Ψtn,m . Since Rm is a scheme, and thus an fpqc-sheaf, Gr "  the morphisms of presheaves Φn : Rn → RnGr and Φn,m : R n,m → Rn induce Gr Gr morphisms of sheaves Φn : Rn → Rn and Φn,m : Rn,m → Rn . Theorem 2.2.11 will show that these morphisms of fpqc-sheaves are isomorphisms. In particular, the functor Rn is represented by the ring scheme RnGr . In order to prove this statement, we will need the following lemma. Lemma 2.2.10 (Lipman 1976, lemma 0.1). — Let k be a perfect field of characteristic p > 0, and let A be a k-algebra. There exists a faithfully flat A-algebra B such that B p = B. Proof. — Let (ai )i∈I be a family of elements of A that generates A as a kalgebra. Let A0 be the polynomial ring k[(Ti )i∈I ], and let f : A0 → A be the unique homomorphism of k-algebras such that f (Ti ) = ai for every i ∈ I; it is surjective. Q Let B0 be the ring of Puiseux polynomials k[(Ti 0 )i∈I ], that is, the k(I) algebra associated with the monoid Q0 . Since k is perfect, the pth power map issurjective on B0 . Moreover, B0 is a free A0 -module, since the monomials i∈I Tiai where 0  ai < 1 for every i ∈ I (and ai = 0 for all but finitely many i) form a basis of B0 as an A0 -module. Now consider the A-algebra B = B0 ⊗A0 A. It is a free as A-module; since it is nonzero, it is faithfully flat. Finally, the property (B0 )p = B0 implies that B p = B. Theorem 2.2.11. — Let R be a complete discrete valuation ring of mixed characteristic (0, p). Let π be a uniformizer of R, and let e be its absolute ramification index. For all integers m, n ∈ N such that m  (n + 1)e, the morphisms Gr are isomorphisms. Φn : Rn → RnGr and Φn,m : Rn,m → Rn,m In particular, the fpqc-sheaves Rn and Rm,n are isomorphic and representable by a k-ring scheme which is isomorphic, as a k-scheme, to An+1 . k "n → R Gr satisProof. — We first prove that the canonical morphism Φn : R n "n . Let F be an fies the universal property of the fpqc-sheaf associated with R "n → F be a morphism of presheaves; we must show fpqc-sheaf, and let α ˜: R that there exists a unique morphism of fpqc-sheaves α : RnGr → F such that α ˜ = α ◦ Φn . Let A be a k-algebra. By Lemma 2.2.10, there exists a faithfully

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flat A-algebra B such that B p = B. This gives rise to a diagram α(A) ˜

Rn (A)

Φn (A)

RnGr (A)

α(A)

Rn (B)

Φn (B)

RnGr (B)

α(B)

F (A)

F (B),

α(B) ˜

where the two dotted arrows are yet to be constructed, the middle vertical arrow identifies RnGr (A) with the equalizer of the two maps from RnGr (B) to RnGr (B ⊗A B), and the rightmost vertical arrow identifies F (A) with the equalizer of the two maps fro F (B) to F (B ⊗A B). Since B p = B, the ˜ ◦ map Φn (B) is a bijection (Lemma 2.2.6, b)), and we may set α(B) = α(B) Φn (B)−1 . Since Φn (A) is surjective, an elementary diagram chase then shows that there exists a unique map α(A) such that the diagram is commutative. It is straightforward to show that α(A) does not depend on the choice of B and that this construction defines a morphism of sheaves α : RnGr → F . By construction, it satisfies α ˜ = α ◦ Φn . Consequently, the morphism Φn "n induces an isomorphism from the fpqc-sheafification Rn of the presheaf R Gr to the fpqc-sheaf defined by the k-scheme Rn . The case of the functor Φn,m is similar. Example 2.2.12. — As illustrated by Example 2.2.4, the sheafification "n → Rn and R  morphisms R n,m → Rn are not isomorphisms in general. However, they may be isomorphisms in some cases. Denote by e the absolute ramification index of R. a) Let us assume that R is absolutely unramified, that is, e = 1. With the notation of Lemma 1.3.7, one has = 0, q = n + 1, and r = 0; hence RnGr = Wn+1 . On the other hand, one has R = W (k) and Rm = Wm (k), "n and R  so that it follows from the definition of the functors R n,m that the Gr "  morphisms Rn → Rn,m → Rn are isomorphisms. b) Let us then assume that n  e − 1. With the notation of Lemma 1.3.7, one has = n, r = n + 1, and q = 0, so that Rn is isomorphic to k[π]/(π n+1 ) as a W (k)-algebra. In this case, for every k-algebra A, one has n+1  ), R n,1 (A) = W1 (A) ⊗W (k) Rn = A ⊗k Rn  A[π]/(π

and this isomorphism is functorial in A. In particular, as a set-valued functor, n+1   , so that R R n,1 is representable by the scheme Ak n,1 is already an fpqcsheaf. c) As a last case, we assume that e divides n + 1 (this also includes the case where e = 1). We now have = e − 1, q = (n + 1)/e and r = 0, and

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231

Rn = Wq (k)[π]/(π e ). For every k-algebra A, one thus has e  R n,q (A) = Wq (A) ⊗W (k) Rn = Wq (A)[π]/(π ),

and the morphism Φn,q is an isomorphism. Remark 2.2.13. — The ring schemes RnGr were introduced by Greenberg (1961). While their definition is simple and explicit, they present the disadvantage that their construction depends on the choice of a uniformizer π. The more intrinsic presentation of the functor Rn as an fpqc-sheafification, due to Lipman (1976), makes the verification of some compatibilities automatic. (2.2.14). — For every k-algebra A and every integer n  0, the structural morphism k → A induces a ring morphism Rn (k) = Rn → Rn (A) that gives Rn (A) the structure of an Rn -algebra. Thus Rn is a k-scheme in Rn -algebras. We will now study how Rn (A) behaves under multiplication with elements of R. Let a ∈ R be any nonzero element and let d = ord(a); multiplication by a induces a group morphism from Rn to Rn+d and a morphism of group "n → R  functors μ (a : R n+d . It gives rise to a morphism of k-group schemes μa : Rn → Rn+d . Lemma 2.2.15. — Let a ∈ R be a nonzero element and let d = ord(a). For every reduced k-algebra A, the morphism of R-modules μa (A) is injective. Proof. — By lemma Proof. — This is clear if a is a unit, for then μa is an isomorphism. Let π be a uniformizer in R; writing a = uπ d , with u ∈ R× , we have μa = μu ◦ μdπ , so that it suffices to treat the case where a = π. To that aim, we will use the isomorphism of Rn with RnGr ; see §2.2.5. Let e be the absolute ramification index of R. Set = min(e − 1, n), and let q and r be defined by n + 1 = qe + r and 0  r < e. Similarly, let  , q  , and r be defined by  = min(e − 1, n + 1), n + 2 = q  e + r , and 0  r < e. First assume that n  e − 2. By Example 2.2.12, b), we can identify μπ (A) with the morphism A[π]/(π n+1 ) → A[π]/(π n+2 ) defined by multiplication with π, which is always injective. Let us then assume that n  e − 1, so that  = e − 1 = . Multiplication by π induces a commutative diagram of R-modules α

W (A) ⊗ W (k) R

W (A)e

Wq+1 (A)r ⊗ Wq (A)e−r

μπ (A)

π

W (A) ⊗ W (k) R

RnGr (A)

W (A)e

Wq+1 (A)r+1 ⊗ Wq (A)e−r−1 β

Gr Rn+1 (A).

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In this diagram, the horizontal morphisms are ring morphisms; the left horizontal ones are defined by the W (k)-module basis {1, π, . . . , π e−1 } of R, the following ones are given by the canonical projections, and the right horizontal morphisms are deduced from the definition of RnGr . Let P = T e +ae−1 T e−1 +· · ·+a0 be the minimal polynomial of π over W (k), so that π e = −ae−1 π e−1 − · · · − a0 . By definition of the ring structure on RnGr (A), we thus have μπ (A) ◦ α(x0 + πx1 + · · · + π e−1 xe−1 ) = β(−a0 xe−1 + π(x0 − a1 xe−1 ) + · · · + π e−1 (xe−2 − ae−1 xe−1 )) for every (x0 , . . . , xe−1 ) ∈ W (A)e . Assume that this expression vanishes. We first have a0 xe−1 = 0 in Wq+1 (A). Recall that P is an Eisenstein polynomial, so that a0 , . . . , ae−1 are divisible by p in W (k) and a0 has p-adic valuation equal to 1. In particular, pxe−1 = 0 in Wq+1 (A). Since p = V F , by equation (1.1.14.2), and V is injective, this implies that F (xe−1 ) = 0 in Wq (A). Since A is reduced, it follows from the definition of F that xe−1 = 0 in Wq (A). Then ai xe−1 = 0 in Wq+1 (A) for every i ∈ {1, . . . , e − 1}, so that 0 = β(0, x0 , . . . , xe−2 ). Consequently, x0 , . . . , xr−1 = 0 in Wq+1 (A) and xr , . . . , xe−1 = 0 in Wq (A), so that α(x0 + πx1 + · · · + π e−1 xe−1 ) = 0. This shows that μπ (A) is injective, as claimed. (2.2.16). — For every pair (m, n) of integers such that m  n  0, the truncation morphism Rm → Rn given by reduction modulo mn+1 induces a morphism of functors R(m → R(n and thus a morphism of k-ring schemes Rm → Rn , which we call the truncation morphism. Fixing a uniformizer π, the standard presentations of Rn+1 and Rn were compared explicitly in §1.3.9. Define integers , q, and r by the relations = min(n, e − 1), n + 1 = eq + r, and 0  r < e, where e is the absolute ramification index of R. When n  e − 1, one has = e − 1, and the truncation morphism Rn+1 → Rn can be identified with the morphism Gr Rn+1

RnGr

r+1 Wq+1 ×k Wqe−r−1

r Wq+1 ×k Wqe−r ,

where the lower horizontal morphism is the truncation morphism Wq+1 → Wq on the (r + 1)-th summand and the identity on all other summands.

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233

Similarly, when n  e − 2, one has = n and the truncation morphism Rn+1 → Rn identifies with the morphism Gr Rn+1

RnGr

An+2 k

An+1 , k

where the lower horizontal morphism is the projection onto the first n + 1 coordinates. In both cases, the truncation Rn+1 → Rn is a linear projection An+2 → An+1 on the level of the underlying schemes. k k (2.2.17). — The limit of the projective system (Rn )n0 of affine ring schemes is representable by an affine k-ring scheme R∞ , whose k-algebra of regular functions O(R∞ ) is the inductive limit of the rings O(Rn ). As a k-scheme, R∞ is isomorphic to AN k . "n (A) = W (A) ⊗W (k) Rn → For every integer n, the tautological map R Rn (A) induces a surjective ring homomorphism W (A) ⊗W (k) R → Rn (A). These morphisms are compatible with the truncation morphisms and furnish a canonical ring homomorphism W (A) ⊗W (k) R → R∞ (A). Proposition 2.2.18. — If A is a k-algebra such that Ap = A, this morphism is an isomorphism. In particular, R∞ (k)  R. Proof. — By construction, the ring W (A) is complete for the V -adic topology. Since Ap = A, the map F is surjective; hence equation (1.1.14.2) implies that the image of [p] coincides with the image of V . Consequently, W (A) is p-adically complete. Since R is a finitely generated W (k)-module, the canonical morphism W (A) ⊗W (k) R → lim W (A) ⊗W (k) Rn ← − n is then an isomorphism. On the other hand, for each integer n  0, the hypothesis Ap = A and Lemma 2.2.6, b) imply that the canonical morphism W (A) ⊗W (k) Rn → Rn (A) is an isomorphism. By composition, we thus obtain an isomorphism W (A) ⊗W (k) R → lim W (A) ⊗W (k) Rn → lim Rn (A) = R∞ (A), ← − ← − n n which coincides with the morphism considered in the statement of the proposition.

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2.3. Basic Properties of the Ring Schemes Rn (2.3.1). — We first recall the assumptions and notation from the previous sections. Let R be a complete discrete valuation ring, let k be its residue field, and let m be its maximal ideal. For every integer n  0, we let Rn = R/mn+1 . If R has equal characteristic, we fix a ring section of the canonical projection R → k. If R has mixed characteristic, we assume that k is perfect. We have constructed a projective system (Rn )n0 of k-ring schemes whose limit is the k-ring scheme R∞ . There are canonical isomorphisms of rings R∞ (k)  R and Rn (k)  Rn for all integers n  0. For every integer n, the morphism R∞ (k) → Rn (k) identifies with the reduction morphism R → Rn . For all integers m, n such that m  n, the morphism Rm (k) → Rn (k) identifies with the reduction morphism Rm → Rn . The ring scheme R0 is canonically isomorphic to the ring scheme A1k with the usual addition and multiplication. The choice of a uniformizer π in R determines isomorphisms of k-schemes for every n  0, under which the truncation morphisms corRn → An+1 k → An+1 , as well as an isomorphism respond to linear projections Am+1 k k N R∞ → Ak = Spec(k[X0 , X1 , . . .]). For every k-algebra A, the map Rn (k) → Rn (A) endows the ring Rn (A) with a structure of an Rn -algebra, and the map R∞ (k) → R∞ (A) endows R∞ (A) with a structure of an R-algebra. Proposition 2.3.2. — Let k  be a field extension of k. If R has mixed characteristic, we assume that k  is perfect. a) The ring R(k  ) is a complete discrete valuation ring with residue field k  , and the canonical morphism R(k) → R(k  ) is an extension of ramification index one. b) Assume that k  is a separable extension of k. Let R be a complete discrete valuation ring with residue field k  , and let R → R be an unramified extension that induces the given field extension k → k  . Then there exists a unique morphism of R-algebras f : R∞ (k  ) → R which induces the identity morphism on the residue fields. This morphism f is an isomorphism. Proof. — First assume that R has equal characteristic, and fix an isomorphism of k-algebras R  k[[t]]. By construction, R(A) = A[[t]] for every k-algebra A. In particular, R(k  ) is isomorphic to k  [[t]], which proves a). With the notation of b), the ring R has equal characteristic as well, and t is still a uniformizer. Since k  is a separable extension of k, it follows from Proposition 1 of Bourbaki (2006, chapter IX, §3, no 3) that we can choose a field of representatives in R that contains the field k. Consequently, R is isomorphic to k  [[t]] as an R-algebra. The only endomorphism of the R-algebra k  [[t]] that induces the identity on the residue field is the identity on k  [[t]]

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235

(note that such an endomorphism is automatically continuous with respect to the t-adic topology, since it sends t to itself). Suppose now that R has mixed characteristic. By Proposition 2.2.18, we have R∞ (k  ) = W (k  ) ⊗W (k) R. Let π be a uniformizing element of R, and let P ∈ W (k)[T ] be its minimal polynomial. By §1.3.6, P is an Eisenstein polynomial; since the extension W (k) → W (k  ) has ramification index 1, the polynomial P remains Eisenstein when viewed as a polynomial in W (k  )[T ]. Writing R∞ (k  )  W (k  ) ⊗W (k) (W (k)[T ]/(P ))  W (k  )[T ]/(P ). we deduce from §1.3.6 that R∞ (k  ) is a complete discrete valuation ring, and that the image of π is a uniformizing element. Consequently, the canonical morphism R∞ (k) → R∞ (k  ) is an extension of complete discrete valuation rings of ramification index 1. This proves a). By Theorem 1.3.4, the natural square W (k )

R

W (k 9 )

R9

commutes. This yields a finite injective local homomorphism of complete discrete valuation rings f : R∞ (k  ) = W (k  ) ⊗W (k) R → R which must be an isomorphism since it has ramification index one and it induces an isomorphism on the residue fields. On the other hand, Theorem 1.3.4 implies that any R-morphism W (k  ) ⊗W (k) R → R inducing the identity map on the residue fields coincides with f on W (k  ) and, hence, is equal to f . Corollary 2.3.3. — Let k  be a field extension of k. If R has mixed characteristic, we assume that k  is perfect. Put R = R∞ (k  ). Then we can  associated with R . There are natural form the ring schemes Rn and R∞  isomorphisms of k -ring schemes: ∼

Rn ⊗k k  − → Rn for every integer n  0, and ∼

 R∞ ⊗k k  − → R∞ .

These isomorphisms are compatible with the truncation morphisms. Proof. — Let n be a nonnegative integer. First, assume that R has equal characteristic. We can identify Rn with Rn ⊗k k  , since R has ramification index one over R by Proposition 2.3.2. For every k  -algebra A, this yields a canonical isomorphism of rings A ⊗k Rn ∼ = A ⊗k Rn that is compatible with the truncation maps. The k  -ring scheme Rn ⊗k k  represents the functor A → Rn (A) on the category of k  -algebras. Hence, by the Yoneda lemma,

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we get an isomorphism of k-ring schemes Rn ⊗k k  → Rn that commutes with the truncation morphisms. Taking the projective limit, we also get an ∼  → R∞ . isomorphism R∞ ⊗k k  − In the mixed characteristic case, we can apply a similar argument. For every k  -algebra A, we have a canonical isomorphism of rings W (A) ⊗W (k ) Rn ∼ = W (A) ⊗W (k) Rn that is compatible with the truncation maps. Hence, " n and the restriction of R(n to we get an isomorphism between the functor R  the category of k -algebras. Taking the fpqc-sheafification, this yields again an isomorphism of k-ring schemes Rn ⊗k k  → Rn that commutes with the ∼  truncation morphisms and an isomorphism R∞ ⊗k k  − → R∞ by passing to the limit. Here we used that fpqc-sheafification commutes with the restriction to the category of k  -algebras. 2.4. The Ideal Schemes Jnm Lemma 2.4.1. — Let n ∈ N and let m ∈ N ∪ {+∞} be such that m  n. The functor Jnm : Algk → Sets,

A → ker(Rm (A) → Rn (A))

is representable by a closed subscheme of Rm . If m ∈ N, then Jnm is isomorphic, as a k-scheme, to Akm−n . Proof. — This is obvious except in the case where R is of mixed characteristic and absolutely ramified; in that case, it follows from §1.3.9 and the Gr ∼ description of the ring scheme Rm = Rm in §2.2.5. Proposition 2.4.2. — Let A be a k-algebra. a) For every integer m  0, the truncation morphism (2.4.2.1)

R∞ (A) → Rm (A)

is surjective, and its kernel contains mm+1 . If R has equal characteristic, or R has mixed characteristic (0, p) and A = Ap , then this kernel is generated by mm+1 . b) Let n and m be integers such that n  m  0, and let α be any integer such that α(m + 1)  n + 1. Let I = Jnm (A) be the kernel of the truncation morphism Rn (A) → Rm (A). One has mm+1 ⊂ I, I α = 0, and mn−m I = 0. If R has equal characteristic or A is reduced, then I consists precisely of the elements in Rn (A) that are killed by mn−m . Proof. — Once a uniformizer has been chosen in R, we can identify the morphism Rn → Rm with a linear projection Ank → Am k , for all integers m and n such that n  m  0. In particular, the morphism Rn (A) → Rm (A) is surjective for every k-algebra A. Since R∞ (A) = lim Rn (A), this implies that ←− the morphism R∞ (A) → Rm (A) is surjective, as well. Since mm+1 is the kernel of the morphism from R∞ (k) to Rm (k), we also observe that mm+1 is contained in the kernel of the projection R∞ (A) → Rm (A).

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237

Let us now assume that R has equal characteristic. Then the choice of a uniformizer t in R induces an isomorphism R  k[[t]], and we can identify the morphism R∞ (A) → Rm (A) with the projection A[[t]]→ A[t]/(tm+1 ) whose kernel is generated by tm+1 . Similarly, the truncation morphism Rn (A) → Rm (A) is given by A[t]/(tn+1 ) → A[t]/(tm+1 ), and its kernel I = (tm+1 ) satisfies both I α = 0 and I = {a ∈ A[t]/(tn+1 ) ; tn−m a = 0}. This proves assertions a) and b) in the equal characteristic case. In the rest of the proof, we thus assume that R has mixed characteristic. Let p be the characteristic of k, and let π be a uniformizer in R. If R  W (k), we take π = p. Let n and m be integers such that n  m  0. In view of the isomorphisms W (A) ⊗W (k) Rm = W (A) ⊗W (k) (Rn /(π m+1 )) = (W (A) ⊗W (k) Rn )/(π m+1 ), we observe that the kernel I˜ of the truncation morphism "n (A) = W (A) ⊗W (k) Rn → W (A) ⊗W (k) Rm = R + R m (A) is generated by π m+1 . As a consequence, it satisfies I˜α = (π)n−m I˜ = 0. By exactness of the construction of the fpqc-sheaf associated with an fpqc˜ This implies that I α = (π)n−m I = presheaf, I is the sheaf associated with I. 0. + If A = Ap , then R(∞ (A) = R∞ (A) and R m (A) = Rm (A) for every m. This shows that the kernel of the morphism R∞ (A) → Rm (A) is generated by π m+1 in this case. It remains to prove that I = {a ∈ Rn (A) ; π n−m a = 0} if A is reduced. We already saw that π n−m I = 0. Consequently, by induction on n, it suffices to show that if a ∈ Rn+1 (A) satisfies πa = 0, then a ∈ ker(Rn+1 (A) → Rn (A)). Let a ∈ Rn+1 (A) be such that πa = 0, and let b be its image in Rn (A). Since A is reduced, the morphism x → πx from Rn (A) to Rn+1 (A) is injective (§2.2.15); it maps b to πb = πa = 0; consequently, b = 0. (2.4.3). — Let m be the maximal ideal of R. Let i be an integer. If i  0, then we set k(i) = (m/m2 )⊗i ; this is a k-vector space of dimension 1. If i < 0, then we set k(i) = Homk (k(−i), k). The choice of a uniformizer π in R determines an isomorphism k(j)  k for every j ∈ Z, but we will use the vector spaces k(j) to make the constructions below independent of the choice of π. This is important, for instance, when one wants to study the action of automorphisms of R on the Greenberg schemes. (2.4.4). — For every integer m  0 and every ring A of characteristic p > 0, m we denote by p A the ring A with A-algebra structure given by A→

pm

A,

m

a → ap .

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(2.4.5). — Let n ∈ N. We will now give a precise description of the structure of the ideal scheme Jnn+1 . We denote it by In for simplicity. Let In = In (k) = Ker(Rn+1 → Rn ). One has mIn = 0, so that we can view In as a vector space over the residue field R/m = k. The map k(n + 1) → In ,

v0 ⊗ . . . ⊗ vn → v0 · . . . · vn

is an isomorphism of k-vector spaces. (2.4.6). — Let us carry out a similar construction on the level of the ideal scheme In . As a k-scheme without additional structure, In is isomorphic to A1k . Let A be a k-algebra. It follows from Proposition 2.4.2 that the multiplication map Rn+1 (A) × In (A) → In (A),

(r, i) → r · i

factors through R0 (A) × In (A), so that we can view In as a module over the k-ring scheme R0 = A1k . First assume that R has equal characteristic. In this case, the map (2.4.6.1) A ⊗k k(n + 1) → In (A),

a ⊗ v0 ⊗ . . . ⊗ vn → a · v0 · . . . · vn

is an isomorphism of A-modules. In particular, In (A) is a free A-module of rank one. Assume then that R has mixed characteristic (0, p), and let e be its absolute ramification index. Let α ∈ N be the smallest integer such that (α + 1)e  n + 2, and let β be the remainder of the Euclidean division of n + 1 by e. Using the Wα+1 (A)-module structure of Rn+1 (A), we construct a morphism of abelian groups ψ : A ⊗k k(β) → In (A),

a ⊗ v1 ⊗ . . . ⊗ vβ → V α (a) · v1 · . . . · vβ ,

where, we recall, V denotes the Verschiebung map. It is now clear from §1.3.9 and the description of the ring scheme RnGr ∼ = Rn in §2.2.5 that ψ defines an isomorphism of A-modules (2.4.6.2)



A ⊗k k(β) → In (A).

Proposition 2.4.7. — Let n ∈ N be an integer. Let A be a k-algebra, let f be an element of A, and let f( ∈ Rn (A) be any element which lifts f . Then the ring morphism Rn (A) → Rn (Af ) factors uniquely through a morphism Rn (A)f( → Rn (Af ). This morphism is an isomorphism. Proof. — The statement is clear if R has equal characteristic, so let us assume that R has mixed characteristic. By Proposition 2.4.2, the morphism Rn (Af ) → Af has nilpotent kernel; since f( maps to the invertible element f , it is invertible as well. Consequently, the morphism Rn (A) → Rn (Af ) factors uniquely through a morphism ψn (A) : Rn (A)f˜ → Rn (Af ). To show that this morphism ψn (A) is an isomorphism, we observe that the same argument furnishes, for every A-algebra B, an analogous morphism

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239

ψn (B) : Rn (B)f˜ → Rn (Bf ). Moreover, these morphisms ψn (B) are functorial in B and hence define a morphism of functors ψn : Fn → Gn , where Fn and Gn are the functors on A-algebras given by Fn (B) = Rn (B)f( and Gn (B) = Rn (Bf ). We will show that ψn is an isomorphism. Since Rn is a sheaf for the fpqc-topology, exactness of localization implies that both functors Fn and Gn are fpqc-sheaves. By Lemma 2.2.10, for every A-algebra B, there exists a faithfully flat B-algebra, C, such that C p = C. To show that the morphism of sheaves ψn is an isomorphism, it thus suffices to show that ψn (B) is an isomorphism when B is an A-algebra such that B p = B. In this case, one has Rn (B) = W (B) ⊗W (k) Rn , and Rn (Bf ) = W (Bf ) ⊗W (k) Rn , because (Bf )p = Bf . We now proceed by induction on n. The result is obvious if n = 0, so we assume that n > 0 and that the result holds for all strictly smaller values of n. Let us choose a uniformizer π in R and consider the commutative diagram of Rn (B)-modules with exact rows: 0

Rn−1 (B)f

π·

ψn−1 (B)

0

Rn−1 (Bf )

Rn (B)f ψn (B)

π·

Rn (Bf )

Bf

0

Id

Bf

0.

By the induction hypothesis, ψn−1 (B) is an isomorphism. It then follows from the snake lemma that ψn (B) is an isomorphism as well. Remark 2.4.8. — Proposition 2.4.7 does not apply to R∞ . Assume, for example, that R = k[[t]], so that one has R∞ (A) = A[[t]] for every k-algebra A. Let f ∈ A. Then R∞ (Af ) = Af [[t]]= (A[[t]])f . In fact, the geometry of the ring R∞ (A) is better understood in the framework of formal schemes; see §3.3. (2.4.9). — Let A be a k-algebra. We endow the ring R∞ (A) = limn Rn (A) ←− with the limit of the discrete topologies on the rings Rn (A). In other words, the family of ideals (Jn∞ (A)) is a basis of neighborhoods of 0. Assume that R has equal characteristic, and let t be uniformizer of R. Then R∞ (A)  A[[t]] and the topology of R∞ (A) is the t-adic topology. Assume now that R has mixed characteristic (0, p). If A = Ap , it follows from Proposition 2.4.2 that the topology on R∞ (A) coincides with the padic topology. However, it is strictly coarser than the p-adic topology when A = Ap . Proposition 2.4.10. — Let A be a k-algebra. a) The ring morphism R → R∞ (A) is continuous. ∞ (A) is an ideal of definition b) For every integer m  0, the ideal Jm of R∞ (A). The topological ring R∞ (A) is admissible.

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Proof. — a) By definition of the topology on R∞ (A), it is separated and ∞ (A) form a fundamental system of open neighcomplete, and the ideals Jm borhoods of 0 in R∞ (A). ∞ (A) contains mm+1 for every m  0, By Proposition 2.4.2, the ideal Jm so that the ring morphism R → R∞ (A) is continuous. b) Let m  0. We need to prove that for every integer n  0, there exists ∞ (A))N ⊂ Jn∞ (A). By Proposition 2.4.2, it an integer N  1 such that (Jm suffices to choose N such that (m + 1)N  n + 1. Then R∞ (A) is admissible, by definition of an admissible topological ring.

§ 3. GREENBERG SCHEMES Let R be a complete discrete valuation ring and denote by k its residue field. If R has mixed characteristic, we assume that k is perfect. If R has equal characteristic, we choose a ring morphism k → R such that the composition with the projection R → k is the identity on k. Using the ring schemes Rn that we have defined in the previous section, we will construct for every R-scheme X a k-scheme Gr∞ (X), the Greenberg scheme of X, that parameterizes points on X with coordinates in extensions of R of ramification index one. It plays a central role in the theory of motivic integration, since it is the space on which the motivic measure is defined. We keep the notations from section 0.0.1: we write m for the maximal ideal of R, and we set Rn = R/mn+1 for every integer n  0.

3.1. Greenberg Schemes as Functors (3.1.1). — Let Y be a k-scheme; let |Y | denote its underlying topological space. Let n ∈ N. Since Rn is a k-scheme in Rn -algebras, the assignment U → Homk (U, Rn ) defines a sheaf in Rn -algebras on the topological space |Y |, which we denote by Rn (OY ). Let y ∈ Y . The stalk of the sheaf Rn (OY ) at the point y is the ring lim Homk (U, Rn ), where U ranges over the affine open neighborhoods of y; −→ this stalk Rn (OY )y is thus equal to the ring Rn (OY,y ). By Proposition 2.4.2, the canonical morphism Rn (OY,y ) → OY,y is surjective, and its kernel is a nilpotent ideal; consequently, Rn (OY )y is a local ring. We have thus defined a locally ringed space in Rn -algebras hn (Y ) = (|Y |, Rn (OY )). For n = 0, one has R0 = A1k , so that R0 (OY ) is nothing but the structure sheaf OY of the scheme Y ; hence one has h0 (Y ) = Y .

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241

Let f : X → Y be a morphism of k-schemes. Let U be an open subset of Y . The natural map Homk (U, Rn ) → Homk (f −1 (U ), Rn ) induces a morphism of sheaves f −1 Rn (OY ) → Rn (OX ) which is a morphism of sheaves in local rings. This gives rise to a morphism of locally ringed spaces: hn (f ) : hn (X) → hn (Y ). Thus, we have defined a functor hn from the category of k-schemes to the category of locally ringed spaces in Rn -algebras. For every pair (m, n) of integers such that 0  m  n, the truncation morphism Rn → Rm induces a morphism of functors hm → hn . Example 3.1.2. — For every open subset U of Y , the section k → R of R → k that we have fixed induces a morphism of rings Γ(U, R0 (Y )) → Γ(U, Rn (Y )), hence a morphism of schemes hn (Y ) → Y . Together with the structural morphism hn (Y ) → Spec(Rn ), this yields a morphism of Rn schemes ϕ : hn (Y ) → Y ⊗k Rn . When Y is affine, say Y = Spec(A), the morphism ϕ coincides with the morphism induced by the isomorphism of rings A ⊗k Rn → Rn (A); it is thus an isomorphism. Since the functor hn respects open immersions, it follows that ϕ is an isomorphism for every kscheme Y . In other words, the functor hn is isomorphic to the functor given by Y → Y ⊗k Rn . Lemma 3.1.3. — a) Let A be a k-algebra and let Y = Spec(A). Then hn (Y ) is isomorphic to Spec(Rn (A)). b) For every k-scheme Y , the locally ringed space in Rn -algebras hn (Y ) is an Rn -scheme. c) If f : Y → X is an open immersion of k-schemes, then the morphism hn (f ) : hn (Y ) → hn (X) is an open immersion of Rn -schemes. Proof. — Let us first prove a). Morphisms from the locally ringed space (|Y |, Rn (OY )) to the affine scheme Spec(Rn (A)) correspond to ring morphisms from Rn (A) to the ring Γ(|Y |, Rn (OY )) = Homk (Y, Rn ) = Homk (Spec(A), Rn ) = Rn (A), and Rn -morphisms correspond to morphisms of Rn -algebras. Let ψ : hn (Y ) →  Spec(Rn (A)) be the unique morphism corresponding to the identity of Rn (A). We shall show that ψ is an isomorphism. Consider the commutative diagram of locally ringed spaces h0 (Y ) = Y Id

Spec(R0 (A)) = Y

hn (Y ) ψ

Spec(Rn (A)).

The morphism Spec(R0 (A)) → Spec(Rn (A)) induced by the truncation morphism Rn (A) → R0 (A) = A is a homeomorphism, because the kernel of

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Rn (A) → R0 (A) is a nilpotent ideal. Moreover, the map of topological spaces underlying the morphism h0 (Y ) → hn (Y ) is the identity. Consequently, ψ is a homeomorphism. Now let f be an element of A, and consider the open subscheme U = Spec(Af ) of Y . Let f( be any element of Rn (A) which maps to f under the truncation morphism Rn (A) → R0 (A). The open subset ψ(U ) of Spec(Rn (A)) is then equal to Spec(Rn (A)f˜), and the morphism of rings OSpec(Rn (A)) (ψ(U )) → Ohn (Y ) (U ) induced by ψ coincides with the isomorphism Rn (A)f( → Rn (Af ) from Proposition 2.4.7. Thus ψ is an isomorphism of ringed spaces. It is clear from its definition that the functor hn sends an open immersion f of k-schemes to an open immersion hn (f ) of ringed spaces. Thus b) is an immediate consequence of a), by the very definition of a scheme, and c) follows from the fact that hn (f ) is a morphism of locally ringed spaces. (3.1.4). — Let X be an R-scheme. For every integer n  0, we consider the functor  HomSchR (hn (Y ), X). Grn (X) : Schok → Sets, Y → As usual, when A is a k-algebra, we will write Grn (X)(A) instead of Grn (X)(Spec(A)). Since hn (Y ) is an Rn -scheme, one has Grn (X)(Y ) = HomSchR (hn (Y ), X ⊗R Rn ), so that the functor Grn (X) only depends on the Rn -scheme X ⊗R Rn . For n = 0, one has h0 (Y ) = Y and R0 = k, so that the functor Gr0 (X) is (the functor associated with) the special fiber X ⊗R k of X. (3.1.5). — For all integers m, n with m  n  0, the morphism of functors hn → hm induces a truncation morphism: m θn,X : Grm (X) → Grn (X).

(3.1.5.1)

When m, n, q are integers such that m  n  q  0, one has n m m θq,X ◦ θn,X = θq,X .

(3.1.5.2) Moreover,

n θn,X = IdGrn (X) .

(3.1.6). — Let Gr∞ (X) : Schok → Sets,

Y → lim HomSchR (hn (Y ), X) ← − n

be the projective limit of the projective system (Grn (X))n0 . For every integer n  0, we denote by (3.1.6.1)

∞ θn,X : Gr∞ (X) → Grn (X)

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243

the natural projection map. For all integers m, n such that m  n  0, one has (3.1.6.2)

m ∞ ∞ ◦ θm,X = θn,X . θn,X

Proposition 3.1.7. — Let X be an R-scheme. For every k-algebra A, the canonical map X(R∞ (A)) → lim X(Rn (A)) = Gr∞ (X)(A) ←− n

is a bijection. Proof. — Let I be the kernel of the morphism from R∞ (A) to R0 (A). By Proposition 2.4.10, it is an ideal of definition of R∞ (A), and the topology of R∞ (A) is the I-adic topology. The proposition reduces to the fact that the canonical map X(R∞ (A)) → lim X(R∞ (A)/I n ) ← − n is a bijection, which holds by Theorem 3/3.3.6. (3.1.8). — Let f : X → Y be a morphism of R-schemes. For every integer n  0, the maps X(hn (T )) → Y (hn (T )) given by ϕ → ϕ ◦ f , for all k-schemes T , define a morphism of functors (3.1.8.1)

Grn (f ) : Grn (X) → Grn (Y ).

These morphisms commute with the truncation morphisms: if m, n are integers such that m  n  0, then one has (3.1.8.2)

m m Grn (f ) ◦ θn,X = θn,Y ◦ Grm (f ).

Passing to the limit, we obtain a morphism (3.1.8.3)

Gr∞ (f ) : Gr∞ (X) → Gr∞ (Y )

such that (3.1.8.4)

∞ ∞ Grn (f ) ◦ θn,X = θn,X ◦ Gr∞ (f )

for every integer n ∈ N. Let n ∈ N∪{+∞}. One has Grn (IdX ) = IdGrn (X) . Moreover, if g : Y → Z is a second morphism of R-schemes, then one has Grn (g ◦ f ) = Grn (g) ◦ Grn (f ). Example 3.1.9. — Assume that R has equal characteristic. In example 3.1.2, we have identified hn (Y ) with Y ⊗k Rn , for every R-scheme Y , functorially in Y . In turn, this identifies the functor Grn (X) with the functor given by Y → HomSchR (hn (Y ), X) = HomSchR (Y ⊗k Rn , X) = HomSchRn (Y ⊗k Rn , X ⊗k Rn ).

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Consequently, Grn (X) is canonically isomorphic to the Weil restriction of X ⊗k Rn with respect to the finite morphism Spec(Rn ) → Spec(k); see Definition 3/1.2.1. Assume, moreover, that X = X0 ⊗k R, for some k-scheme X0 . Let us choose a uniformizer t ∈ R and identify R with k[[t]]. This identifies HomR (Spec(A[t]/(tn+1 )), X) with Homk (Spec(A[t]/(tn+1 )), X0 ), for every k-algebra A. The Greenberg functor Grn (X) thus coincides with the jet scheme Ln (X0 /k), compatibly with the truncation morphisms. As a consequence, the Greenberg functor Gr∞ (X) coincides with the arc scheme L∞ (X0 /k). Lemma 3.1.10. — For every n ∈ N ∪ {+∞}, the functor Grn (X) is a Zariski sheaf on the category of k-schemes. In particular, these functors are completely determined by their restrictions to the category of affine k-schemes. Proof. — It suffices to treat the case of Grn (X), for n ∈ N, because a limit of Zariski sheaves is again a Zariski sheaf. Let thus Y be a k-scheme, and let (Yi )i∈I be an open covering of Y . For every i ∈ I, let fi ∈ Grn (X)(Yi ); assume that fi and fj coincide in Grn (X)(Yi ∩ Yj ). By construction, fi is a morphism ϕi : hn (Yi ) → X, and one has ϕi |hn (Yi ∩Yj ) = ϕj |hn (Yi ∩Yj ) . By Lemma 3.1.3, (hn (Yi )) is an open covering of hn (Y ), and hn (Yi ∩ Yj ) = hn (Yi ) ∩ hn (Yj ) for every i, j. Consequently, there exists a unique morphism ϕ : hn (Y ) → X whose restriction to hn (Yi ) coincides with ϕi . The corresponding element f of Grn (X)(Y ) is the unique element inducing fi in Grn (X)(Yi ), for every i ∈ I. This concludes the proof. Proposition 3.1.11. — Let X, Y be two R-schemes, and let p1 : X×S → Y → X, p2 : X ×R Y → Y be the two projections. For every n ∈ N ∪ {+∞}, the morphism of functors (Grn (p1 ), Grn (p2 )) : Grn (X ×R Y ) → Grn (X) ×k Grn (Y ) is an isomorphism. Proof. — The Greenberg functor Grn (·), viewed as a functor on the category of k-schemes, resp. of k-algebras if n = +∞, is a right adjoint to the functor hn , resp to the functor A → R∞ (A). Consequently, it commutes with all categorical limits. The proposition thus follows as a particular case; see Remark 3/1.2.8. Example 3.1.12. — a) Assume that X = A1R . Then uniformizer π of R determines, for every integer n  0, an (see section 2.3.1). By construction, k-schemes Rn  An+1 k A1R (hn (A)) = O(hn (A)) = Rn (A), for every k-algebra A.

the choice of a isomorphism of Grn (A1R )(A) = This furnishes

§ 3. GREENBERG SCHEMES

245

in particular an isomorphism of k-schemes Grn (X)  An+1 . For all intek m corresponds, under gers m, n such that m  n, the truncation map θn,X these isomorphisms, to a linear projection that only keeps m + 1 coordinates. Passing to the limit, we see that Gr∞ (X) is isomorphic to the infinitedimensional affine space Spec(k[x0 , x1 , . . .]). b) More generally, let X = AdR . It follows from the previous example d(n+1) and from Proposition 3.1.11 that Grn (X)  Ak , the truncation maps being induced by linear projections. Similarly, Gr∞ (X) is isomorphic to Spec(k[(xi,n )(i,n)∈{1,...,d}×N ]). c) Let us assume that k has characteristic p > 0 and that R = W (k). Let X be the closed subvariety of A2R = Spec(R[x, y]) defined by the equation x2 + py 3 = 0. Let n ∈ N. Then Grn (X) is the subfunctor of 2(n+1)

Grn (A2R ) = Ak

= Spec(k[x0 , y0 , . . . , xn , yn ])

defined by the equation (x0 , . . . , xn )2 + p(y0 , . . . , yn )3 = 0 in Wn+1 (A), for all k-algebras A. This shows that Grn (X) is a closed subscheme of the scheme Grn (A2R ). For example, when n = 1, we get the equations ) x20 = 0, p 2x0 x1 + y03p = 0, so that Gr1 (X)  Spec(k[x0 , y0 , x1 , y1 ]/(x20 , 2xp0 x1 + y03p )). Likewise, Gr∞ (X) is the closed subscheme of Gr∞ (A2R ) = Spec(k[x0 , y0 , x1 , y1 , . . .]) defined by the equation (x0 , x1 , . . .)2 + p(y0 , y1 , . . .)3 = 0 in W (A), for all k-algebras A. This yields the infinite list ⎧ x20 ⎪ ⎪ p ⎪ ⎪ 2x0 x1 + y03p ⎨ p 3p 2 S2 (x0 , 0, 2x0 x1 , y0 , P2 (x0 , x0 , x1 , x1 , x2 , x2 ), 2 ⎪ ⎪ ⎪ 3y02p y1p ) ⎪ ⎩ ... in the variables x0 , y0 , x1 , y1 , . . .

of equations = =

0, 0,

= 0,

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3.2. Representability of the Greenberg Schemes (3.2.1). — The aim of this section is to prove that, for every R-scheme X and every integer n  0, the functors Grn (X) and Gr∞ (X) are representable by k-schemes. We have already seen two examples in Example 3.1.12. As an auxiliary result, we first show that these functors behave well with respect to formally étale morphisms (in particular, open immersions). Lemma 3.2.2. — Let f : X → Y be a formally étale morphism of Rschemes. For all elements m  n in N ∪ {+∞}, the natural morphism m ) : Grm (X) → Grm (Y ) ×Grn (Y ) Grn (X) (Grm (f ), θn,X

is an isomorphism. In particular, the natural morphism m ) : Grm (X) → Grm (Y ) ×(Y ⊗R k) (X ⊗R k) (Grm (f ), θ0,X

is an isomorphism. Proof. — It is enough to prove the result for m, n ∈ N, the case m = ∞ then follows by passing to the limit. It suffices to show that, for every k-algebra A, the natural map (3.2.2.1)

X(Rm (A)) → Y (Rm (A)) ×Y (Rn (A)) X(Rn (A))

is a bijection. It follows from Proposition 2.4.2 that the ring morphism Rm (A) → Rn (A) is a surjection with nilpotent kernel, so that (3.2.2.1) is a bijection by the infinitesimal lifting criterion for formally étale morphisms (ÉGA IV4 , 17.1.1). Theorem 3.2.3. — the functor

a) Let X be an R-scheme. For every integer n  0,

Grn (X) : Schok → Sets,

Y → HomSchR (hn (Y ), X)

m is representable by a k-scheme. The truncation morphisms θn,X are affine morphisms of k-schemes. b) If X is separated, resp. affine, then so is Grn (X). c) If X is of finite type over R, then Grn (X) is of finite type over k. d) It f : X → Y is an open, resp. closed immersion of R-schemes, then Grn (f ) is an open, resp. closed immersion of k-schemes.

Proof. — We first prove that Grn (X) is representable by an affine k-scheme if X is affine. We choose a presentation O(X)  R[(xs )s∈S ]/I for some set S and some ideal I in R[(xs )s∈S ]. The choice of a uniformizer π . This isomorphism in R determines an isomorphism of k-schemes Rn ∼ = An+1 k n+1 (different from the usual one, if defines a k-ring scheme structure on Ak n > 0) and allows us to identify an element a in the R-algebra Rn (A) with a tuple (a0 , . . . , an ) of elements in A, for every k-algebra A.

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For every polynomial f ∈ R[(xs )s∈S ], there exists a unique family (f0 , . . . , fn ) in k[(xs,0 , . . . , xs,n )s∈S ] such that f (b) = (f0 ((bs,0 , . . . , bs,n )s∈S ), . . . , fn ((bs,0 , . . . , bs,n )s∈S )) for every k-algebra A and every b ∈ (Rn (A))S . We denote by I  the ideal  with f ∈ I and in k[(xs,0 , . . . , xs,n )s∈S ] generated by the polynomials fm m ∈ {0, . . . , n}. Let X  be the affine k-scheme defined by X  = Spec(k[(xs,0 , . . . , xs,n )s∈S ]/I  ). For every k-algebra A and every element b ∈ X(R∞ (A)) ⊂ (Rn (A))S , the maps X(Rn (A)) → X  (A) : b → (bs,0 , . . . , bs,n )s∈S define an isomorphism of functors from Grn (X) to X  . In particular, Grn (X) is representable. If X is of finite type over R, then we can take a presentation as above, where S a finite set, so that Grn (X) is of finite type over k. Now let X be an arbitrary R-scheme. Then Lemma 3.2.2 and the fact that Grn (X) is a sheaf for the Zariski topology imply that the functor Grn (X) is representable: the corresponding scheme can be constructed by gluing the k-schemes Grn (U ) where U runs through a cover of X by affine open subschemes (see Proposition 3/1.1.10). Lemma 3.2.2 also implies that Grn (·) respects open immersions. If f : X → Y is a closed immersion of R-schemes, then the property that Grn (f ) be a closed immersion can be checked locally on the target Grn (Y ). We may thus assume that Y , and hence X as well, is affine. In that case, it is obvious from the above construction that Grn (f ) is a closed immersion. m are affine: It follows from Lemma 3.2.2 that the truncation morphisms θn,X if we cover X by affine open subschemes U , then the k-schemes Grn (U ) form a covering of Grn (X) by affine open subschemes, by the first part of the proof, and by Lemma 3.2.2 we can identify the restriction of Grm (X) → Grn (X) over Grn (U ) with the truncation morphism Grm (U ) → Grn (U ), which is affine. The same argument shows that Grn (X) is of finite type over k if X is of finite type over R. If X is separated, then the closed subscheme X ⊗R k = Gr0 (X) of X is n is affine. separated, so that Grn (X) is separated since θ0,X Corollary 3.2.4. — Let X be an R-scheme. a) The functor Gr∞ (X) is representable by a k-scheme. ∞ : Gr∞ (X) → b) For every integer n ∈ N, the truncation morphism θn,X Grn (X) is affine. c) If X is separated, resp. affine, then so is Gr∞ (X). d) If f : X → Y is an open, resp. closed, immersion of R-schemes, then Gr∞ (f ) is an open, resp. closed, immersion of k-schemes.

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m Proof. — Since the truncation morphisms θn,X are affine, the projective limit

Gr∞ (X) = lim Grn (X) ←− n0

is representable by a k-scheme, and the truncation morphisms θn,X are affine; see (ÉGA IV3 , 8.2.3). If X is affine, then Gr∞ (X) is simply the spectrum of the colimit of the direct system of rings O(Grn (X)). If X is separated, then Gr∞ (X) is separated because θ0∞ : Gr∞ (X) → X ⊗R k is affine. By Lemma 3.2.2, Gr∞ (·) respects open immersions. Now assume that f : X → Y is a closed immersion. We will show that Gr∞ (f ) is a closed immersion. This property is local on the target Gr∞ (Y ), so that we may assume that X and Y are affine, by Lemma 3.2.2. Then all the schemes Grn (X) and Grn (Y ) are affine. The projective limit of a system of closed immersions of affine schemes is again a closed immersion, because the direct limit of a system of surjective ring morphisms is surjective, by right exactness of the direct limit functor. Beware that, even if X is of finite type over R, the scheme Gr∞ (X) is not of finite type over k, in general; see Example 3.1.12. (3.2.5). — For every n in N, the functor Grn is called the Greenberg functor of level n, and Grn (X) is called the Greenberg scheme of X of level n. Likewise, Gr∞ is called the Greenberg functor (of level ∞), and Gr∞ (X) is called the Greenberg scheme of X. By definition, the functor Grn is right adjoint to the functor hn , so that for every R-scheme X, the k-scheme Grn (X) comes equipped with a universal family hn (Grn (X)) → X which corresponds to the identity morphism on Grn (X).

3.3. Greenberg Schemes of Formal Schemes In this section, we extend the definition of the Greenberg schemes to locally Noetherian R-adic formal schemes. (3.3.1). — Recall that a locally Noetherian R-adic formal scheme is a locally ringed space over Spec(R) such that Xn := X ⊗R Rn is a locally Noetherian Rn -scheme for every integer n  0 and X = lim Xn = (|X0 |, lim OXn ) −→ ←− n0

n0

in the category of locally ringed spaces in R-algebras. (In this case, the topology on OX is completely determined by the R-algebra structure.) (3.3.2). — Let A be a k-algebra. Since the topological ring R∞ (A) is admissible by Proposition 2.4.10), we may consider its formal spectrum Spf(R∞ (A)) as defined in (ÉGA I, §10.1). Since the kernels of the truncation morphism R∞ (A) → Rn (A) form a fundamental system of open

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neighborhoods of 0 in R∞ (A), we have Spf(R∞ (A)) = lim Spec(Rn (A)) −→ n0

in the category of topologically ringed spaces (where we put the discrete topology on Rn (A)). The continuous morphism of admissible rings R → R∞ (A) defines a morphism of formal schemes Spf(R∞ (A)) → Spf(R). More generally, for every k-scheme Y , the locally ringed space in Ralgebras defined by the inductive limit limn hn (Y ) admits a natural structure −→ of a formal scheme over Spf(R). Example 3.3.3. — Let X be a locally Noetherian R-scheme. Then its formal m-adic completion X = lim (X ⊗R Rn ) −→ n0

is a locally Noetherian R-adic formal scheme. Let A be a k-algebra. By Proposition 2.4.10, one has Gr∞ (X)(A) = lim HomSchRn (Spec(Rn (A)), X ⊗R Rn ) ←− n0

= HomForSchR (Spf(R∞ (A)), X) This gives a natural interpretation of the Greenberg scheme Gr∞ (X) in terms of the formal scheme X. Conversely, applied to the formal scheme X, the constructions below will recover the Greenberg schemes of X, so that the study of Greenberg scheme of R-adic formal schemes encompasses, as a particular case, that of Greenberg schemes of (locally Noetherian) R-schemes. (3.3.4). — Let X be a locally Noetherian R-adic formal scheme. For every integer n  0, we set (3.3.4.1)

Grn (X) = Grn (Xn ).

We call this k-scheme the Greenberg scheme of X of level n. Grn (Xn ) = Grn (Xm ) for every integer m  n. For all integers m  n  0, the truncation morphism (3.3.4.2)

One has

m : Grm (X) = Grm (Xm ) → Grn (Xm ) = Grn (X) θn,X

n = idGrn (X) and is an affine morphism of k-schemes. Moreover, one has θn,X q q m θn,X ◦ θm,X = θn,X for all integers q, m, n such that q  m  n. Let n ∈ N. If X is separated, then Xn is separated so that Grn (X) is separated. If X is of finite type (resp. locally of finite type) over R, then Xn is of finite type (resp. locally of finite type) over R, so that Grn (X) is of finite type (resp. locally of finite type) over k.

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(3.3.5). — Let us define Gr∞ (X) as the projective limit of the system (Grn (X)) with respect to the truncation morphisms. For every k-algebra A, one has an identification Gr∞ (X)(A) = lim X(Rn (A)) = HomForSchR (Spf(R∞ (A)), X), ← − n functorially in A. More generally, for every k-scheme Y , one has (3.3.5.1) Gr∞ (X)(Y ) = lim HomSchR (hn (Y ), X) = HomFormSchR (lim hn (Y ), X). ← − −→ n n0

m Since the morphisms θn,X are affine, the functor Gr∞ (X) is represented by a k-scheme, which we call the Greenberg scheme of X (of level ∞). This scheme Gr∞ (X) is endowed with truncation morphisms

(3.3.5.2)

∞ : Gr∞ (X) → Grn (X). θn,X

m ∞ ∞ ◦ θm,X = θn,X . For all integers m, n such that m  n, one has θn,X

(3.3.6). — Let f : X → Y be a morphism of locally Noetherian R-adic formal schemes. By base change, it induces for every integer n  0 a morphism fn : Xn → Yn of Rn -schemes, hence a morphism of k-schemes (3.3.6.1)

Grn (f ) : Grn (X) → Grn (Y).

These morphisms commute with the truncation morphisms: one has Grn (f )◦ m m θn,X = θn,Y ◦ Grm (f ) for all m, n ∈ N such that m  n. Passing to the limit, we obtain a morphism of k-schemes (3.3.6.2)

Gr∞ (f ) : Gr∞ (X) → Gr∞ (Y)

such that ∞ ∞ = θn,Y ◦ Gr∞ (f ) Grn (f ) ◦ θn,X

for every n ∈ N. (3.3.7). — The points of Gr∞ (X) can be interpreted as follows. Let k  be a field extension of k, and assume that k  is perfect if R has mixed characteristic. Then by Proposition 2.3.2, R∞ (k  ) is a complete discrete valuation ring with residue field k  , of ramification index one over R. If k  is separable over k, it is the only such extension up to unique isomorphism. By definition of the Greenberg schemes, there is a canonical bijection ∼

→ X(R∞ (k  )). Gr∞ (X)(k  ) − Consequently, we can think of the k-scheme Gr∞ (X) as parameterizing points of X with values in extensions of R of ramification index one. Each point x of Gr∞ (X) gives rise to an element ψx of X(R∞ (κx )), where κx denotes the residue field of X at x if R has equal characteristic and the perfect closure of this residue field if R has mixed characteristic.

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(3.3.8). — Let k  be a field extension of k, and assume that k  is perfect if R has mixed characteristic. Put R = R∞ (k  ). By the definition of the Greenberg schemes, the isomorphisms of ring schemes in corollary 2.3.3 induce isomorphisms of k  -schemes Grn (X) ⊗k k  → Grn (X ⊗R R ) Gr∞ (X) ⊗k k  → Gr∞ (X ⊗R R ) which are compatible with the truncation morphisms. 3.4. Néron Smoothenings of Formal Schemes (3.4.1). — The process of Néron smoothening was introduced by Néron (1964) as a step in the construction of Néron models of abelian varieties over discretely valued fields. It can be viewed as a weak form of resolution of singularities for schemes and formal schemes over R, which has the advantage that it is valid without any restrictions on the characteristic or the dimension. Moreover, smoothenings are constructed through a compact algorithm that is governed by an elementary invariant, called Nérons defect of smoothness. The canonical reference for Néron smoothenings of R-schemes is Bosch et al. (1995). Néron smoothenings of formal schemes also lead to so-called weak Néron models of K-analytic spaces, which will play a crucial rule in the theory of motivic integration on analytic spaces in section 7/5. A brief introduction to K-analytic spaces is given in section A/3. All K-analytic spaces are assumed to be strictly K-analytic and Hausdorff. We say that a morphism f : Y → X of K-analytic spaces is an analytic domain immersion if it induces an isomorphism from Y to a locally closed analytic domain of X. (3.4.2). — Let X be a K-analytic space. A weak Néron model of X is a smooth formal R-scheme of finite type X, endowed with an analytic domain immersion Xη → X such that the map Xη (K  ) → X(K  ) is bijective for every finite unramified extension K  of K. Thus we can view X as a smooth R-model of the unramified points on X. Let X be a formal R-scheme of finite type. A Néron smoothening of X is a morphism of formal R-schemes of finite type h : Y → X such that hη : Yη → Xη is an analytic domain immersion and the pair (Y, hη ) is a weak Néron model for Xη . The latter condition is equivalent to saying that the map Y(R ) → X(R ) is a bijection for every finite unramified extension R of R. (3.4.3). — Let X and X be weak Néron models of X. A morphism of weak Néron models X → X is a morphism of formal R-schemes h : X → X such that hη commutes with the immersions of Xη and Xη into X. Note that such a morphism is unique if it exists, since hη is fully determined and X is flat over R. If h exists, we say that X dominates X. Likewise, if Y → X and Y → X are Néron smoothenings, then a morphism of Néron smoothenings Y → Y is a morphism of formal X-schemes Y → Y. Again, such a morphism is unique if it exists (since it is also a morphism of

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weak Néron models of Xη ), and in that case we say that Y → X dominates Y → X. Note that Y → Y is still a Néron smoothening. Definition 3.4.4. — We say that a K-analytic space X is néronian if there exists a compact analytic domain U in X such that U is quasi-smooth over K and the inclusion map U (K  ) → X(K  ) is bijective for every finite unramified extension K  of K. Theorem 3.4.5 (Bosch-Schlöter). — a) A K-analytic space X has a weak Néron model if and only if X is néronian. In particular, every compact quasi-smooth K-analytic space has a weak Néron model. b) A formal R-scheme of finite type X possesses a Néron smoothening Y → X if and only if Xη is néronian. In that case, there exists a composition Y → X of admissible blow-ups such that Sm(Y) → X is a Néron smoothening, where Sm(Y) denotes the R-smooth locus of Y. Proof. — a) Assume that X has a weak Néron model, say X. Then, Xη is a compact quasi-smooth analytic domain of X such that Xη (K  ) = X(K  ) for every finite unramified extension K  of K. Consequently, X is néronian. Conversely, let U be a compact quasi-smooth analytic domain of X such that U (K  ) = X(K  ) for every finite unramified extension K  of K; let us prove that X admits a weak Néron model. Since every weak Néron model of U is also a weak Néron model of X, we may assume that X is compact, and thus has a formal R-model of finite type over R. It thus suffices to prove assertion b). This result is proven in Bosch and Schlöter (1995) by means of an adaptation of the Néron smoothening algorithm described in Bosch et al. (1990). In this algorithm, one modifies X by means of well-chosen admissible blow-ups in such a way that, in the end, all R -points on X are contained in the R-smooth locus Sm(X), for all finite unramified extensions R of R. Weak Néron models and Néron smoothenings are by no means unique, as is illustrated by the following example. Nevertheless, we will see in section 7/5 that one can extract some interesting invariants from a weak Néron model that are independent of the choice of the model, using motivic integration. Example 3.4.6. — Let X be the closed unit disk M (K{T }). Then X = Spf(R{T }) is a smooth formal R-model of X, and thus a fortiori a weak Néron model. Blowing up X at the origin of its special fiber, we obtain a new formal R-model X which is no longer smooth but which is still regular. The nonsmooth locus Xsing consists of a unique point x, and the completed local ring at this point is of the form Spf(R[[u, v]]/(uv−π)) where π is a uniformizer in R. Now it is clear that, for every extension R of R of ramification index one, none of the points in X (R ) passes through x, since otherwise there would exist elements u0 and v0 in the maximal ideal of R such that u0 v0 = π, which contradicts the fact that π generates the maximal ideal of R . It follows that

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Sm(X ) is another weak Néron model of X. Its special fiber is isomorphic to the disjoint union of the affine line A1k and the torus Gm,k . Proposition 3.4.7. — a) Let X be a néronian K-analytic space. If X and X are weak Néron models of X, then there exists a weak Néron model of X that dominates both X and X . b) Let X be a formal R-scheme of finite type such that Xη is néronian. If Y → X and Y → X are Néron smoothenings, then there exists a Néron smoothening of X that dominates both Y and Y . Proof. — a) We denote by X  the intersection of Xη and Xη in X. This is a compact analytic domain in Xη that contains all the unramified points of X. Applying an admissible blow-up to X followed by a Néron smoothening, we can find a weak Néron model X of X  that dominates X; this is also a weak Néron model of X. Then, performing another admissible blow-up followed by a Néron smoothening, we can arrange that the immersion Xη → Xη extends to a morphism of formal R-schemes X → X . Now X is a weak Néron model of X dominating X and X . b) It suffices to take a Néron smoothening of the fibered product of Y and Y over X. Proposition 3.4.8. — If X is a regular formal R-scheme of finite type, then the open immersion Sm(X) → X is a Néron smoothening. Proof. — Let R be a finite unramified extension of R. We must show that every R -point x on X is contained in Sm(X). Since R is étale over R, the formal scheme X ⊗R R is again regular. Moreover, Sm(X ⊗R R ) = Sm(X) ⊗R R . For schemes, this follows from flat descent of smoothness (ÉGA IV4 , 17.7.2); the result for formal schemes then follows from the fact that X is smooth over R at a point y if and only if the scheme Xn is smooth over Rn at y for every n  0. Thus, we may assume that R = R. Then it follows from Bosch et al. (1990, 3.1/2) that the completed local ring of X at x0 ∈ X(k) is isomorphic to a formal power series ring R[[t1 , . . . , td ]]. Hence, X is smooth over R at x0 , so that x lies in Sm(X).

3.5. Néron Smoothening and Greenberg Schemes Proposition 3.5.1. — Let h : Y → X be a morphism of formal R-schemes of finite type, and assume that Y is smooth and that hη is an analytic domain immersion. a) If R has equal characteristic, then h is a Néron smoothening if and only if the map Gr∞ (Y)(k  ) → Gr∞ (X)(k  ) induced by h is a bijection for every separable extension k  of k.

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b) If R has mixed characteristic, then h is a Néron smoothening if and only if the map Gr∞ (Y)(k  ) → Gr∞ (X)(k  ) induced by h is a bijection for every perfect extension k  of k. Proof. — It follows immediately from the definitions that h is a Néron smoothening if and only if the map α : Gr∞ (Y)(k  ) → Gr∞ (X)(k  ) is a bijection for every finite separable extension k  of k. We must show that this implies that the latter map is still bijective for any infinite separable extension k  of k (assumed to be perfect in the mixed characteristic case). Let k  be such an extension and set R = R(k  ). This is an extension of R of ramification index one. If we denote by K  the quotient field of R , then we can identify α with the map (Yη ⊗K K  )(K  ) → (Xη ⊗K K  )(K  ) induced by the base change hη of hη to K  . This map is injective because hη , and thus hη , are immersions. We will prove that α is also surjective. Let y be a point of Gr∞ (X)(k  ). It suffices to show that it lifts to a point x in Gr∞ (Y)(k  ) for some extension k  of k  : then the fact that hη is an open immersion implies that x is already defined over k  . By assumption, Gr∞ (Y)(F ) → Gr∞ (X)(F ) is bijective if F is a finite separable extension of k. Thus if we denote by k s a separable closure of k, then for every integer n  0, the image of the morphism Grn (h) : Grn (Y) → Grn (X) is a constructible subset of Grn (X) that contains all the points of θn (Gr∞ (X)) defined over k s . Recall that θn (Gr∞ (X)) is a constructible subset of Grn (Y) by Greenberg’s approximation theorem (Corollary 5/1.5.7). If we denote by Y the closure of θn (y) in Grn (X), endowed with its induced reduced structure, then the k-smooth locus of Y is nonempty, because the function field of Y is contained in the separable extension k  of k. This implies that Y contains a dense subset of points defined over k s (Bosch et al. 1990, 2.2/13). Moreover, the set Y ∩ θn (Gr∞ (X)) is constructible, and it contains the generic point θn (y) of Y , so that it also contains a dense subset of points defined over k s . We conclude that for every n  0, the intersection of Y with the image of Grn (h) is a dense constructible subset of Y and hence contains θn (y). Now it follows from Corollary 5/1.5.5 that y lies in the image of Gr∞ (h) : Gr∞ (Y) → Gr∞ (X). Corollary 3.5.2. — Let R be an unramified extension of R; if R has mixed characteristic, we assume that the residue field of R is perfect. Let X be a formal R-scheme of finite type such that Xη is néronian. If Y → X is a Néron smoothening, then the morphism of formal R -schemes Y ⊗R R → X ⊗R R obtained by base change is still a Néron smoothening. Proof. — By Proposition 2.3.2, we can identify R with R∞ (k  ), where k  denotes the residue field of R . Since smoothness of formal schemes and the property of being an analytic domain immersion of analytic spaces are

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preserved under base change, the result now follows from §3.3.8 and Proposition 3.5.1. Remark 3.5.3. — One can show that Corollary 3.5.2 is valid for all extensions R → R of ramification index one, without any assumptions on the residue fields of R and R . The corresponding statement for algebraic schemes is proven in Bosch et al. (1990, 3.6.5), and this proof carries over to the formal setting.

§ 4. TOPOLOGICAL PROPERTIES OF GREENBERG SCHEMES 4.1. Irreducible Components of Greenberg Schemes (4.1.1). — Let X be a Noetherian R-adic formal scheme. Let X be its maximal reduced closed formal subscheme, associated with the largest nilideal of OX . (1) Let Xflat be the maximal R-flat closed formal subscheme, defined by the m-torsion ideal of OX . Lemma 4.1.2. — The morphisms of k-schemes Gr∞ (X ) → Gr∞ (X)

and

Gr∞ (Xflat ) → Gr∞ (X)

deduced from the canonical immersions X → X and Xflat → X induce isomorphisms on the associated reduced schemes. In particular, the underlying maps are homeomorphisms. Proof. — Since these morphisms are closed immersions, it suffices to prove that they are surjective. Let thus ξ ∈ Gr∞ (X), let k  be a perfect field extension of its residue field, and let ψ : Spf(R∞ (k  )) → X be a morphism of formal schemes inducing ξ. Since R∞ (k  ) is reduced, the morphism ψ factors through X , hence ξ ∈ Gr∞ (X ). Since R∞ (k  ) is R-flat, this morphism factors through Xflat as well, so that ξ ∈ Gr∞ (Xflat ). This proves the lemma. Lemma 4.1.3. — Let X be a formal R-scheme of finite type, and let (Xi )i∈I be the finite family of the rig-irreducible components (see §A/3.4.7). Then one has  Gr∞ (Xi ). Gr∞ (X) = i∈I

(1) The apparently natural notation X red is classically used to denote the scheme defined by the largest ideal of definition of X.

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Proof. — Let k  be a perfect field extension of k. It suffices to show that every morphism of formal R-schemes ψ : Spf(R∞ (k  )) → X factors through a morphism Spf(R∞ (k  )) → Xi , for some i ∈ I. Let h : X → X be the normalization of X, and let (Xi )i∈I be the family of connected components of X , where Xi is defined as the scheme-theoretic image of Xi , that is, by the coherent ideal Ker(OX → h∗ OXi ). Since R∞ (k  ) is normal, there exists a unique morphism ψ  : R∞ (k  ) → X such that ψ = h ◦ ψ  . Since Spf(R∞ (k  )) is a single point, there exists a unique i ∈ I such that ψ  factors through Xi . This implies that ψ factors through the irreducible component Xi of X. Remark 4.1.4. — The straightforward generalization of Kolchin’s theorem 3/4.3.4 to Greenberg schemes does not hold: there are rig-irreducible formal schemes whose Greenberg scheme is reducible, and even disconnected. For example, let π be a uniformizer in R, and let us consider the affine formal scheme X = Spf(R{x, y}/(π − xy)). It is rig-irreducible, because the ring R{x, y}/(π − xy) is an integral domain. Let Y be the complement of the origin in X (it is the R-smooth locus of X) and denote by f the open immersion Y → X. Then for every extension R of R of ramification index one, the map Y(R ) → X(R ) is bijective, by Example 3.4.6. Hence, the open immersion Gr∞ (Y) → Gr∞ (X) is surjective and thus an isomorphism. The formal scheme Y is smooth and has two connected components, Y1 and Y2 . This gives a decomposition of Gr∞ (Y) as the disjoint sum of nonempty open subschemes Gr∞ (Y1 ) and Gr∞ (Y2 ).

4.2. Constructible Subsets of Greenberg Schemes (4.2.1). — We have recalled in A/1.1 the general definition of a (globally) constructible subset of a scheme. In the case of Greenberg schemes, the following lemma shows that they coincide with the so-called cylinders in motivic integration. Lemma 4.2.2. — Let X be a formal R-scheme of finite type. a) For every integer n ∈ N and every constructible subset D of Grn (X), ∞ −1 ) (D) of Gr∞ (X) is constructible; it is closed (resp. open) the subset (θn,X if D is closed (resp. open). b) For every constructible subset C of Gr∞ (X), there exist an integer n and ∞ −1 ) (D). Moreover, if a constructible subset D of Grn (X) such that C = (θn,X C is closed (resp. open), then D can be taken to be closed (resp. open). c) Every constructible subset of Gr∞ (X) is globally constructible. Proof. — Since X is of finite type, the k-schemes Grn (X) are of finite type as well, and Gr∞ (X) is the projective limit of the projective system (Grn (X)) n+1 . Since they are affine, the result follows along the truncation morphisms θn,X from théorème 8.3.11 of (ÉGA IV3 ), as recalled in Proposition A/1.3.3.

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(4.2.3). — Let X be a formal R-scheme of finite type, and let C be a constructible subset of Gr∞ (X). We will say that C is of level  n if there exists ∞ −1 ) (D). It is clear a constructible subset D of Grn (X) such that C = (θn,X from the definition that a constructible subset of level  n is also of level  m, for every integer m  n. Remark 4.2.4. — Let C be a constructible subset of Gr∞ (X). In Corol∞ (C) is a constructible subset of lary 5/1.5.7 below, we will show that θn,X Grn (X), for every integer n  0. (This is an application of the ArtinGreenberg approximation theorem.) Consequently, C is constructible of level  n if and only if one has C = ∞ −1 ∞ ) (θn,X (C)). (θn,X Remark 4.2.5. — Let f : Y → X be a morphism of Noetherian R-adic formal schemes. For every constructible subset C of Gr∞ (X), f −1 (C) is a constructible subset of Gr∞ (Y); see Theorem A/1.2.4, c). In particular, if Y is a closed formal subscheme of X, the trace on Gr∞ (Y) of a constructible subset of Gr∞ (X) is a constructible subset of Gr∞ (Y). Lemma 4.2.6. — Let X be a formal R-scheme of finite type. Let Z be a closed subscheme of Gr∞ (X); for every integer n, let Zn be the schematic image of Z in Grn (X). Then the canonical morphism Z → limn0 Zn is an ←− isomorphism of schemes. Proof. — By Lemma 3.2.2, we may assume that X is affine. Then Gr∞ (X) is affine and we let A be its ring of functions; similarly, for every integer n  0, Grn (X) is affine and we let An be its ring of functions. The truncation morphisms θnm : Grm (X) → Grn (X) and θn : Gr∞ (X) → Grn (X), for all integers m, n such that m  n, correspond to morphisms of rings ϕm n : An → Am and ϕn : An → A and identify A with the inductive limit limn0 An . −→ Let I be the ideal of Z; for every integer n, let In = ϕ−1 n (I). One then has Zn = V (In ). The morphisms ϕn induce ring morphisms ψn : (An /In ) → m (A/I); the morphisms ϕm n induce ring morphisms ψn : (An /In ) → (Am /Im ). The canonical morphism limn0 (An /In ) → (A/I) is an isomorphism of rings. −→ Consequently, Z → limn0 Zn is an isomorphism of schemes. ←− 4.3. Thin Subsets of Greenberg Schemes Definition 4.3.1. — Let X be a formal R-scheme of finite type, and let d be the dimension of its generic fiber Xη . A subset A of Gr∞ (X) is said to be thin if there exist a finite affine open covering (Ui )i∈I of X, and, for every i ∈ I, a closed formal subscheme Zi of Ui , such that: a) For every i ∈ I, A ∩ Gr∞ (Ui ) ⊂ Gr∞ (Zi ); b) For every i ∈ I, the dimension of the generic fiber (Zi )η is at most d − 1. Otherwise, we say that A is fat.

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(4.3.2). — Let X be a formal R-scheme of finite type. Let X , resp. Xflat , be the maximal reduced, resp. flat, closed formal subscheme of X. Recall Lemma 4.1.2: the inclusions of X and Xflat in X induce homeomorphisms from Gr∞ (X ) and Gr∞ (Xflat ) to Gr∞ (X). Lemma 4.3.3. — Let X be a formal R-scheme of finite type. Let A be a subset of Gr∞ (X). The following assertions are equivalent: a) The set A is thin in Gr∞ (X); b) The set A is thin in Gr∞ (X ); c) The set A is thin in Gr∞ (Xflat ). Proof. — Observe that the dimensions of the generic fibers of Xred and Xflat are equal to that of Xη . Consequently, a thin subset of Gr∞ (Xred ), resp. of Gr∞ (Xflat ), is thin in Gr∞ (X). Conversely, in Definition 4.3.1, we can replace each Zi by its maximal R-flat reduced closed formal subscheme without changing Gr∞ (Zi ); then Zi factors through X and Xflat , so that thinness in Gr∞ (X) implies thinness in Gr∞ (X ) and Gr∞ (Xflat ). Example 4.3.4. — Let π be a uniformizer of R, let m and n be integers such that m  2 and n  0, and let X = Spf(R{X, Y1 , . . . , Yn }/(X m − π)). One has dim(Xη ) = n. However, Gr∞ (X) = ∅, so that Gr∞ (X) is thin. This example illustrates that the property of being thin depends on X and not only on Gr∞ (X). Example 4.3.5. — Let π be a uniformizer of R, let m and n be integers such that m  2 and n  0, and let X = Spf(R{X1 , X2 , Y1 , . . . , Yn }/(X1m − πX2m )). Let Z be the closed formal subscheme of X defined by the ideal (X1 , X2 ). One has dim(Xη ) = n + 1 and dim(Zη ) = n. If R is an extension of R of ramification index one, the only solution of the equation X1m = πX2m in R is (0, 0). This implies the set theoretical equality Gr∞ (X) = Gr∞ (Z); hence Gr∞ (X) is thin. Example 4.3.6. — Let X be the complex Whitney umbrella, i.e., the hypersurface of A3C defined by the polynomial T1 T22 − T32 . Let Z be its singular locus; it is defined by the equations T22 = T1 T2 = T3 = 0 in X, so that it is supported on the T1 -axis in C3 . Put R = C[[t]], and let X and Z be the R-adic formal completions of X ⊗C R and Z ⊗C R. Choose an integer m  1 and let x ∈ Lm (X)(C) be the C-jet of order m given by (t, 0, 0). Then one can check that the constructible subset (θm,X )−1 (x) = (θm,X )−1 (x) is contained in L∞ (Z) = Gr∞ (Z). In particular, (θm,X )−1 (x) is thin in L∞ (X) = Gr∞ (X). Remark 4.3.7. — a) Let X be a k-variety. Let us say that a subset of the space of arcs L∞ (X/k) is algebraically thin if it is contained in a subspace of the form L∞ (Z/k), where Z is a closed subvariety of X such

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that dim(Z) < dim(X). These are the sets called thin in Ein et al. (2004); let us compare this definition with Definition 4.3.1. We thus assume that we are in the equal characteristic case, i.e., R = k[[t]], and let X be the R-adic formal completion of X ⊗k R, so that the Greenberg space Gr∞ (X) identifies canonically with the space of arcs L∞ (X/k) of X. One has dim(Xη ) = dim(X). Every algebraic closed subvariety Z of X gives rise to a closed formal subscheme Z of X such that dim(Zη ) = dim(Z) and the closed subset Gr∞ (Z) of Gr∞ (X) identifies with the closed subset L∞ (Z/k) of L∞ (X/k). Consequently, any algebraically thin subset of L∞ (X/k) is thin. However, the converse does not hold in general. Here is an easy counterexample: set X = A1k = Spec(k[T ]) and consider the k-arc γ defined by T → t. Then the singleton S consisting of γ is not algebraically thin in L∞ (X/k) because the image of γ in X contains the generic point of X (the only algebraically thin arcs on a curve over k are the constant arcs at closed points). On the other hand, γ defines a section of the structural morphism X → Spf(R) and thus gives rise to a closed formal subscheme Z of X of relative dimension 0 over R. The point γ of Gr∞ (X)(k) lies in Gr∞ (Z)(k), so that the set S is thin as a subset of Gr∞ (X). We can also consider the following finer example. Set X = A2k , so that R{T } be any converging power series which X = Spf(R{T1 , T2 }). Let f ∈  n is not algebraic, such as f = n0 tn T 2 , and let Y be the closed formal subscheme of X defined by the ideal (T2 − f (T1 )). Since Yη is an analytic curve, the set Gr∞ (Y) is thin in Gr∞ (X). However, there is no strict algebraic  This subvariety Z of X ⊗k R such that Gr∞ (Y) is contained in Gr∞ (Z). shows that we would get a strictly weaker notion of thinness if we worked with algebraic subschemes of X ⊗k R rather than formal subschemes of its formal completion. b) Let Z be a closed formal R-subscheme of X such that dim(Zη ) < d. By Definition 4.3.1, Gr∞ (Z) is a thin subset of Gr∞ (X). Subsets of Gr∞ (X) which are contained in such a subspace could be called globally thin. This furnishes a stronger notion, because not every closed formal subscheme of an affine open subscheme U of X extends to a closed formal subscheme of X in general. However, this notion is not useful for motivic integration. Remark 4.3.8. — Let us assume that k is perfect, and let X be a formal R-scheme of finite type of pure relative dimension. We shall define below (Definition 5/1.3.1) the singular locus Xsing of X, which is a closed formal subscheme. We will then prove (Proposition 6/2.4.6) that a constructible subset of Gr∞ (X) is thin if and only if it is contained in Gr∞ (Xsing ). ˆ R, for some kAssume, in particular, that R = k[[t]] and that X = X ⊗ ˆ variety X. Then Xsing = (Xsing ) ⊗ R so that a constructible subset of L (X) is thin if and only if it is contained in L (Xsing ).

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4.4. Order Functions and Constructible Sets (4.4.1). — Let X be a formal R-scheme of finite type. Let R be an extension of R of ramification index one, and let ψ : Spf(R ) → X be a morphism of formal schemes. Let x ∈ X be the image of the unique point of Spf(R ) by ψ. For f ∈ OX,x , we define ordψ (f ) = ordR (ψ ∗ (f )),

(4.4.1.1)

where ordR : R → N ∪ {+∞} denotes the normalized discrete valuation of R . By convention, one has ordψ (f ) = +∞ if ψ ∗ (f ) = 0. For f, f  ∈ OX,x , one has the relations ordψ (f + f  )  min(ordψ (f ), ordψ (f  ) ordψ (f f  ) = ordψ (f ) + ordψ (f  ). In other words, the function ordψ is a semi-valuation on the ring OX,x . If R is an extension of R of ramification index one, and ψ  : Spf(R ) → X is the composition of ψ with the natural morphism from Spf(R ) to Spf(R ), one has ordψ = ordψ . (4.4.2). — Let ξ ∈ Gr∞ (X). As in (3.3.7), we associate with ξ a morphism ψξ : Spf(R∞ (κξ )) → X, and we set (4.4.2.1)

ordξ = ordψξ .

(4.4.3). — Let now I be a coherent sheaf of ideals on X. To this sheaf of ideals, we attach an order function ordI : Gr∞ (X) → N ∪ {+∞} on Gr∞ (X), defined by (4.4.3.1)

ordI (ξ) = inf ordξ (f ). f ∈Ix

∞ (ξ) belongs to the support of the One has ordI (ξ)  1 if and only if θ0,X closed subscheme of X0 defined by the ideal I . One has ordI (ξ) = +∞ if and only if ξ belongs to the Greenberg scheme of the closed formal subscheme Z of X defined by I (equivalently, if and only if ψξ factors through Z). In particular, ordI takes values in N if Zη = ∅ (equivalently, if I induces the unit ideal sheaf on Xη ). Let I and J be coherent sheaves of ideals. Let ξ ∈ Gr∞ (X). The following formulas follow from the definition:

(4.4.3.2) (4.4.3.3)

ordI +J (ξ) = min(ordI (ξ), ordJ (ξ)), ordI ·J (ξ) = ordI (ξ) + ordJ (ξ).

Similarly, if I ⊂ J , then one has (4.4.3.4)

ordI (ξ)  ordJ (ξ)

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Example 4.4.4. — In the affine case, the function ordI can be described very concretely. Let us assume that X = Spf(A), where A has a presentation A = R{X1 , . . . , XN }/(f1 , . . . , fm ). In this context, the coherent ideal I corresponds to an ideal I of R{T1 , . . . , Tn } containing (f1 , . . . , fm ). Let (g1 , . . . , gs ) be a finite family which generates I. The closed formal subscheme Y of X defined by I is then equal to Spf(R{X1 , . . . , XN }/(g1 , . . . , gs )). Let ξ ∈ Gr∞ (X) and let ψξ : Spf(R ) → X be the corresponding morphism of formal schemes, where R = R∞ (κξ ). This morphism ψ corresponds to a family x = (x1 , . . . , xN ) ∈ (R )n such that f1 (x) = · · · = fs (x) = 0. Then one has (4.4.4.1)

ordI (ξ) = min(ordR (h(x))) = min (ordR (gi (x))). 1is

h∈I

This is finite if and only if there exists i such that gi (x) = 0, that is, if and only if ξ ∈ / Gr∞ (Y). Example 4.4.5. — Let X, Y be formal schemes of finite type over Spf(R), purely of relative dimension d  0. Let f : Y → X be an R-morphism of formal schemes. Assume that Y is smooth over Spf(R). We will define below the Jacobian ideal Jacf which is a coherent ideal on Y measuring where f is not smooth. The associated function ordJacf will be a crucial ingredient of the change of variables formula in motivic integration. Lemma 4.4.6. — Let f : X → Y be a morphism of formal R-schemes of finite type, and let I be a coherent ideal on Y. One has ordI OX = ordI ◦ Gr∞ (f ) on Gr∞ (X). Proof. — Let us consider an extension R of R of ramification index one and a morphism ψ : Spf(R ) → X, corresponding to a point ξ of Gr∞ (X). ∞ (ξ). The point f (ξ) ∈ Gr∞ (Y) corresponds to the morphism Let x = θ0,X  f ◦ ψ : Spf(R ) → Y. By definition, one then has ordI OX (ξ) =

inf

u∈(f ∗ I )x

ordR (ψ ∗ u) =

inf

v∈If (x)

ordR (ψ ∗ f ∗ v) = ordI (f (ξ)),

as was to be shown. Proposition 4.4.7. — Let X be a formal R-scheme of finite type and let I be a coherent sheaf of ideals on X. Let n  0 be an integer and let C = {x ∈ Gr∞ (X) | ordI (x) > n}. Then C is a closed constructible subset of Gr∞ (X) of level  n. More precisely, one has (4.4.7.1)

∞ −1 ) (Grn (Y)), C = (θn,X

where Y denotes the closed formal subscheme of X defined by I .

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Proof. — Let ξ be a point of Gr∞ (X), associated with a morphism ∞ (ξ) ∈ X. ψξ : Spf(R(κξ )) → X as in (3.3.7). Let x = θ0,X ∞ Let ξn = θn,X (ξ). This is a point of Grn (X), corresponding to the morphism ψξ,n : Rn (κξ ) → X, defined by the composition of ψξ with the canonical morphism Spf(Rn (κξ )) → Spf(R∞ (κξ )). For every f ∈ Ix , one has ord(ψξ∗ f ) > n if and only if ψξ∗ (f ) belongs to ∗ the nth power of the maximal ideal of R∞ (κξ ) and if and only if ψn,ξ (f ) = 0. This implies the asserted formula ∞ −1 ) (Grn (Y)). C = (θn,X

By Theorem 3.2.3, Grn (Y) is a closed subscheme of Grn (X). Consequently, C is a closed constructible subset of level  n, as was to be shown. Corollary 4.4.8. — For every integer n ∈ N, the set {ξ ∈ Gr∞ (X) ; ordI (ξ) = n} is constructible in Gr∞ (X). If I induces the unit ideal on Xη , then the function ordI : Gr∞ (X) → N is bounded. Proof. — The first part of the statement follows at once from Proposition 4.4.7. If I induces the unit ideal on Xη , then the closed formal subscheme of X defined by I has empty generic fiber, so that ordI does not reach the value +∞. Thus all the fibers of ordI are constructible. In particular, ordI takes only finitely many values, by the quasi-compactness of the constructible topology on Gr∞ (X) (see Theorem A/1.2.4).

CHAPTER 5 STRUCTURE THEOREMS FOR GREENBERG SCHEMES

(0.0.1). — Throughout this chapter, we denote by R a complete discrete valuation ring with maximal ideal m and residue field k. For every integer n  0, we set Rn = R/mn+1 . When R has mixed characteristic, we will tacitly assume that k is perfect. In the case of equal characteristic, we choose a section of the ring morphism R → k, which turns R into a k-algebra; then any uniformizer t of R induces a k-isomorphism k[[t]] R of complete discrete valuation rings. (0.0.2). — In order to develop the theory of motivic measures on Greenberg schemes of formal R-schemes, it is important to have a good understanding of the structure of these schemes. First of all, we need to understand the behavior of the truncation morphisms between Greenberg schemes of different levels. This behavior is quite transparent when the formal scheme is smooth over R but much more subtle in the presence of singularities. One of the main tools at our disposal is the Artin–Greenberg approximation theorem, which can be thought of as a geometric version of Hensel’s lemma for singular formal schemes. We also need to understand the structure of the morphisms of Greenberg schemes that are induced by morphisms of formal R-schemes. These results will later allow us to compare the motivic measures on these Greenberg schemes and prove a change of variables formula for motivic integrals, which is the crucial point in most applications of the theory. (0.0.3). — In this chapter, we work within the general framework of Greenberg schemes of formal R-schemes of finite type, introduced by Sebag (2004a) in the context of motivic integration. This generality comprises the following two important special cases. – (Arc schemes). Let X be a k-variety. We set R = k[[t]] and define ˆ k R. Then the Greenberg schemes identify functorially with the jet X=X⊗ schemes:

© Springer Science+Business Media, LLC, part of Springer Nature 2018 A. Chambert-Loir et al., Motivic Integration, Progress in Mathematics 325, https://doi.org/10.1007/978-1-4939-7887-8_5

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Grn (X) = Ln (X/k)

and

Gr∞ (X) = L∞ (X/k).

This is the classical setting of motivic integration, as introduced by Kontsevich (1995) and developed by Denef and Loeser (1999). – (Greenberg schemes of R-varieties, equal characteristic). We still consider R = k[[t]] but start from an R-variety X and define X as its formal completion. Then one has functorial identifications Grn (X)  Grn (X ) for all n. This setting has been first put forward by Looijenga (2002). (0.0.4). — To simplify notation, we will sometimes denote Gr∞ (X), ∞ by Gr(X), Gr(f ), and θn,X . We will also omit the symbol Gr∞ (f ), and θn,X m X from the notations for the truncation morphisms θn,X and θn,X if the formal R-scheme X is clear from the context.

§ 1. GREENBERG APPROXIMATION ON FORMAL SCHEMES 1.1. Fitting Ideals (1.1.1). — Let A be a ring, and let M be an A-module of finite presentation. So, we can find an exact sequence of A-modules ϕ

Ar − → As → M → 0 for some integers r, s  0. The linear map ϕ is represented by an (s × r)matrix L. For every integer i  0, the i-th Fitting ideal Fitti (M ) of M is the ideal of A generated by the minors of rank (s − i) of the matrix L; in particular, Fitti (M ) = 0 if i < s − r since, in that case, s − i > r. By convention, one sets Fitti (M ) = A if i  s. These ideals do not depend on the chosen presentation (see, e.g., Eisenbud (1995), §20.2) and form an ascending sequence Fitt0 (M ) ⊂ Fitt1 (M ) ⊂ . . . . (1.1.2). — It is clear from the definition that the construction of the Fitting ideals commutes with arbitrary base change: if B is an A-algebra, then Fitti (M ⊗A B) is the ideal Fitti (M )B of B for every i  0. One can deduce from Nakayama’s lemma a useful geometric interpretation of the Fitting ideals. Namely, for every integer i and every prime ideal p of A, the following properties are equivalent: (i) One has p ⊃ Fitti (M ); (ii) The Ap -module Mp cannot be generated by i elements; (iii) The κ(p)-vector space M ⊗A κ(p) has dimension > i, where κ(p) is the field of fractions of A/p. Example 1.1.3. — Let us recall that, for every n ∈ N, we denote by Rn the ring R/mn+1 . Let M be a finitely generated R-module, and let π be a

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uniformizer in R. Then M is isomorphic to m  Rni ) R ⊕ ( i=1

for some integers , m  0 and n1 , . . . , nm  0. Thus M has a presentation of the form Rm → R+m → M → 0 where the injective map Rm → R+m is given by (r1 , . . . , rm ) → (0, . . . , 0, π n1 +1 r1 , . . . , π nm +1 rm ). It follows that Fitti (M ) = (0) for i < and Fitt (M ) = mn1 +...+nm +m . Note that the exponent n1 + . . . + nm + m is precisely the length of the torsion submodule of M . (1.1.4). — The definition of the Fitting ideals can be extended to coherent sheaves on locally noetherian formal schemes. Let X be a locally noetherian formal scheme, and let M be a coherent OX -module. For every integer i  0, there exists a unique coherent ideal sheaf Fitti (M ), called the i-th Fitting ideal of M , such that (Fitti (M ))(U) = Fitti (M (U)) for every affine open formal subscheme U of X. Moreover, for every ideal of definition I of X and all nonnegative integers i, n, one has an equality of sheaves of ideals Fitti (M )/I n Fitti (M ) = Fitti (M /I n M ) on the locally noetherian scheme (X, OX /I n ). Let x be a point of X. The following properties are equivalent: (i) The point x belongs to the closed formal subscheme defined by Fitti (M ); (ii) The OX,x -module Mx cannot be generated by i elements; (iii) The κ(x)-vector space Mx ⊗ κ(x) has dimension > i. 1.2. Greenberg Schemes of Smooth Formal Schemes Proposition 1.2.1. — Let X be a smooth formal R-scheme of finite type. The map X(S) → X(Sn ) is surjective for every extension S of R and every integer n  0. Proof. — This is a direct consequence of the infinitesimal lifting criterion for smoothness. Proposition 1.2.2. — Let X be a formal R-scheme of finite type. Assume that X is smooth of pure relative dimension d.

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a) For every integers m, n such that m  n  0, the truncation morphism m θn,X : Grm (X) → Grn (X) (m−n)d

is a locally trivial fibration with fiber Ak . b) For every integer n  0, the truncation morphism θn,X : Gr(X) → Grn (X) d is a locally trivial fibration with fiber (AN k ) .

Proof. — Since X is smooth, every point of X has an affine formal neighborhood U such that there exists an étale morphism to Y = Spf(R{x1 , . . . , xd }), where d is the relative dimension of X at x. Let m ∈ N ∪ {+∞} and let n ∈ N such that n  m. By lemma 4/3.2.2, the diagram Grm (U) m θn,X

Grn (U)

Grm (Spf(R{z1 , . . . , zd })) m θn,Y

Grn (Spf(R{z1 , . . . , zd }))

is commutative and Cartesian. a) If m is finite, example 4/3.1.12 asserts that there exist isomorphisms Grm (Y)  Ad(m+1)k and Grn (Y)  Ad(n+1)k under which the morm identifies with a linear projection. This furnishes an isomorphism phism θn,Y (m−n)d

Grm (U)  Grn (U) ×k Ak

,

m which transforms θn,X into the first projection. b) Assume now that m = ∞. Then there exists isomorphisms Gr∞ (Y)  d(n+1) ∞ AdN and Grn (Y)  Ak under which the truncation morphism θn,Y k identifies with a linear projection. In this case, we obtain an isomorphism d Gr∞ (U)  Grn (U) ×k (A∞ k ) ∞ which transforms θn,X into the first projection.

1.3. The Singular Locus of a Formal Scheme Definition 1.3.1. — Let X be a formal R-scheme of finite type of relative dimension d. We define the Jacobian ideal JacX of X by JacX = Fittd (Ω1X/R ). We denote by Xsing the closed formal subscheme of X defined by the coherent ideal sheaf JacX and call it the singular locus of X. Since the formation of Fitting ideals commutes with arbitrary base change, the formation of Xsing is compatible with base change to arbitrary extensions of R. Proposition 1.3.2. — Let X be a formal R-scheme of finite type of pure relative dimension d.

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a) If X is flat over R, then the complement of Xsing in X is the open formal subscheme of X consisting of the points where X is smooth over R. b) The complement of (Xsing )η in Xη is the K-analytic open subspace of Xη consisting of the points where Xη is quasi-smooth over K. Proof. — If X is flat over R, the formal scheme X is smooth over R at a point x ∈ X if and only if X0 = X ⊗R k is smooth over k at x. The formation of Fitting ideals is compatible with base change, so a) follows from the fact that X0 is smooth over k at x if and only if the OX0 ,x -module Ω1X0 /k,x can be generated by d elements. The analogous property for Xη follows from Ducros (2009, 6.3). Remark 1.3.3. — The expression “singular locus” for Xsing is an abuse of terminology for several reasons: – It is a notion relative to the structural morphism X → Spf(R); – Neither the closed formal subscheme Xsing of X nor its underlying reduced closed formal subscheme is characterized by their support in X; – If X is not flat over R, then Xsing may not contain all of the points at which X is not smooth over R. For example, if X is the disjoint union of Spec(k) and Spf(R), then Xsing is empty. (1.3.4). — In practice, one can compute the Jacobian ideal JacX in the following way. Suppose that x is a point in X(k). Then we can write the completed local ring of X at x as OX,x ∼ = R[[z1 , . . . , zr ]]/(f1 , . . . , f ) for some , r  0. If we denote by Jf (x) the Jacobian matrix

 ∂fi (x) Jf (x) = ∂zj i=1,...,,j=1,...,r of the tuple f = (f1 , . . . , f ) at the point x, then the OX,x -module Ω1X,x ⊗OX,x OX,x is isomorphic to the cokernel of the morphism of free modules O → X,x

r OX,x defined by the matrix Jf (x). By the stability of Fitting ideals under base change, it follows that the ideal JacX ·OX,x is generated by the rank r − d minors of the Jacobian matrix Jf (x).

(1.3.5). — Let X be a formal R-scheme of finite type. We now give an alternative description of its singular locus Xsing , similar to the description of the singular locus of a variety explained in corollary 3/5.3.4. In a slightly different setting, one can find this description in Elkik (1973, §0.2); see also Section 4 in Swan (1998). This description does not require X to be of pure relative dimension over R. (1.3.6). — Assume that X is affine, say X = Spf(A), and choose a finite presentation A = R{z1 , . . . , zr }/I for the topological R-algebra A. If f = (f1 , . . . , f ) is any finite tuple of elements in I of length  r, we denote by Δ(f1 , . . . , f ) the ideal of R{z1 , . . . , zr } generated by all rank

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minors of the Jacobian matrix

Jf =

∂fi ∂zj

 i=1,...,, j=1,...,r

with the convention that Δ(f1 , . . . , f ) is the trivial ideal (1) if = 0. We denote by J the ideal  (1.3.6.1) J =I+ Δ(f1 , . . . , f )((f1 , . . . , f ) : I) f

of R{z1 , . . . , zr }, where the sum is taken over all tuples f = (f1 , . . . , f ) of elements in I of length  r. (Recall that for any two ideals I, J of a ring S, one sets (I : J) = {s ∈ S ; sJ ⊂ I}. This is again an ideal of S.) We denote by JacX the image of J in A = R{z1 , . . . , zr }/I and by HX its radical. The ideal JacX may depend on the chosen presentation of A, but corollary 1.3.8 below insures that HX does not. Proposition 1.3.7. — With the preceding notations, the ideal HX of A is the intersection of the prime ideals p of A such that the complex of Ap -modules 0 → (I/I 2 ) ⊗A Ap →

(1.3.7.1)

r 

 A/R ⊗A Ap → 0, Ap dzi → Ω

i=1

that is deduced from the fundamental complex (A/3.3.5.2) associated with the pair (R{z1 , . . . , zr }, I), is not split exact. Proof. — Let p be a prime ideal of A, and let q be its inverse image in R{z1 , . . . , zr }. For every finite tuple f = (f1 , . . . , f ) of elements of I, the complex (1.3.7.1) sits in a commutative diagram A

Jf

Ar

r

0

(I/I 2 )

⊗A A

A dzi

A/R ⊗A Ap

0.

i=1

First, assume that HX is not contained in p. This means that there exists a tuple (f1 , . . . , f ) of elements in I, with  r, such that Δ(f1 , . . . , f ) is not contained in p and ((f1 , . . . , f ) : I) is not contained in the inverse image q of p in R{z1 , . . . , zr }. Then at least one of the ( × )-minors of the Jacobian matrix Jf does not belong to p; moreover, f1 , . . . , f generate the ideal I locally at the prime q of R{z1 , . . . , zr }. By inspection of the above diagram,  A/R ⊗A Ap is isomorphic to the cokernel of the linear map we conclude that Ω (Ap ) → (Ap )r defined by the Jacobian matrix Jf . Since Jf has a minor of  A/R ⊗A Ap is a maximal rank which is invertible in Ap , this implies that Ω free Ap -module of rank r − . It follows that the complex (1.3.7.1) is split exact.

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Conversely, assume that this complex is a split exact sequence. In partic A/R ⊗A Ap and (I/I 2 ) ⊗A Ap are free, and the sum ular, the Ap -modules Ω of their ranks is equal to r. Let then (f1 , . . . , f ) be a family of elements of I whose images form a basis of (I/I 2 ) ⊗A Ap . By Nakayama’s lemma, the family (f1 , . . . , f ) generates the ideal I locally at the prime ideal q, so that the ideal ((f1 , . . . , f ) : I) is not contained in q. By inspection of the  A/R ⊗A Ap identifies with the cokernel of above diagram, the Ap -module Ω  r the linear map (Ap ) → (Ap ) defined by the Jacobian matrix Jf . Since it is free of rank r − , at least one of the minors of rank of Jf is a unit in Ap . Consequently, HX is not contained in p. Corollary 1.3.8. — a) Let p be an open prime ideal of A. Then p contains HX if and only if X = Spf(A) is smooth over R at the point corresponding to p. b) Let P be a prime ideal of A ⊗R K. Then the inverse image of P in A contains HX if and only if Xη = M (A ⊗R K) is quasi-smooth over K at any point supported at P. c) The ideal HX does not depend on the choice of a finite presentation of A. Proof. — Point c) follows from a) and b), because HX is radical and thus an intersection of prime ideals in A. Split exactness of the complex (1.3.7.1) can be tested after base change over any faithfully flat local homomorphism Ap → B. Assume that p is an open prime ideal of A, and denote by x the point of X corresponding to p. Then the natural local homomorphism Ap → OX,x is faithfully flat, since this homomorphism is an isomorphism on the respective completions and the source and target are noetherian. Therefore, our criterion for p to contain HX follows from the Jacobian criterion for smoothness for formal R-schemes (Tarrío et al. 2007, 4.15). The case where P is a prime ideal of A ⊗R K can be proven in the same way. By Berkovich (1993, 2.1.1), the natural map M (A ⊗R K) → Spec(A ⊗R K) is surjective; let x be any point in the preimage of P. Then the local homomorphism (A ⊗R K)P → OXη ,x is faithfully flat by (Berkovich 1993, 2.1.4) and the result follows from the Jacobian criterion for quasi-smoothness for K-analytic spaces; see Ducros (2018, 4.2.1). Corollary 1.3.9. — Let X be a flat affine formal R-scheme of finite type of pure relative dimension. The ideal HX is the radical of the Jacobian ideal c ⊂ JacX . JacX of X. Thus, there exists an integer c  1 such that HX Proof. — The first assertion is a consequence of proposition 1.3.2 and corollary 1.3.8; the second follows, since the ring O(X) is noetherian. Corollary 1.3.10. — Let X = Spf(A) be a formal R-scheme of finite type such that HX = (0) and A ⊗R K = {0}.

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a) If K has characteristic zero, then A ⊗R K is not reduced. b) If K has characteristic p > 0, then there exists a finite extension K  of K such that (K  )p ⊂ K and A ⊗R K  are not reduced. Proof. — By corollary 1.3.8, the affinoid K-analytic space Xη = M (A ⊗R K) is nowhere quasi-smooth. In other words, the local ring OXη ,x is not geometrically regular for any point x of Xη . Then it follows from proposition 2.1.1 and theorem 2.2.1 in Berkovich (1993) that A ⊗R K is not geometrically regular at any of its prime ideals. Consider in particular a minimal prime ideal p of A ⊗R K (such an ideal exists since A ⊗R K = (0)); then the local ring (A ⊗R K)p is an inseparable K-algebra. If K has characteristic zero, this means that (A ⊗R K)p and therefore A ⊗R K are not reduced. If K has characteristic p > 0, we conclude that there exists a finite extension K  of K such that (K  )p ⊂ K and A ⊗R K  are not reduced. 1.4. An Application of Hensel’s Lemma Theorem 1.4.1. — Let X be a formal R-scheme of finite type of pure relative dimension; assume that X is a local complete intersection. Then for all integers n  e  0, we have the following property: if x is an element of X(Rn+e ) whose image in X(Re ) does not belong to Xsing (Re ), then the image of x in X(Rn ) can be lifted to an element of X(R). n+e Proof. — Let x0 = θ0,X (x). Since X is a local complete intersection, there exist an integer r   0 and a regular sequence (f1 , . . . , f ) of elements in R[[z1 , . . . , zr ]] such that the completed local ring OX,x0 is isomorphic as an R-algebra to R[[z1 , . . . , zr ]]/(f1 , . . . , f ). The relative dimension of X over R at x0 is equal to d = r − . Let a = (a1 , . . . , ar ) ∈ (m)r be a lifting of (z1 (x), . . . , zr (x)) ∈ (m/mn+e+1 )r . By assumption, fi (a) ∈ mn+e+1 for every i ∈ {1, . . . , }. We have seen in (1.3.4) that the ideal JacX ·OX,x0 is generated by the × -minors of the Jacobian matrix of the tuple of power series (f1 , . . . , f ). n+e Since θe,X (x) ∈ Xsing (Re ), there exists such a minor Δ such that

Δ(a) ≡ 0 mod me+1 . By lemma 1/1.3.3, there exists b = (b1 , . . . , br ) ∈ (m)r such that aj ≡ bj (mod mn+1 ) for every i ∈ {1, . . . , r} and fi (b) = 0 for every i ∈ {1, . . . , }. This tuple b corresponds to a point y ∈ X(R) whose image in X(Rn ) equals that of x. This concludes the proof. (1.4.2). — Let X be a formal R-scheme of finite type of pure relative dimension d, and let JacX be its Jacobian ideal. In §4/4.4.3, we have attached to JacX a function on Gr(X); we call it the order of the Jacobian and denote it by ordjacX . It can be computed in the following way: every point x of Gr(X) corresponds to a point in X(R ) for some extension R of R of ramification

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index one. Since the formation of Fitting ideals commutes with base change, we have ordjacX (x) =

inf

f ∈Fittd (x∗ Ω1X/R )

ordR (f ) = lengthR (R / Fittd (x∗ Ω1X/R )).

If x is contained in the singular locus of X, then ordjacX (x) = +∞. Otherwise, x∗ Ω1X/R is an R -module of rank d, and it follows from example 1.1.3 that ordjacX (x) = lengthR (x∗ Ω1X/R )tors . For every integer e  0, we define a constructible subset of Gr(X) by (1.4.2.1) Gr(e) (X) = {x ∈ Gr(X) | ordjacX (x)  e} = (θe )−1 (Gre (X) Gre (Xsing )). Likewise, for every integer n  e, we set (1.4.2.2)

Grn(e) (X) = (θen )−1 (Gre (X)

Gre (Xsing )).

If Xη is quasi-smooth, then JacX induces the unit ideal on Xη because the formation of Fitting ideals commutes with base change to K. In that case, ordjacX takes values in N. Thus Gr(e) (X) = Gr(X) when e is sufficiently large, because the constructible topology on Gr(X) is quasi-compact by theorem A/1.2.4. Corollary 1.4.3. — Let X be a formal R-scheme of finite type of pure relative dimension. If X is a local complete intersection, then (e)

θn (Gr(e) (X)) = θnn+e (Grn+e (X)) for all integers n  e  0. Proof. — The inclusion (e)

θn (Gr(e) (X)) ⊂ θnn+e (Grn+e (X)) is obvious. Conversely, take ξ ∈ Grn+e (X). Let k  be the residue field of ξ if R has equal characteristic and its perfection otherwise; let S = R(k  ). The point ξ corresponds to a point x ∈ X(Sn+e ) whose image in X(Se ) does not ˆ S, there exists a belong to Xsing . By the preceding theorem, applied to X ⊗ point y ∈ X(S) whose image in X(Sn ) coincides with that of x. The point y n+e ∞ corresponds to a point ψ of Gr(X) such that θn,X (ψ) = θn,X (ξ). Since n  e, (e)

n+e ∞ one has also θe,X (ψ) = θe,X (ξ), so that η ∈ Gr(e) (X). This concludes the proof.

1.5. Greenberg’s Approximation Theorem Theorem 1.5.1 (Greenberg approximation). — Let X be a formal Rscheme of finite type.

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a) Assume that K has characteristic zero. Then there exist integers γ, δ > 0 such that, for every integer n  0 and every extension S of R of ramification index one, the images of the truncation maps X(S) → X(Sn )

and

X(Sγn+δ ) → X(Sn )

coincide. b) When K has characteristic p > 0, the same statement holds, provided one only considers unramified extensions S. In particular, it holds for S = R. Proof. — For notational convenience, we say that a pair (γ, δ) of positive integers is suitable for X if the conclusion of the theorem holds: for every integer n  0 and every extension S of R of ramification index one (resp. every unramified extension S, if K has positive characteristic), the images of the truncation maps X(S) → X(Sn ) and X(Sγn+δ ) → X(Sn ) coincide. It will be convenient to denote such truncation maps by θn and θnγn+δ , in accordance with the notations for the truncation morphisms of Greenberg schemes. We will prove the theorem by noetherian induction on X. We thus assume that the theorem holds for every strict closed formal subscheme of X. Since X is quasi-compact, we may assume that X is affine, say, X = Spf(A) with A = R{z1 , . . . , zr }/I for some integer r  0 and some ideal I of R{z1 , . . . , zr }. We will make use of the ideal HX defined in §1.3.5. a) Let us first assume that A ⊗R K is the trivial ring. In this case, there exists an integer n0  0 such that mn0 +1 A = (0). This implies that X(Sn ) = ∅ for every integer n  n0 . In this case, the theorem is true for γ = 1 and δ = n0 . In the rest of the proof, we suppose that A ⊗R K = 0. b) We then assume that HX is not the zero ideal. Let Y be the closed formal subscheme of X defined by HX . By the induction hypothesis, there exists a pair (γ, δ) of positive integers that is suitable c ⊂ JacX . for Y. By corollary 1.3.9, there exists an integer c  1 such that HX We will show that the pair (2cγ, 2cδ) is suitable for X. Let S be an extension of R of ramification index one, let n ∈ N and let x ∈ X(S2cγn+2cδ ); we must show that the point x = θn (x) ∈ X(Sn ) can be lifted to X(S). Assume first that θγn+δ (x) ∈ Y(Sγn+δ ); by the induction hypothesis, there exists a point y ∈ Y(S) such that θn (y) = θn (x), hence the desired result in this case. We may thus assume that θγn+δ ∈ Y(Sγn+δ ). Thus, there exist a tuple f = (f1 , . . . , f ) of elements of I, with  r; a rank minor, Δ, of the Jacobian matrix Jf ; and an element h of ((f1 , . . . , f ) : I) such that Δ(x) ≡ 0 and h(x) ≡ 0 mod mcγn+cδ+1 S. Let X be the closed formal subscheme of Spf(R{z1 , . . . , zr }) defined by the ideal (f1 , . . . , f ). Note that X is a closed formal subscheme of X . By lemma 1/1.3.3, there exists a point y in X (S) such that x and y have the same image in X (Scγn+cδ ). In particular, y is a lift of x . Let us now prove that y ∈ X(S). Let g be an element of I. Then gh belongs to the ideal (f1 , . . . , f ), so that (gh)(y) = 0. On the other hand, we

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know that h(y) ≡ h(x) ≡ 0 mod mcγn+cδ+1 S. Thus g(y) = 0 for every g ∈ I. This means that y lies in X(S). c) Let us now settle the case where HX = (0). If K has characteristic zero, it follows from corollary 1.3.10 that A ⊗R K is not reduced; let us then set R = R and K  = K. If K has characteristic p > 0, this corollary asserts that there exists a finite extension K  of K such that (K  )p ⊂ K and A ⊗R K  are not reduced; let R be the integral closure of R in K  . In both cases, R is a finite extension of R such that X = X ⊗R R is not reduced. Let Y be the maximal reduced closed formal subscheme of X . We observe that the Weil restriction functor RR /R extends naturally to affine formal R -schemes. Indeed, for every affine formal R -scheme of finite type Z, the functor A → Z(A ⊗R R ) from the category of R-adic algebras to the category of sets is represented by the affine formal R-scheme RR /R (Z) = lim R(R ⊗R Rn )/Rn (Z ⊗R (R ⊗R Rn )). −→ n0

We consider the tautological morphism of formal R-schemes X → RR /R (X ) defined by the natural map X(A) → X (A ⊗R R ) for every R-adic algebra A. Applying the Weil restriction functor RR /R to the closed immersion Y → Y, we obtain a closed immersion of formal R-schemes RR /R (Y ) → RR /R (X ). We then define a closed formal subscheme Y of X by setting Y = RR /R (Y ) ×RR /R (X ) X. It follows directly from the definitions that the closed immersion Y ⊗R R → X = X ⊗R R factors through Y . Thus Y = X, because Y = X . By the induction hypothesis, there exists a pair (γ, δ) of integers that is suitable for Y. We denote by N the nilradical of X . Since X is noetherian, there exists an integer q > 0 such that N q = (0). Let us now prove that the pair (qγ, qδ) is suitable for X. Let S be an extension of R of ramification index one; if K has positive characteristic, we assume that S is unramified. Let S  = S ⊗R R . By lemma 4/1.2.5, S  is an extension of R of ramification index one, and it is unramified if K has positive characteristic. Let e be the ramification index of R over R, and let n be a positive integer. Since the ramification index of S  over S is equal to e, the truncation maps and the definition of X lead to a commutative diagram: X(Sqn−1 )

9 X9 (Seqn−1 )

X(Sn−1 )

9 X9 (Sen−1 ).

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  Since N q = (0), the image of the map X (Seqn−1 ) → X (Sen−1 ) is con   tained in Y (Sen−1 ) = RR /R (Y )(Sn−1 ). This implies that the image of X(Sqn−1 ) → X(Sn−1 ) is contained in Y(Sn−1 ). In particular, for every integer n  0, the image of X(Sqn ) → X(Sn ) is contained in Y(Sn ). qγn+qδ Let now x ∈ X(Sqγn+qδ ) and let y = θγn+δ (x). By what precedes, one has y ∈ Y(Sγn+δ ). By the definition of the pair (γ, δ), there exists a point y  ∈ Y(S) such that θn (y  ) = θnγn+δ (y). One then has θn (y  ) = θnqγn+qδ (x). This proves that the pair (qγ, qδ) is suitable for X and concludes the proof of theorem 1.5.1.

Corollary 1.5.2. — Let X be a formal R-scheme of finite type. If X(Rn ) is nonempty for every integer n  0, then X(R) is nonempty. Remark 1.5.3. — The crucial ingredient of the proof of theorem 1.5.1 is Hensel’s lemma (lemma 1/1.3.3), and it is known that this lemma holds not only for complete discrete valuation rings but for Henselian discrete valuation rings as well. Let then R be an excellent Henselian discrete valuation ring, let K be its field of fractions, and let k be its residue field. Minor modifications of the given proof of theorem 1.5.1 show that it admits an algebraic counterpart where the formal scheme X is replaced by an R-scheme X of finite type and where one only considers extensions S of R which are Henselian and whose residue field is a separable extension of k if K has characteristic p > 0. The excellence condition is needed in the final part of the proof of theorem 1.5.1 to ensure that the extension R (integral closure of R in the finite extension K  ) is finite over R. This result (when S = R) was first proven by Greenberg (1966). An  points of X (R)  can important consequence is that X (R) is dense in X (R): be approximated with arbitrary m-precision by R-valued points of X . The Artin approximation theorem is the generalization of this theorem to more general Henselian local rings of dimension > 1. The first case was indeed due to Artin (1969) who treated in particular the case of rings of power series over an excellent discrete valuation ring. The case of a general excellent Henselian local ring is due to Popescu (1986); see also Spivakovsky (1999). We refer to the surveys of Teissier (1995) and Popescu (2000) for further information. Corollary 1.5.4. — Let X be a formal R-scheme of finite type. Then there exist positive integers γ, δ such that, for every integer n  0, θn (Gr(X)) = θnγn+δ (Grγn+δ (X)). Proof. — We only need to prove the inclusion θnm (Grm (X)) ⊂ θn (Gr(X)), since the other inclusion is obviously true. By §4/3.3.8, we may assume that the residue field k of R is perfect. Let then γ, δ be positive integers for which the conclusion of theorem 1.5.1 holds. Let m = γn + δ and let ξ ∈ Grm (X), and let k  be its residue field, resp. a perfect extension of its residue field

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in the mixed characteristic case. Let S = R(k  ) and let x ∈ X(Sm ) be the Sm -valued point of X corresponding to ξ. By theorem 1.5.1, there exists a point x ∈ X(S) such that θn (x ) = θnm (x). This point y corresponds to a point ξ  ∈ Gr(X) such that θn (ξ  ) = θnm (ξ). Thus θnm (ξ) ∈ θn (Gr(X)). Corollary 1.5.5. — Let h : Y → X be a morphism of formal R-schemes of finite type. Then a point x of Gr(X) lies in the image of Gr(h) : Gr(Y) → Gr(X) if and only if θn,X (x) lies in the image of Grn (h) : Grn (Y) → Grn (X) for every integer n  0. Proof. — Let κ be the perfect closure of the residue field of Gr(X) at x. By base change to R(κ) and using §4/3.3.8, we reduce to the case where R = R(κ). Pulling back the morphism h through the base change x : Spf(R) → X, we may assume moreover that X = Spf(R). Then the statement we have to prove takes the following form: if Grn (Y) is nonempty for every n  0, then Gr(Y) is nonempty. This follows immediately from corollary 1.5.4. Definition 1.5.6. — Let X be a formal R-scheme of finite type. For every integer n  0, we denote by γX (n) be the smallest integer m such that m  n m (Grm (X)). The function γX so defined is called the and θn,X (Gr(X)) = θn,X Greenberg function of X. By corollary 1.5.4, the Greenberg function γX can be bounded from above by an affine function n → γn + δ. When X is R-smooth, it follows from theorem 1.4.1 that γX (n) = n for all n  0. Corollary 1.5.7. — Let X be a formal R-scheme of finite type. a) Let n ∈ N, let D be a subset of Grn (X), and let C = (θn )−1 (D). Then one has (1.5.7.1)

θn (C) = θnγX (n) ((θnγX (n) )−1 (D)).

b) Let C be a constructible subset of Gr(X). Then θn (C) is a constructible subset of Grn (X), for every n ∈ N. Proof. — a) The left-hand side of equation (1.5.7.1) consists of the points in D that lift to Gr(X), and its right-hand side consists of the points that lift to GrγX (n) (X). The equality thus follows from corollary 1.5.4 and the definition of the Greenberg function. b) By lemma 4/4.2.2, there exist an integer m  0 and a constructible n −1 ) (D) subset D of Grm (X) such that C = (θm )−1 (D). Replacing D by (θm  q −1 if n  m, we assume that m  n. Let q = γX (m) and let D = (θm ) (D ); it is a constructible subset of Grq (X). By part a), one then has q (D )) = θnq (D ). θn (C) = θnm (θm (C)) = θnm (θm

It thus follows from Chevalley’s theorem (see theorem A/1.2.4, b)) that θn (C) is constructible.

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Corollary 1.5.8. — Let S be an unramified extension of R, and assume that S has perfect residue field if R has mixed characteristic. If n ∈ N and x is a point in X(Rn ) that lifts to X(S), then x lifts to X(R ) for some finite unramified extension R of R. Proof. — Let m = γX (n), and let ξ be the point of Grn (X) corresponding to x. The assumption on x implies that the fiber of θnm : Grm (X) → Grn (X) over ξ has a point defined over a separable extension of k. The closure of this point in the fiber is a generically smooth scheme of finite type over k, hence possessing a point ξ  defined over a finite separable extension k  of k, by Bosch et al. (1995, 2.2/13). Let R = R(k  ); it is a finite unramified  ). By extension of R and the point x ∈ X(Rn ) lifts to a point of X(Rm theorem 1.5.1 and the definition of the Greenberg function, it then also lifts to a point x ∈ X(R ). We also record the following technical result for later use. Proposition 1.5.9. — Let X be a formal R-scheme of finite type. Let Z be a closed formal subscheme of X. Let γZ be the Greenberg function of Z. Let C be a constructible subset of Gr(X) of level  n. Then, for every positive integer m, we have the following properties: a) The set θm,Z (C ∩ Gr(Z)) is a constructible subset of Grm (Z). b) If m  n, then (GrγZ (m) (Z))) = θm,Z (C ∩ Gr(Z)). θm,X (C ∩ θγ−1 Z (m),X Proof. — Recall that the morphisms Grm (Z) → Grm (X) and Gr(Z) → Gr(X) commute with the truncation morphisms. By lemma 4/4.2.5, the intersection C ∩ Gr(Z) is a constructible subset of Gr(Z), so that a) follows from corollary 1.5.7. Let us now prove b). It is obvious that (GrγZ (m) (Z))) ⊃ θm,Z (C ∩ Gr(Z)), θm,X (C ∩ θγ−1 Z (m),X so that we only need to prove the converse inclusion. Let xm be a point of (GrγZ (m) (Z))). θm,X (C ∩ θγ−1 Z (m),X (GrγZ (m) (Z)) be such that xm = θm,X (x). Then Let x ∈ C ∩ θγ−1 Z (m),X θγZ (m),X (x) is a lift of xm to GrγZ (m) (Z). By the definition of the Greenberg function γZ , there exists a point z ∈ Gr(Z) such that θm,Z (z) = xm . Since m  n and C is constructible of level  n, the equality θm,X (x) = θm,X (z) implies that z ∈ C. Consequently, we have θm,X (C ∩ θγ−1 (GrγZ (m) (Z))) ⊂ θm,Z (C ∩ Gr(Z)), Z (m),X as was to be shown.

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§ 2. THE STRUCTURE OF THE TRUNCATION MORPHISMS In this section, we will analyze in detail the structure of the truncation morphisms θnm between Greenberg schemes of a formal R-scheme X. These results will be essential for the construction of motivic integrals on formal schemes and for the proof of their change of variables formula. 2.1. Principal Homogeneous Spaces and Affine Bundles (2.1.1). — Let S be a scheme, and let X : SchoS → Sets and G : SchoS → Groups be presheaves on the category of S-schemes with values in the category of sets and of groups, respectively. For example, X and G could be (the functors associated with) an S-scheme and and S-group scheme, respectively. Suppose that G acts on X, that is, suppose we are given a morphism of functors τ : G ×S X → X inducing, for every S-scheme T , a left group action of G(T ) on X(T ). We say that X is a formally principal homogeneous space with translation space G if the morphism (τ, p2 ) : G ×S X → X ×S X is an isomorphism. In other words, for every S-scheme T such that X(T ) = ∅, the group G(T ) acts simply transitively on X(T ). More precisely, for every element x ∈ X(T ), the map G(T ) → X(T ),

g → τ (g, x)

is a G(T )-equivariant bijection. One says that the formally principal homogeneous space X is trivial if X(S) = ∅. Then every x ∈ X(S) induces a G-equivariant isomorphism of functors from G to X. If G is a group scheme over S and X is a formally principal homogeneous space as above, then we say that X is a principal homogeneous space if X is representable by a faithfully flat S-scheme locally of finite presentation. In this case, G is also flat and locally of finite presentation over S, because G ×S X is isomorphic to X ×S X and the properties of being flat and locally of finite type can be checked after faithfully flat base change. (2.1.2). — Let S be a scheme and let E be a quasi-coherent OS -module. We consider the contravariant functor on the category of S-schemes to the category of A1S -modules that maps an S-scheme f : T → S to the module HomOT (f ∗ E , OT ) over A1S (T ) = OT (T ). By (ÉGA II, 1.7.8), this presheaf is represented by an S-scheme of A1S -modules V(E ) = Spec(SymOS (E )),

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where SymOS (E ) denotes the symmetric algebra of E over OS . We call V(E ) the vector scheme associated with E . (1) If E is locally free of finite rank, then V(E ) is faithfully flat (even smooth) over S; we then say that V(E ) is a vector bundle. It follows from the definition that every morphism of schemes g : S  → S induces a canonical isomorphism of schemes of A1S  -modules V(g ∗ E ) → V(E ) ×S S  . (2.1.3). — By (ÉGA II, 1.7.11), the map E → V(E ) defines a contravariant functor from the category of quasi-coherent OS -modules to the category of schemes of A1S -modules. If E → E  is a surjective morphism of quasi-coherent OS -modules, then the morphism V(E  ) → V(E ) is a closed immersion. In that case, we call V(E  ) a subvector scheme of V(E ) or a subvector bundle if E and E  are locally free of finite rank. (2.1.4). — Let S be a scheme and let E be a locally free OS -module of finite rank. A principal homogeneous space X with translation space V(E ) is called an affine bundle over S. If, moreover, E is free, then we call X a free affine bundle. Proposition 2.1.5. — Let S be a scheme, and let E be a locally free OS module of finite rank. Let X be an affine bundle over S with translation space V(E ). Then X has a section over every affine open subscheme of S. In particular, it is locally trivial for the Zariski topology. The proof requires some basic knowledge of Grothendieck topologies and cohomology theory; the reader who is unfamiliar with these notions can skip the proof and simply accept the result. Proof. — We may suppose that S is affine. Affine bundles over S with trans1 (S, V(E )). By lation space V(E ) are classified by the cohomology group Hfppf Milne (1980, III.3.7), this cohomology group is computed as follows: 1 1 Hfppf (S, V(E )) = HZar (S, Hom OS (E , OS )).

Since S is affine, the latter cohomology group is trivial. 2.2. Truncation Morphisms and Principal Homogeneous Spaces (2.2.1). — Let us first recall some basic facts in the theory of deformations of morphisms. Let B be a ring, let I be an ideal of B such that I 2 = 0, and let A = B/I; let p : B → A be the canonical surjection. Observe that the natural B-module structure on I gives rise to an A-module structure: for every a ∈ A, and every x ∈ I, one sets a · x = bx, where b is any element of B such that p(b) = a. (1) In

(ÉGA II) the terminology fibré vectoriel (vector bundle) is used instead, but we prefer to reserve this name for the case where E is locally free, as is common in the literature.

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Let C be a ring and let f : C → A be a morphism of rings. We are interested in describing the set Homf (C, B) of morphisms g : C → B such that p ◦ g = f . Assume that it is nonempty. Let then g, g  : C → B be two elements of Homf (C, B). The map u =  g − g : C → B is group morphism. Since p ◦ g = p ◦ g  = f , the image of u is contained in I. For all x, y ∈ C, one has u(xy) = g  (xy) − g(xy) = g  (x)g  (y) − g(x)g(y) = g  (x)u(y) + u(x)g(y), which we rewrite as u(xy) = f (x) · u(y) + f (y) · u(x), using the A-module structure on I. Such a map is called an I-valued f derivation on C; let Derf (C, I) denote the abelian group of I-valued f derivations on C. Conversely, let g : C → B be a ring morphism such that p ◦ g = f , and let u be an f -derivation on C. The same computation proves that g  = g + u is a ring morphism from C to B such that p ◦ g  = f . The map Derf (C, I) × Homf (C, B) → Homf (C, B),

(u, g) → g + u

is a group action on Homf (C, B). What precedes shows that if Homf (C, B) is nonempty, then Derf (C, I) acts simply transitively on Homf (C, B). On the other hand, recall that the C-module of differentials Ω1C is endowed with a universal derivation d : C → Ω1C such that for every derivation u ∈ Derf (C, I), there exists a unique A-linear map u : A ⊗C,f Ω1C → I satisfying u ◦ d = u. We can thus rewrite the preceding action as HomA (A ⊗C Ω1C , I) × Homf (C, B) → Homf (C, B). (2.2.2). — The above construction can be generalized to a relative set-up and globalized as follows. Let S be a scheme, let I ⊂ OS be a quasi-coherent ideal sheaf such that I 2 = 0, and let j : T → S be the closed immersion defined by I . Since I 2 = 0, one can naturally view I as a quasi-coherent OT -module. Now assume that S is equipped with a morphism S → Y to a scheme Y , and let X be another scheme over Y . Let f : T → X be a morphism of Y -schemes, and let Homf (S, X) be the set of morphisms of Y -schemes g : S → X such that g ◦ j = f . One has a natural group action HomOT (f ∗ Ω1X/Y , I ) × Homf (S, X) → Homf (S, X). If Homf (S, X) is nonempty, then HomOT (f ∗ Ω1X/Y , I ) acts simply transitively on Homf (S, X). (2.2.3). — Let now X be a formal R-scheme of finite type. Let m and n be integers such that 0  n  m  2n + 1. We will endow the scheme Grm (X) with a structure of formally principal homogeneous space whose translation space is a vector scheme associated with a quasi-coherent module on Grn (X).

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(2.2.4). — Recall that, for every k-scheme Y , we have introduced in section 4/3.1 an Rn -scheme hn (Y ) = (|Y |, Rn (OY )). By the definition of the Greenberg scheme Grn (X), one has adjunction identifications Hom(hn (Y ), X) = Hom(Y, Grn (X)) and Hom(hm (Y ), X) = Hom(Y, Grm (X)), which are functorial in the k-scheme Y . Let Y be a k-scheme. Since n  m  2n+1, the quasi-coherent ideal sheaf Jnm (OY ) on hm (Y ) has square zero. Let f : hn (Y ) → Xn be a morphism of Rn -schemes. Deformation theory furnishes a canonical group action Homhn (Y ) (f ∗ Ω1Xn /Rn , Jnm (OY )) × Homf (Y, Grm (X)) → Homf (Y, Grm (X)), which is simply transitive if Homf (Y, Grm (X)) = ∅. (2.2.5). — We now consider the universal case where Y = Grn (X), and let γ : hn (Grn (X)) → Xn correspond to the identity morphism of Grn (X) = Grn (Xn ). In the adjunction identification, for every k-scheme Y , a morphism u : Y → Grn (X) corresponds to f = γ ◦ hn (u) : hn (Y ) → X. m on the category of Grn (X)-schemes by We define an abelian presheaf Tn,X (2.2.5.1)

m : SchGrn (X) → Ab, Tn,X

(u : Y → Grn (X)) → HomOhn (Y ) (hn (u)∗ γ ∗ Ω1Xn /Rn , Jnm ).

It is a sheaf for the Zariski topology on (SchGrn (X) ). The results in (2.2.4) can be reformulated in the following way. Proposition 2.2.6. — Let X be a formal R-scheme of finite type. For every integer n  0 and every integer m such that n  m  2n + 1, the scheme Grm (X) is in a canonical way a formally principal homogeneous space over m . Grn (X) with translation space Tn,X (2.2.7). — Let us look at the special case where m = n + 1. To simplify the notation, we pose In = Jnn+1 . In this case, the Rn+1 -module structure on In factors through the quotient R0  A1k of Rn . Let moreover u : Y → Grn (X) be a morphism of k-schemes, and let f = γ ◦ hn (u) : hn (Y ) → Xn be the corresponding morphism of Rn -schemes. The restriction of f to the n ◦ u : Y → X0 . We closed subscheme h0 (Y ) = Y of hn (Y ) factors through θ0,X thus obtain a canonical isomorphism n+1 n Tn,X (Y ) → HomOY (u∗ (θ0,X )∗ Ω1X0 /k , In ).

If R has mixed characteristic and absolute ramification index e  1, then we denote by FS the absolute Frobenius morphism on S, by α the unique integer such that Rn+1 has characteristic pα+1 , and by β the remainder of the Euclidean division of n + 1 by e. If R has equal characteristic, then we denote by FS the identity of S, and we set α = 0 and β = n + 1. We have defined in section 4/2.4 a canonical isomorphism of OS -modules (2.2.7.1)

(FSα )∗ OS ⊗k k(β) → In .

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It induces an isomorphism of abelian groups n+1 n (S) → HomOS (f ∗ (θ0,X )∗ (FXα0 )∗ Ω1X0 /k ⊗k k(−β), OS ). Tn,X n+1 Consequently, the abelian presheaf Tn,X is represented by the vector scheme

(2.2.7.2)

n Vn,X = V((θ0,X )∗ (FXα0 )∗ Ω1X0 /k ⊗k k(−β))

on Grn (X). Beware, however, that the OS -module structure on n )∗ (FXα0 )∗ Ω1X0 /k ⊗k k(−β), OS ) Vn,X (S) = HomOS (f ∗ (θ0,X

is different from the one on n HomOS (f ∗ (θ0,X )∗ Ω1X0 /k , In )

if R has mixed characteristic. Corollary 2.2.8. — Let us assume that X is smooth of pure relative dimension d. a) Assume that X is affine and that Ω1X0 /k is free. Then Vn,X is a free vector bundle of rank d, and Grn+1 (X) is a trivial affine bundle over Grn (X) with translation space Vn,X . b) The sheaf Vn,X is a vector bundle of rank d, and Grn+1 (X) is a locally trivial affine bundle over Grn (X) with translation space Vn,X . This refines corollary 1.2.2. Proof. — It follows from equation 2.2.7.2 that Vn,X is a vector bundle of rank d on Grn (X) and is free if Ω1X0 /k is free. Then assertion a) is a direct consequence of propositions 2.1.5 and 2.2.6. Since X is covered by affine open formal subschemes U such that Ω1U0 /k is free, assertion b) follows. Corollary 2.2.9. — Let X be a formal R-scheme of finite type, let n ∈ N n (ξ) and let m = dimκ(x) (Tx X0 ). If the fiber and let ξ ∈ Grn (X). Let x = θ0,X n+1 −1 (θn,X ) (ξ) is nonempty, it is isomorphic to the κ(ξ)-scheme Am κ(ξ) . n+1 Proof. — By theorem 2.2.6, the fiber of θn,X over ξ is a formally principal homogeneous bundle over Spec κ(ξ) with translation space (Vn,X ) ⊗ κ(ξ). It is representable, by construction, and flat because κ(ξ) is a field. Moreover, Vn,X ⊗ κ(ξ) is a κ(ξ)-vector space of rank m. Assume that the fiber is nonempty. Then proposition 2.1.5 implies that it is isomorphic to Am κ(ξ) , as was to be shown.

Corollary 2.2.10. — Let R be an unramified extension of R; if R has mixed characteristic, then assume that the residue field of R is perfect. Let X be a formal R-scheme of finite type. Then for every integer n  0, every  ) can also be lifted to X(Rn+1 ). element of X(Rn ) that can be lifted to X(Rn+1

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 Proof. — Let x ∈ X(Rn ) be an element that admits a lift x ∈ X(Rn+1 );  let ξ and ξ be the corresponding points of Grn (X) and Grn+1 (X). Then n+1  n+1 (ξ ) = ξ. Consequently, the fiber θn,X : Grn+1 (X) → ξ ∈ Grn (X)(k) and θn,X Grn (X) over the k-rational point ξ of Grn (X) is nonempty and hence isomorphic to an affine space over k. In particular, it contains a k-rational point. n+1  (ξ ) = ξ. This Thus, there exists a point ξ  ∈ Grn+1 (X)(k) such that θn,X   points ξ corresponds to a point x ∈ X(Rn+1 ) that lifts x.

2.3. The Images of the Truncation Morphisms (2.3.1). — Let X be a formal R-scheme of finite type. Corollary 2.2.9 den+1 scribes the nonempty fibers of the truncation morphisms θn,X : Grn+1 (X) → Grn (X). In this section, we study the analogous question after restriction to the image θn+1 (Gr(X)) of the truncation morphism. The answer is immediate in the smooth case, by surjectivity of θn+1 , but much more subtle in the presence of singularities. We will answer the question in general under the assumption that we are not too close to the singular locus Xsing of X. Theorem 2.3.11, the main result of this section, will turn out to be one of the cornerstones of the theory of motivic integration on singular formal schemes. We begin with a general dimension estimate. Proposition 2.3.2. — Let X be a formal R-scheme of finite type of relative dimension d. Then for every n  0, the fibers of the truncation map (2.3.2.1)

n+1 : θn+1,X (Gr(X)) → θn,X (Gr(X)) θn,X

are constructible sets of dimension at most d. In particular, the dimension of the constructible set θn,X (Gr(X)) is at most d(n + 1). Proof. — By corollary 1.5.7, the sets θn+1 (Gr(X)) and θn (Gr(X)) are constructible subsets of Grn+1 (X) and Grn (X). Let x ∈ Gr(X) and let Yx be the fiber of the morphism (2.3.2.1) over θn,X (x). It is a constructible subset of n+1 −1 (θn,X ) (θn,X (x)). Let us prove that dim(Yx )  d. Let κ be the perfect closure of the residue field of x. By base change to R(κ), we reduce to the case where k = κ; then we can identify x with a point in x ∈ X(R). We may also suppose that X is affine, say, X = Spf R{z1 , . . . , zr }/(f1 , . . . , f ). The point x corresponds to a tuple (x1 , . . . , xr ) in X(R) ⊂ Rr . Let π be a uniformizer in R. For every i ∈ {1, . . . , }, let us set gi (u1 , . . . , ur ) = fi (x1 + π n+1 u1 , . . . , xr + π n+1 ur )

∈ R{u1 , . . . , ur }.

Let Y = Spf(R{u1 , . . . , ur }/(g1 , . . . , g )), and let Y be its largest R-flat closed formal subscheme. Now consider the morphism of formal R-schemes h : Y → X defined by zj → xj + π n+1 uj for j = 1, . . . , r. On the generic fibers, it induces an analytic domain immersion hη : Yη → Xη . Since the

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kernel of the morphism of ring schemes Rn+1 → R0 is killed by π n+1 , by proposition 4/2.4.2, the morphism Grn+1 (h) : Grn+1 (Y ) → Grn+1 (X) factors through Gr0 (Y ). Moreover, its image contains Yx . Since Y is R-flat and its generic fiber has dimension at most d, we conclude that Gr0 (Y )  Y ⊗R k has dimension at most d. Consequently, dim(Yx )  d. The last assertion follows by induction on n. (2.3.3). — We shall later need the following extension of Hensel’s lemma (lemma 1/1.3.3). Lemma. — We fix integers r   0 and n  e  0. Let f = (f1 , . . . , f ) be a finite family of elements of R[[z1 , . . . , zr ]]. Let S be an extension of R of ramification index one, and let n be its maximal ideal. Let c be an r-tuple of elements in n such that f (c) = 0. Assume that c mod nn+1 lies in (R/mn+1 )r ⊂ (S/nn+1 )r and that there exists an ( × )-minor Δ of the Jacobian matrix

 ∂fi (c) Jf (c) = ∂zj i=1,...,, j=1,...,r of f such that Δ ≡ 0 mod ne+1 . Then there exists an r-tuple b of elements of m such that f (b) = 0 and c ≡ b mod nn+1 . The analogous statement holds when the power series fi belong to R{z1 , . . . , zr } and c is an element of S r . Proof. — Let π be a uniformizer in R. Let a in Rr be such that a ≡ c mod nn+1 . As in the proof of lemma 1/1.3.3, it is sufficient to find an element u ∈ Rr such that f (a + π n+1 u) ≡ 0 mod mn+e+1 . We again consider the Taylor expansion f (a + π n+1 u) = f (a) + π n+1 Jf (a)u + π 2n+2 ε(u), where ε(u) is an -tuple of elements in R[[u1 , . . . , ur ]]. Let M be the R-module given by M = π n+1 Jf (a)Rr + (mn+e+1 ) ⊂ R . Since 2n + 2 > n + e + 1, there exists an element u ∈ Rr as desired if and only if f (a) ∈ M . Now, observe that the element v = (c − a)/π n+1 ∈ S r satisfies f (a + π n+1 v) = f (c) = 0, so that f (a) ∈ π n+1 Jf (a)S r + (nn+e+1 ) . In other words, f (a) belongs to the submodule M ⊗R S of S  . Since the morphism R → S is faithfully flat, the natural morphism (R /M ) → (S  /M ⊗R S) = (R /M ) ⊗R S

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is injective, hence f (a) ∈ M . If the series fi belong to R{z1 , . . . , zr } and c is an element of S r , we can reduce to the previous situation by applying the coordinate change zj = zi −cj where cj is any element of R such that cj − cj belongs to m. Proposition 2.3.4. — Let X be a flat formal R-scheme of finite type of pure relative dimension. There exists an integer c  1 such that the following statement is true. For every pair (n, e) of integers such that n  ce  0, every extension R of R of ramification index one and every point x ∈ X(Rn ) such that θen (x) ∈ Xsing (Re ), the following properties are equivalent: (i) There exists y ∈ X(R ) such that x = θn,X (y); (ii) There exist an extension R of R of ramification index one and a n+ce  point z ∈ X(Rn+ce ) such that x = θn,X (z). Proof. — Since the problem is local on X, we can assume that X is affine. Then there exist an integer r  0 and an ideal I of R{z1 , . . . , zr } such that X = Spf(A), where A = R{z1 , . . . , zr }/I. In this situation, we have defined in §1.3.6 an ideal JacX (see equation (1.3.6.1)) of A and its radical HX . By corollary 1.3.9, the ideals JacX and JacX/R have the same radical; since A is noetherian, there exists an integer c  1 such that (JacX )c ⊂ JacX . Let now R , n, and e be as in the statement. The implication (i)⇒(ii) is obvious, so let us assume that we are in the situation of (ii). Since θe (x) ∈ Xsing (Re ) and (JacX )c ⊂ JacX , the point θce (x) does not belong to the closed formal subscheme defined by JacX . Consequently, there exist an integer , a family f = (f1 , . . . , f ) of elements of I, a rank r minor Δ of the Jacobian matrix Jf , and an element h of ((f1 , . . . , f ) : I) such that Δ(x) ≡ 0 and h(x) ≡ 0 modulo mce+1 R . Denote by Y the closed formal subscheme of Spf(R{z1 , . . . , zr }) defined by the ideal (f1 , . . . , f ). Since f1 , . . . , f ∈ I, X is a closed formal subscheme of Y. Since n  ce, Hensel’s lemma (lemma 1/1.3.3) implies that there exists a point y in Y(R ) such that θn,Y (y) = x. Then, by lemma 2.3.3, there exists a point y  in Y(R ) such that x = θn,Y (y  ) in Grn (Y)(R ). Let g ∈ I. Then gh belongs to the ideal (f1 , . . . , f ), so that gh(y  ) = 0. On the other hand, we have h(y  ) ≡ h(x) ≡ 0 mod mce+1 R . In particular, h(y  ) = 0, hence g(y  ) = 0. This proves that y  ∈ X(R ) and concludes the proof. Definition 2.3.5. — Let X be a flat formal R-scheme of pure relative dimension. The smallest integer c  1 for which proposition 2.3.4 holds will be referred to as the Elkik–Jacobi constant of X.

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Lemma 2.3.6. — Let X be a flat formal R-scheme of finite type of pure relative dimension d. Let e  0 be an integer and let x ∈ Gr(e) (X)(k); thus x ∈ X(R) and θe (x) ∈ / Xsing (Re ). a) The length of the torsion R-module (x∗ Ω1X/R )tors is at most e. b) The R-module M = (x∗ Ω1X/R )/(x∗ Ω1X/R )tors is free of rank d. c) Let q : x∗ Ω1X/R → M be the canonical projection. Let n be an integer such that n  e, and let τ : Jn2n+1 → Jnn+1 be the canonical projection. The morphisms q and τ induce a diagram HomR (M, Jnn+1 ) q

HomR (x 1 /R , Jn2n+1 )

τ

HomR (x 1 /R , Jnn+1 ).

The morphism q ∗ is injective, and its image coincides with the image of τ∗ . Recall that for all integers m, n such that m  n, we defined Jnm as the kernel of the canonical surjection from Rm to Rn . Proof. — Since x ∈ Gr(e) (X)(k), one has ordjacX (x)  e. Since the formation of Fitting ideals commutes with base change, this implies that Fittd (x∗ Ω1X/R ) ⊃ me . Moreover, (x∗ Ω1X/R ) ⊗R K  K d because the generic fiber of X is quasi-smooth of dimension d at the K-point corresponding to x. Assertions a) and b) thus follow from example 1.1.3. The morphism q is surjective; hence q ∗ is injective. Let f : x∗ Ω1X/R → Jn2n+1 be an R-morphism. Since the length of (x∗ Ω1X/R )tors is at most e and n  e, one has π n a = 0 for every a ∈ (x∗ Ω1X/R )tors and every uniformizer π in R. Consequently, π n f ((x∗ Ω1X/R )tors ) = 0, hence f ((x∗ Ω1X/R )tors ) ⊂ π n+2 R2n+1 since R2n+1 = R/π 2n+2 R. Therefore, τ (f ((x∗ Ω1X/R )tors )) = 0, and there exists a unique morphism g : M → Jnn+1 such that g ◦ q = τ ◦ f ; equivalently, τ∗ (f ) = q ∗ (g). This proves that the image of τ∗ is contained in the image of q ∗ . Conversely, let g : M → Jnn+1 be an R-morphism. Since τ is surjective and M is a free R-module, there exists a morphism f : M → Jn2n+1 such that τ ◦ f = g ◦ q, so that q ∗ (g) belongs to the image of τ∗ . This concludes the proof of the lemma. (2.3.7). — Let X be a flat formal R-scheme of finite type of pure relative dimension d; let x ∈ X(R). Assume that x ∈ Gr(Xsing ) and let e = ordjacX (x). Let n be an integer such that n  e. Let Yx = (θnn+1 )−1 (θn (x)) ⊂ Grn+1 (X), and let Vx = HomR (x∗ Ω1X/R , Jnn+1 ). Observe that Yx is not emtpy, since it contains θn+1 (x). By proposition 2.2.6, the scheme Yx admits a canonical structure of an affine bundle over Spec(k) with translation space Vx .

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Let Yx = θn+1 (Gr(X)) ∩ Yx be the set of elements of Yx that lift to Gr(X). By corollary 1.5.7, this is a constructible subset of Yx ; it contains the kpoint θn+1 (x). Let also Vx = HomR (x∗ Ω1X/R /(x∗ Ω1X/R )tors , Jnn+1 ). By lemma 2.3.6, it is mapped isomorphically by q ∗ to a k-vector subspace of dimension d of Vx . We view Vx as a subvector bundle of Vx over Spec(k). Proposition 2.3.8. — Let c be the Elkik–Jacobi constant of X. Let us assume that n  ce. a) For every extension k  of k, the action of Vx ⊗k k  on Yx (k  ) induces a simply transitive action of q ∗ (Vx ) ⊗k k  on Yx (k  ). b) The constructible set Yx is a closed subset of Yx ; we endow it with its induced structure of a reduced scheme. c) The k-scheme Yx is isomorphic to Adk . Proof. — For simplicity of notation, we omit the indices x. Let us prove a). By base change from R = R(k) to R(k  ), we reduce to the case k  = k. (Note that this does not affect the Elkik–Jacobi constant.) Let us retain the notation from lemma 2.3.6. We shall denote by (v, y) → v · y the action of an element v ∈ V on a point y ∈ Y (k). We first prove that Y  (k) is stable under the action of q ∗ (V  ). Let y ∈ X(R) be such that θn (y) = θn (x). Let g ∈ V  and let us prove that q ∗ (g)·θn+1 (y) ∈ Y  (k). By lemma 2.3.6, there exists a morphism f : x∗ Ω1X/R → Jn2n+1 such that τ ◦ f = g ◦ q. By naturality of the action defined by deformation theory, one has 2n+1 q ∗ (g) · θn+1 (y) = τ∗ (f ) · θn+1 (y) = θn+1 (f · θ2n+1 (y)),

so that q ∗ (g) · θn+1 (y) lifts to X(R2n+1 ). Since n  ce, proposition 2.3.4 implies that q ∗ (g) · θn+1 (y) lifts to X(R), as was to be shown. It remains to prove that the action of V  on Y  is transitive. Let y ∈ X(R) be such that θn (y) = θn (x). Then the two points θ2n+1 (y) and θ2n+1 (x) lift θn (x); hence there exists an element f ∈ Hom(x∗ Ω1X/R , Jn2n+1 ) such that f · θ2n+1 (x) = θ2n+1 (y). By lemma 2.3.6, there exists a morphism g ∈ V  such that q ∗ (g) = τ∗ (f ). One then has 2n+1 q ∗ (g) · θn+1 (x) = τ∗ (f ) · θn+1 (x) = θn+1 (f · θ2n+1 (x)) 2n+1 = θn+1 (θ2n+1 (y)) = θn+1 (y).

This concludes the proof of part a). In particular, the assumptions of lemma 2.3.9 below are satisfied, so that part b) follows. Lemma 2.3.9. — Let κ be a field with algebraic closure κa . Let V be a vector bundle over Spec(κ), and let Y be an affine bundle over Spec(κ) with translation space V . Let V  be a subvector bundle of V , and consider a nonempty constructible subset Y  of Y . Assume that Y  (κa ) is stable under

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the action of V  (κa ) on Y (κa ) and that V  (κa ) acts transitively on Y  (κa ). Then the following properties hold. a) The set Y  is closed in Y . If we endow Y  with its induced reduced structure, then Y  is a closed subscheme of Y that is stable under the action of V  on Y . b) Assume either that κ is perfect or that Y  (κ) is nonempty. Then Y  is an affine bundle over Spec(κ) with translation space V  . In particular, there exists an isomorphism of κ-schemes Y   Adκ where d = dim(V  ). Proof. — a) Let us fix a point y ∈ Y  (κa ) and consider the translation isomorphism v → v + y. τy : V ⊗κ κa → Y ⊗κ κa , By our assumptions, Y  (κa ) is the image of V  (κa ) under this isomorphism of schemes; in particular, Y  (κa ) is a closed subset of Y (κa ). Since Y  is constructible in Y , this implies that Y  is closed in Y . We endow Y  with its induced reduced structure. By restriction, the action τ : V ⊗κ Y → Y of V on Y induces a morphism τ  : V  ⊗κ Y  → Y . By hypothesis, the image of (V  ⊗κ Y  )(κa ) → Y (κa ) by τ  is contained in Y  (κa ). Since V  ⊗κ Y  is reduced, the morphism τ  factors through the reduced closed subscheme Y  of Y . b) Let us first assume that Y  (κ) = ∅ and let y ∈ Y  (κ). The translation morphism τy : V → Y , v → v + y, induces a commutative diagram V

V

τy

τy

Y

Y

in which the vertical morphisms are closed immersion and the morphism τy is an isomorphism. In particular, τy is a closed immersion. Since the schemes V  and Y  are reduced and τy is surjective at the level of κa -points, the morphism τy is an isomorphism. We now treat the case where κ is perfect. By part a), the action of V on Y and the action of V  on Y  induce a commutative diagram V

⊗κ Y

τ

V

⊗κ Y

τ

Y

⊗κ Y

Y

⊗κ Y

in which the vertical morphisms are closed immersions and the morphism τ is an isomorphism. In particular, τ  is a closed immersion; by a) it is surjective at the level of κa -points. To prove that Y  is an affine bundle over Spec(κ) with translation space V  , we need to prove that τ  is an isomorphism. The source and target of τ are reduced because κ is perfect. Moreover, τ is bijective on the level of κa -points by a). Consequently, τ is an isomorphism.

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We have proven that Y  is an affine bundle over Spec(κ) with translation space V  . All affine bundles over Spec(κ) are trivial by proposition 2.1.5. In particular, Y  is isomorphic to Adκ with d = dim(V  ). Remark 2.3.10. — Let us illustrate that the hypothesis in b) cannot be omitted. Let κ be an imperfect field, and let us take V = Y = A1κ . Let V  = {0}, and let Y  be a single purely inseparable closed point on A1κ . Then all of the assumptions of lemma 2.3.9 hold, except the hypothesis in b), because Y  (κ) = ∅ and κ is imperfect. The scheme Y  is not an affine bundle with translation space V  . Indeed, it would otherwise be trivial and hence have a κ-point. Theorem 2.3.11. — Let X be a flat formal R-scheme of finite type of pure relative dimension d, and let c  1 be its Elkik–Jacobi constant. Let e ∈ N and let x ∈ Gr(e) (X). Let n be an integer such that n  ce. a) The set Yx = (θnn+1 )−1 (θn (x)) ∩ θn+1 (Gr(X)) is closed in the fiber Yx = n+1 −1 (θn ) (θn (x)). We endow Yx with its induced reduced structure. b) Assume that R has equal characteristic, and let κ be the residue field of θn (x). Then the κ-scheme Yx is isomorphic to Adκ . c) Assume that R has mixed characteristic, and let κ be the perfection of the residue field of θn (x). Then the κ-scheme Yx ⊗ κ is isomorphic to Adκ . d) In particular, dim(Yx ) = d. Proof. — By base change to R(κ), we can assume that κ = k and that θn (x) ∈ Grn (X)(k) = X(Rn ). The point x of Gr(X) corresponds to a point in X(R ) for some extension R of R of ramification index one, and this point is a lift of θn (x). It then follows from proposition 2.3.4 that θn (x) lifts to an R-valued point of Gr(X). We may thus replace x by such a point and assume that x ∈ X(R). Now the theorem is a consequence of proposition 2.3.8.

§ 3. GREENBERG SCHEMES AND MORPHISMS OF FORMAL SCHEMES The most important result in the theory of motivic integration is the change of variables formula. This formula is based on a precise description of the fibers of the morphisms Grn (h) induced by a morphism of formal R-schemes of finite type h : Y → X. Such a description will be established in theorem 3.2.2. First, we need to make some preparations. 3.1. The Jacobian Ideal and the Function ordjacf (3.1.1). — Let h : Y → X be a morphism of flat formal R-schemes of finite type of pure relative dimension d. Let R be an extension of R and let ψ be

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a point in Y(R ). Then h induces a morphism     (3.1.1.1) α : (h ◦ ψ)∗ ΩdX/R /(torsion) → ψ ∗ ΩdY/R /(torsion) of free R -modules of finite rank. We define the order of the Jacobian of h along ψ by means of the formula ordjach (ψ) = lengthR Coker(α). This construction gives rise to a function ordjach : Gr(Y) → N ∪ {∞} : y → ordjach (ψy ) where ψy is defined as in (4/3.3.7). If hη is étale, then α becomes an isomorphism after tensoring with the quotient field of K  of R , so that the cokernel of α is torsion and ordjach only takes values in N. If Y is smooth over R, then we also define the Jacobian ideal sheaf of h as the coherent ideal sheaf Fitt0 (Ω1Y/X ) on Y; it is denoted by Jach . If we denote by Z the closed formal subscheme of Y defined by Jach , then Y Z is the largest open formal subscheme of Y where Ω1Y/X vanishes, and Yη Zη is the largest K-analytic open subspace of Yη where Ω1Yη /Xη vanishes. Example 3.1.2. — Let h : Y → X be a morphism of smooth formal Rschemes of finite type of pure relative dimension d; assume that hη is étale. In this case, the fundamental sequence

is exact and provides a resolution of Ω1Y/X by locally free OY -modules of rank d. We can use this resolution to compute the Fitting ideals of Ω1Y/X . In particular, Jach = Fitt0 (Ω1Y/X ) is the determinant ideal of the morphism h∗ Ω1X/R → Ω1Y/R . This is a locally principal ideal sheaf on Y. Lemma 3.1.3. — Let h : Y → X be a morphism of flat formal R-schemes of finite type of pure relative dimension d, and assume that Y is smooth over R. Then for every extension R of R and every point ψ in Y(R ), one has ordjach (ψ) = lengthR ψ ∗ Ω1Y/X . Moreover, Jach is the annihilator of the cokernel of the morphism h∗ ΩdX/R → ΩdY/R and ordjach coincides with the order function associated with the coherent ideal sheaf Jach (see (4/4.4.3)). Proof. — The first assertion follows immediately from the definition: we have an exact sequence of R -modules

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and we can identify the cokernel of the morphism α in (3.1.1.1) with the cokernel of Dh(ψ), because the R -module ψ ∗ ΩdY/R has no torsion by our assumption that Y is smooth over R. Now we prove the remaining assertions. Since Y is smooth over R, the coherent sheaf Ω1Y/R is locally free. Since the statement is local on Y, we may assume that Ω1Y/R is free and that there exist an integer e and a surjective e → h∗ Ω1X/R . Then we can use the exact morphism of OY -modules ϕ : OY sequence e OY → Ω1Y/R → Ω1Y/X → 0 to compute the Fitting ideals of Ω1Y/X . By definition, the 0-th Fitting ideal e → ψ ∗ Ω1Y/R where is the sum of the determinant ideals of the morphisms OY I runs through the subsets of {1, . . . , e} of cardinality d. Equivalently, it is the annihilator of the cokernel of the morphism ∧d ϕ :

d ,

e OY → ΩdY/R ∼ = OY

induced by ϕ; but this cokernel is precisely the cokernel of the morphism h∗ ΩdX/R → ΩdY/R e because ∧d ϕ factors through the surjection ∧d OY → h∗ ΩdX/R . Now the fact that ΩdY/R is a line bundle and the formation of cokernels commutes with base change immediately implies that ordjach coincides with the order function associated with Jach .

Proposition 3.1.4. — Let h : Y → X be a morphism of flat formal Rschemes of finite type of pure relative dimension d. Assume that Y is smooth over R and that hη is étale. Then the function ordjach : Gr(Y) → N is constructible (i.e., it has constructible fibers) and bounded. If X is also smooth over R, then ordjach is constant on Gr(U), for every connected component U of Y. Proof. — Since Y is smooth, ordjach is the order function associated with the coherent ideal sheaf Jach on Y, by lemma 3.1.3. Such a function has constructible fibers at all finite values, by corollary 4/4.4.8; but ordjach has values in N by our assumption that hη is étale. Thus ordjach has constructible fibers; then it takes only finitely many values, because the constructible topology on Gr(Y) is quasi-compact by theorem A/1.2.4. Now assume that X is also smooth over R. Then the ideal sheaf Jach is locally principal, by example 3.1.2, and it induces the identity ideal sheaf on Yη because hη is étale. It follows that the restriction of Jach to each connected component U of Y is generated by mi for some integer i  0; then ordjach is constant with value i on Gr(U).

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291

Proposition 3.1.5 (Chain rule). — Let g : Z → Y and h : Y → X be morphisms of flat formal R-schemes of finite type of pure relative dimension d. Assume either that gη is étale or that Yη and Zη are quasi-smooth over K. Then ordjach◦g = (ordjach ◦ Gr(g)) + ordjacg . Proof. — Let R be an extension of R, and let ψ be a point in Z(R ). Consider the sequence of R -modules     β (h ◦ g ◦ ψ)∗ ΩdX/R /(torsion) −−→ (g ◦ ψ)∗ ΩdY/R /(torsion)   α −−→ ψ ∗ ΩdZ/R /(torsion). It induces a short exact sequence

The morphism α also induces a surjective morphism of R -modules

We need to show that lengthR Coker(α ◦ β) = lengthR Coker(β) + lengthR Coker(α). If gη is étale, then γ is injective, and thus an isomorphism, because α becomes an isomorphism after tensoring with the quotient field K  of R . In this case, the result follows from the additivity of the length in short exact sequences of R -modules. Now assume that Yη and Zη are quasi-smooth. Then α is a morphism of free R -modules of rank 1. Thus either α is injective, which again implies that γ is an isomorphism; or α = 0, in which case the cokernels of α ◦ β and α have infinite length, and the result holds as well. Lemma 3.1.6. — Let h : Y → X be a morphism of flat formal R-schemes of finite type of pure relative dimension d. Assume that Y is smooth over R. Let y ∈ Y(R), let e = ordjach (y), and e = ordjacX (h(y)); assume that e, e = +∞. a) The canonical sequence of R-modules Dh(y)

0 → (h(y)∗ Ω1X/R )tors → h(y)∗ Ω1X/R −−−−→ y ∗ Ω1Y/R → y ∗ Ω1Y/X → 0 is exact. b) One has lengthR ((h(y)∗ Ω1X/R )tors ) = e

and

lengthR (y ∗ Ω1Y/X ) = e.

c) For every R-module M and every morphism α : h(y)∗ Ω1X/R → me M such that (h(y)∗ Ω1X/R )tors ⊂ Ker(α), there exists a morphism β : y ∗ Ω1Y/R → M such that β ◦ Dh(y) = α.

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Proof. — We start from the fundamental exact sequence h∗ Ω1X/R → Ω1Y/R → Ω1Y/X → 0 of OY -modules induced by the morphism h. Pulling it back through y, we obtain an exact sequence of R-modules: h(y)∗ Ω1X/R → y ∗ Ω1Y/R → y ∗ Ω1Y/X → 0. Since Y is smooth of pure relative dimension d over R, the module y ∗ Ω1Y/R is free of rank d. Our assumption that e is finite is equivalent to saying that h(y) ∈ Xsing (R), so that the module h(y)∗ Ω1X/R has rank d, as well. We have already proven the expressions for e and e in lemma 3.1.3 and section (1.4.2), respectively. Since Ω1Y/R is torsion-free, the kernel of the morphism h∗ Ω1X/R → Ω1Y/R contains the torsion module (h∗ Ω1X/R )tors . Comparing ranks, we get an equality. This proves a) and b). Let us now prove c). Let M be an R-module, and let α : h(y)∗ Ω1X/R → me M be a morphism of R-modules. By hypothesis, α factors through a morphism α ¯ : h(y)∗ Ω1X/R /(torsion) → me M . Let π be a uniformizer of R. There exist a basis (v1 , . . . , vd ) of y ∗ Ω1Y/R and a family (e1 , . . . , ed ) of nonnegative integers such that (π e1 v1 , . . . , π ed vd ) is a basis of h(y)∗ Ω1X/R )/(torsion). Then e1 + · · · + ed = length(y ∗ Ω1Y/X ) = e. For every i, let mi ∈ M be such that α ¯ (π ei vi ) = π ei mi ; this is possible since ei  e and the image of α ¯ is contained in π e M . Let β : y ∗ Ω1Y/R → M be the unique morphism such that β(vi ) = mi for every i. One has β(π ei vi ) = α ¯ (π ei vi ) by construction; hence β ◦ Dh(y) = α, as was to be shown. Proposition 3.1.7. — Let h : Y → X be a morphism of flat formal Rschemes of finite type of pure relative dimension d; assume that Y is smooth over R. Let y ∈ Gr(Y); let e = ordjach (y) and e = ordjacX (h(y)). For every integer n  max(2e, e ) and every point x ∈ Gr(X) such that θn,X (h(y)) = θn,X (x), there exists a point y  ∈ Gr(Y) such that h(y  ) = x and θn−e,Y (y  ) = θn−e,Y (y). Proof. — Let κ be a common perfect extension of the residue fields of x and y. By base change to R(κ), we reduce to the case where x and y correspond to elements of X(R) and Y(R), which we still denote by x and y. Let us first show the existence of an element z ∈ Y(Rn+1 ) such that n+1 (z) = θn−e,Y (y) and h(z) = θn+1,X (x). θn−e,Y The points θ2n+1,X (h(y)) and θ2n+1,X (x) of X(R2n+1 ) live in the fiber of the truncation morphism X(R2n+1 ) → X(Rn ) over θn,X (h(y)) = θn,X (x). The kernel Jn2n+1 = mn+1 /m2n+2 of the morphism R2n+1 → Rn has square 0. By deformation theory, these two points are deduced one from the other by the action of some element α of HomR (h(y)∗ Ω1X/R , mn+1 /m2n+2 )

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293

(see proposition 2.2.6). Since the length of the torsion submodule of h(y)∗ Ω1X/R is equal to e , the image α of α in 

HomR (h(y)∗ Ω1X/R , mn+1 /m2n+2−e ) maps (h(y)∗ Ω1X/R )tors to 0. By lemma 3.1.6, this morphism α is induced by a morphism β in 

HomR (y ∗ Ω1Y/R , mn+1−e /m2n+2−e ). Since n  e , we may consider the image β  of β in HomR (y ∗ Ω1Y/R , mn+1−e /mn+2 ) and form the point z ∈ Y(Rn+1 ) which is deduced from θn+1,Y (y) by the action of β  . Since the image of β  is contained in mn+1−e /mn+2 , one has θn−e,Y (z) = θn−e,Y (y). Moreover, the point h(z) is deduced from h(θn+1,Y (y)) by the action of the image of α in HomR (h(y)∗ Ω1X/R , mn+1 /mn+2 ), so that h(z) = θn+1,X (x). Since Y is smooth over R, there exists a point zn+1 ∈ Y(R) such that θn+1,Y (zn+1 ) = z; then θn+1,X (h(zn+1 )) = θn+1,X (x) and θn−e,Y (zn+1 ) = θn−e,Y (y). Since n − e  e, one has ordjach (zn+1 ) = e. We then reiterate this process from the pair (x, zn+1 ) and construct in this way a sequence (zm )mn of points in Y(R) such that zm = y and θm+1,X (h(zm )) = θm+1,X (x)

and θm−e−1,Y (zm ) = θm−e−1,Y (zm−1 )

for every m > n. In particular, θn−e,Y (zm ) = θn−e,Y (y) for every m. This sequence converges to a point y  ∈ Y(R) such that h(y  ) = x and θn−e,Y (y  ) = θn−e,Y (y). 3.2. Description of the Fibers Definition 3.2.1. — Let h : Y → X be a morphism of flat formal Rschemes of finite type. Let B be a subset of Gr(Y); we say that the map h is Gr-injective on B if Gr(h)|B is injective and if, moreover, for every point y ∈ B, the corresponding field extension κ(h(y)) → κ(y) is (i) An isomorphism, if R has equal characteristic; (ii) Purely inseparable, if R has unequal characteristic. If, moreover, A = h(B), then we say that Gr(h) : B → A is Gr-bijective. Equivalently, Gr-injectivity (resp. Gr-bijectivity) means that the map B(k  ) → A(k  ) is injective (resp. bijective) for every field extension k  of k in case R has equal characteristic, resp. for every perfect field extension in case R has inequal characteristic.

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Theorem 3.2.2. — Let h : Y → X be a morphism of flat formal R-schemes of finite type. Assume that X and Y have pure relative dimension d over R and that Y is smooth over R. Let c be the Elkik–Jacobi constant of X. Let m, e, e be nonnegative integers such that m  max(e, e − e). Let B ⊂ Gr(Y) be a constructible subset of level  m such that, on B, the function ordjach is constant with value e. Let A = h(B) ⊂ Gr(X) and  assume that A ⊂ Gr(e ) (X). a) The set A is a constructible subset of level  m + e in Gr(X). b) Assume that the map h : B → A is injective. Let y, z ∈ B and let n ∈ N be such that θn,X (h(y)) = θn,X (h(z)) in Grn (X). If n  max(2e, m + e), then θn−e,Y (y) = θn−e,Y (z) in Grn−e (Y). c) Assume that the map h : B → A is Gr-bijective. Let n be an integer such that n  max(2e, m + e, ce ), let x ∈ θn,X (A), and let Bx = θn,Y (B) ∩ Grn (h)−1 (x). Then Bx is a closed subset of Grn (h)−1 (x); we endow it with its induced reduced schematic structure. Assume that R has equal characteristic and let F be the residue field of x; then Bx is isomorphic to AeF . Assume that R has mixed characteristic, and let F be a perfect extension of the residue field of x; then (Bx ⊗ F )red is isomorphic to AeF . Proof. — a) Let B  = θm+e,Y (B); by corollary 1.5.7, this is a constructible subset of Grm+e (Y). Let A = h(B  ) ⊂ Grm+e (X) and let us show that A = (θm+e )−1 (A ). The inclusion A ⊂ (θm+e,X )−1 (A ) is obvious, so that it suffices to prove the converse inclusion. Let x be a point of Gr(X) such that θm+e (x) ∈ A . By definition, there exists a point y ∈ B such that θm+e,X (h(y)) = θm+e,X (x). By proposition 3.1.7, applied to n = m + e  max(2e, e ), there exists a point y  ∈ Gr(Y) such that h(y  ) = x and θm,Y (y  ) = θm,Y (y). Since B is a constructible subset of level  m, this implies that y  ∈ B, so that x ∈ A. b) Applying proposition 3.1.7 to y and x = h(z), we see that there exists a point y  ∈ Gr(Y) such that h(y  ) = h(z) and θn−e,Y (y) = θn−e,Y (y  ). Since n − e  m and B is constructible of level  m, one has y  ∈ B. Then, the injectivity of h on B implies that y  = z. Consequently, θn−e,Y (y) = θn−e,Y (z), as was to be shown. c) The hypothesis that h : B → A is Gr-bijective is preserved by base change to R(F ); we may thus assume that F = k. It follows from proposition 2.3.4 that there exists a k-rational point x ∈ Gr(X) such that θn,X (x ) = x. By part a), one has x ∈ A, because n  m+e. Since Gr(h) induces a bijection from B(k) to A(k), there exists a k-rational point y  ∈ B(k) such that h(y  ) = x . Let y = θn,Y (y  ); this is a k-rational point of Bx . n n By part b), one has θn−e,Y (z) = θn−e,Y (y) for every z ∈ Bx . Since the truncation map θn,Y is surjective and B is constructible of level  m  n − e,

§ 3. GREENBERG SCHEMES AND MORPHISMS OF FORMAL SCHEMES

we have

295

n n (θn−e )−1 (θn−e,Y (y)) ⊂ θn,Y (B)

and we can write n n Bx = (θn−e,Y )−1 (θn−e,Y (y)) ∩ Grn (h)−1 (x). n n )−1 (θn−e,Y (y)). In particular, Bx is a closed subset of the fiber Z = (θn−e,Y We endow Bx with its induced reduced schematic structure. We know by proposition 2.2.6 that Z is a principal homogeneous space over Spec(k) whose translation space is given by the functor

Algk → Ab,

n A → HomR (y ∗ Ω1Y/R , Jn−e (A)).

n n The analogous property holds for the fiber Y = (θn−e,X )−1 (θn−e,X (x)) of n n θn−e,X over θn−e,X (x). Thus the reduced k-scheme Bx represents the functor from the category RedAlgk of reduced k-algebras to the category of sets that associates with a reduced k-algebra A the kernel of the morphism n n (A)) → HomR (x∗ Ω1X/R , Jn−e (A)). HomR (y ∗ Ω1Y/R , Jn−e n (A)), this kernel is isomorphic By left exactness of the functor HomR (·, Jn−e to n HomR (y ∗ Ω1Y/X , Jn−e (A)), functorially in A. To conclude the proof, we show that there exists a bijection n HomR (y ∗ Ω1Y/X , Jn−e (A)) → Ae

which is functorial in the reduced k-algebra A. Since the function ordjach is constant on B, with value e, the length of the R-module y ∗ Ω1Y/X is equal to e (lemma 3.1.6). Let us write it as a direct sum y ∗ Ω1Y/X 

r 

R ei ,

i=1

where e1 , . . . , er are nonnegative integers such that e1 + · · · + er = e − r; in particular, ei  e − 1 for every i. On the other hand, for every integer q such that 0  q  e − 1, the map α → α(1) induces a bijection n n (A)) to the set of elements a ∈ Jn−e (A) such that from HomR (Rq , Jn−e π q+1 a = 0. Since A is reduced, proposition 4/2.4.2 asserts that this set is n (A), so that we obtain a isomorphism of R-modules precisely Jn−q−1 

n n HomR (Rq , Jn−e (A)) − → Jn−q−1 (A),

defined for every reduced k-algebra A and functorial in A. It then follows from lemma 4/2.4.1 that this functor is isomorphic to the functor defined by the affine space Aq+1 k . Consequently, the functor n A → HomR (y ∗ Ω1Y/X , Jn−e (A))

on the category of reduced k-algebras is isomorphic to the functor associated with Aek , as was to be shown.

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Corollary 3.2.3. — Let h : Y → X be a morphism of flat formal R-schemes of finite type of pure relative dimension d such that Y is smooth over R, Xη is quasi-smooth over K, and hη is étale. Then for every constructible subset B of Gr(Y), the set Gr(h)(B) is constructible in Gr(X). Proof. — The assumption that Xη is quasi-smooth over K implies that  Gr(X) = Gr(e ) (X) when e is sufficiently large. Since ordjach is constructible and bounded on Gr(Y) by proposition 3.1.4, we may assume that it is constant on B. Then the result follows directly from theorem 3.2.2. Corollary 3.2.4. — Assume that k is perfect. Let h : Y → X be a morphism of flat formal R-schemes of finite type of pure relative dimension d such that Xη and Yη are quasi-smooth over K and hη is étale. Then the function ordjach : Gr(Y) → N is constructible and bounded. Proof. — Let g : Z → Y be a Néron smoothening of Y. Then ordjach ◦ Gr(g) = ordjach◦g − ordjacg by proposition 3.1.5. Both functions in the right-hand side are constructible and bounded, by proposition 3.1.4, so that ordjach ◦ Gr(g) is constructible and bounded, as well. Since k is perfect, the map Gr(h) is surjective, by proposition 4/3.5.1. Now it follows from corollary 3.2.3 that ordjach is constructible and bounded. Example 3.2.5. — Assume that R is of mixed characteristic (0, p) and absolutely unramified; thus R = W (k), with k a perfect field of characteristic p > 0. We set X = Spf(R{x}) and Y = Spf(R{y}), and we consider the morphism h : Y → X defined by x → py. The map h : Gr(Y) → Gr(X) is injective, and its image is the constructible subset A = (θ0 )−1 (o) in Gr(X), where o denotes the origin of the special fiber X0 = Spec(k[x]) of X. The function ordjach is constant with value e = 1 on Gr(Y). For every integer n  0, the morphism h induces a morphism of Greenberg schemes h : Grn (Y) → Grn (X) that is explicitly given by  p yi−1 if i  1, Spec(k[y0 , . . . , yn ]) → Spec(k[x0 , . . . , xn ]), xi → 0 if i = 0. If F is a perfect field and a = (a0 , . . . , an ) is an F -valued point on Grn (X) = Spec(k[x0 , . . . , xn ]), with n > 0, then the fiber of h over a is the non-reduced scheme 1/p

p Spec(F [y0 , . . . , yn ]/((y0 − a1 )p , . . . , (yn−1 − a1/p n ) ))

whose maximal reduced closed subscheme is isomorphic to Spec(F [yn ])  A1F .

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Example 3.2.6. — Let X and Y be formal R-schemes of finite type of pure relative dimension d, and assume that Y is smooth over R and connected. Let h : Y → X be a morphism of formal R-schemes such that hη : Yη → Xη is an analytic domain immersion. Choose an integer e  0 and denote by B  the inverse image of Gr(e ) (X) in Gr(Y). Since hη is quasi-étale, the sheaf Ω1Y/X vanishes on Yη , so that (Jach )η = OYη . Because we assumed that Y is smooth and connected, there exists an integer e  0 such that the Jacobian ideal Jach is generated by me . Consequently, ordjach is constant with value e on Gr(Y). Thus theorem 3.2.2 asserts that the image of the map Gr(Y) → Gr(X) is a constructible subset A of Gr(X). Let us show that h satisfies the conditions of theorem 3.2.2, c). If R has equal characteristic, then we have to show that for every field extension k  of k, the map Gr(Y)(k  ) → Gr(X)(k  ) is injective. By base change to the complete discrete valuation ring R(k  ), we may assume that k = k  . Then we can identify the map Gr(Y)(k  ) → Gr(X)(k  ) with the map Y(R) = Yη (K) → X(R) = Xη (K) which is injective because hη is an analytic domain immersion. A similar argument applies in the case where R has unequal characteristic (considering only perfect extensions k  ). A case of particular importance is the one where k is perfect, h : X → X is a Néron smoothening as defined in section 4/3.4, and Y is a connected component of X . In that case, Gr(X ) → Gr(X) is bijective, by proposition 4/3.5.1. This example will be of crucial importance to define motivic invariants of K-analytic spaces in section 7/5. 3.3. Codimension of Constructible Sets in Greenberg Spaces Lemma 3.3.1. — Let X be a smooth formal R-scheme of finite type. For every integer n  0 and every constructible subset A of Grn (X), one has θn−1 (A) = θn−1 (A). In particular, the Zariski closure of a constructible subset of Gr(X) is constructible as well. Proof. — It suffices to prove the first statement. The inclusion θn−1 (A) ⊃ θn−1 (A) follows immediately from the continuity of θn , so that we only need to prove that θn−1 (A) ⊂ θn−1 (A). It is enough to show that the truncation map −1 (U ) θn is open. The topology on Gr(X) is generated by opens of the form θm with m a nonnegative integer and U an open subset of Grm (X). Replacing n −1 ) (U ) and m by n if m < n, we reduce to the case where m  n. U by (θm Since X is smooth, the truncation map θm is surjective, so that −1 (U )) = θnm (U ). θn (θm

Smoothness of X also implies that θnm is smooth, and, in particular, open; thus θnm (U ) is open. This concludes the proof.

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Definition 3.3.2. — Let X be a formal R-scheme of finite type. One sets  Grnondeg (X) = Gr(X) Gr(Xsing ) = Gr(e) (X). e∈N

Points of Grnondeg (X) are called nondegenerate. Proposition 3.3.3. — Let X be a formal R-scheme of finite type of pure relative dimension d, and let c be its Elkik–Jacobi constant. Assume that Gr(Xsm ), the Greenberg scheme of the R-smooth locus of X, is dense in Grnondeg (X). Let A be a nonempty constructible subset of Gr(X). Let e ∈ N be such that every irreducible component of A meets Gr(e) (X). Then, for every integer n  ce such that A is of level  n, the (topological) codimension of A ∩ Grnondeg (X) in Grnondeg (X) is equal to (3.3.3.1)

codim(θn (A ∩ Gr(e) (X)), θn (Gr(e) (X))) = (n + 1)d − dim(θn (A ∩ Gr(e) (X))).

Proof. — Our assumption that Gr(Xsm ) is dense in Grnondeg (X) implies that Grn (Xsm ) = θn (Gr(Xsm )) is dense in θn (Gr(e) (X)) for all n  e. Since X is of pure relative dimension d, we know that Grn (Xsm ) has pure dimension (n + 1)d for every integer n  0. Thus θn (Gr(e) (X)) has pure dimension (n + 1)d for all n  e, which shows the equality of the two terms of the formula. We will now prove that the codimension in (3.3.3.1) is independent of the choices of e and n. Let m ∈ N be such that m  ce and let Am be a −1 (Am ). Let n be an integer constructible subset of Grm (X) such that A = θm such that n  m. By theorem 2.3.11 one has dim(θn (A ∩ Gr(e) (X))) = dim(θm (A ∩ Gr(e) (X))) + (n − m)d, so that codim(θn (A ∩ Gr(e) (X)), θn (Gr(e) (X))) = codim(θm (A ∩ Gr(e) (X)), θm (Gr(e) (X))). Let us denote this integer by q. Let A0  A1  · · ·  As be a strictly increasing chain of irreducible closed subsets of Gr(X) such that A0 is the closure of an irreducible component of A. Let e ∈ N be such that A0 ∩ Gr(e) (X) = ∅. By lemma 4/4.2.6, there exists an integer m  ce such that θn (Ai−1 )  θn (Ai ) for every i ∈ {1, . . . , s} and every n  m. This implies that q  s, so that q  codim(A ∩ Grnondeg (X), Grnondeg (X)). Conversely, let n and e be integers such that n  ce and A = (θn )−1 (θn (A)), and let Bn0  Bn1  · · ·  Bnq be a strictly increasing chain of irreducible closed subsets of θn (Gr(e) (X)) such that Bn0 is the closure of an irreducible component of θn (A ∩ Gr(e) (X)). For every i

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in {0, . . . , q}, we denote by ηi the generic point of Bni . Then the set (θnn+1 )−1 (ηi ) ∩ θn+1 (Gr(e) (X)) is irreducible by theorem 2.3.11, and we define i Bn+1 to be its closure in θn+1 (Gr(e) (X)). The sets Bin+1 again form a strictly 0 increasing chain of irreducible closed subsets of θn+1 (Gr(e) (X)), and Bn+1 is (e) the closure of an irreducible component of θn+1 (A ∩ Gr (X)). i )n , and we Proceeding i a sequence (Bm  inductively,i we obtain for every set Ai = mn (θm )−1 (Bm ). The sets Ai are irreducible closed subsets of Gr(e) (X) and satisfy the inclusions Ai−1  Ai for i ∈ {1, . . . , q}. Moreover, A0 is the closure of an irreducible component of A ∩ Gr(e) (X), hence q  codim(A ∩ Gr(e) (X), Gr(e) (X)). Since this holds for every e and Grnondeg (X) is the union of the open subschemes Gr(e) (X), this implies that q  codim(A ∩ Grnondeg (X), Grnondeg (X)), as was to be shown. Remark 3.3.4. — The condition that Gr(Xsm ) is dense in Grnondeg (X) is  kR satisfied, for instance, when R has equal characteristic and X = X ⊗ for some k-variety X. Indeed, in that case, Gr(Xsm ) = L∞ (Xsm ) and Grnondeg (X) = L∞ L∞ (Xsing ), so that the property follows from lemma 3/4.2.2. Remark 3.3.5. — In the literature, the integer codim(A ∩ Grnondeg (X), Grnondeg (X)) is called the codimension of the constructible set A. However, the following example, due to Ishii and Reguera (2013), shows that when X is not smooth, the topological codimension of A ∩ Grnondeg (X) may differ from the codimension of A in Gr(X). Let X be the hypersurface of A3k = Spec(k[x, y, z]) defined by the equation 2 x + y 2 + z 2 = 0. (We assume that the characteristic of k is = 2.) Its singular locus is then reduced to the origin, and its minimal desingularization π : Y → X is such that E = π −1 (0) is an irreducible curve. Let ν = ordE be the corresponding divisorial valuation on X, and let WX (ν) be the irreducible closed constructible subset of L∞ (X) defined in definition 7/2.2.3. Then one may check that the topological codimension of WX (ν) is equal to 1, but codim(WX (ν)) = 2. See (Ishii and Reguera 2013, §2.8) for details. Proposition 3.3.6. — Let X and Y be flat formal R-schemes of finite type of pure relative dimension d, and let h : Y → X be a morphism. Assume that Y is smooth over R. Let B be a closed irreducible constructible subset of Gr(Y) and let A = h(B). Then A is constructible and closed; moreover, codim(A ∩ Grnondeg (X)) = codim(B) + ordB (Jach ). Proof. — Let e = ordB (Jach ) = inf x∈B ordjach (x). Let p be an integer, and let Bp be an irreducible closed subscheme of Grp (Y) such that B = −1 θp,Y (Bp ); without loss of generality, we assume that p  e. Then there exists

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a dense open subscheme Bp of Bp such that ordjach (x) = e for every x ∈ −1 (Bp ). Observe that B  is an irreducible dense open constructible B  = θp,Y subset of B and ordjach (x) = e for every x ∈ B  . Let m = p + e; let −1 m −1  m −1 ) (Bp ) and Bm = (θp,Y ) (Bp ) so that B = θm,Y (Bm ) and Bm = (θp,Y −1     B = θm,Y (Bm ). Let then Am = h∗ (Bm ) and Am = h∗ (Bm ); one has −1 Am = Am . By theorem 3.2.2, one thus has h∗ (B  ) = θm,X (Am ), so that −1 A = h∗ (B) = h∗ (B  ) = θm,X (Am ).

This shows that A is a closed constructible subset of Gr(X). By theorem 3.2.2  ) = dim(Am ) + e, so that dim(Bm ) = dim(Am ) + e as again, one has dim(Bm well. Using proposition 3.3.3, this implies that codim(A ∩ Grnondeg (X)) = (m + 1) dim(X) − dim(Am ) = (m + 1) dim(Y) − dim(Bm ) + e = codim(B) + e = codim(B) + ordB (Jach ).

3.4. Example: Contact Loci in Arc Spaces (3.4.1). — Let k be a field and let X be a k-variety. Let λ : A1 ×k L∞ (X) → L∞ (X) denote the action of the monoid scheme A1k on L∞ (X) that was introduced in corollary 3/3.6.5. −1 Let C be a subset of L∞ (X); let then C  = λ(pr−1 1 (Gm ) ∩ pr2 (C)) and −1  C = λ(pr2 (C)) be the “orbits” of C under the Gm - and the A1 -actions. One says that C is homogeneous if C = C  ; observe that C  and C  are homogeneous and that C  = C  ∪ s∞,X (θ0,X (C)). We make the following additional observations concerning a subset C of L∞ (X) and the associated homogeneous subsets C  and C  . Lemma 3.4.2. — Let k be a field, let X be a k-variety, and let C be a subset of L∞ (X); let C  and C  be the orbits of C under the Gm - and A1 -actions. a) If C is constructible, then C  and C  are constructible as well. b) If C is irreducible, then C  and C  are irreducible as well. c) If C is homogeneous and closed, then C = C  = C  . Moreover, C contains s∞,X (θ0,X (C)), and θ0,X (C) is closed in X. d) If C is an irreducible component of L∞ (X), then C is homogeneous. Proof. — a) Assume that C is constructible, and let m ∈ N be such that C = (θm,X )−1 (D) for some constructible subset D = θm,X (C)) of Lm (X). By defining D and D analogously to C  and C  , one observes that C  = (θm,X )−1 (D ) and C  = (θm,X )−1 (D ), whence the claim.

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b) Since Gm,k is a geometrically irreducible k-scheme, the preimage of C in Gm,k ×k L∞ (X) is irreducible, so that C  is irreducible. The case of C  is analogous. c) For every γ ∈ C, the point s∞,X (θ0,X (γ)) is the constant arc γ0 based at the origin of γ; thus γ0 = λ(0, γ) = limt→0 λ(t, γ). Since C is homogeneous and closed, γ0 lies in C. The relation θ0,X (C) = s−1 ∞,X (C) also implies that θ0,X (C) is closed in X. d) This follows from b), since C ⊂ C  ⊂ C  . Remark 3.4.3. — Assume that k is perfect. We will prove that a nonempty closed homogeneous constructible subset C of L∞ (X) is not thin. Let n  0 be an integer and A be a closed constructible subset of Ln (X) such that C = θn−1 (A). By lemma 3.4.2, c), there exists a closed point x ∈ X such that the constant arc s∞,X (x) at x belongs to C. Let p : Y → X be the normalization of the blow-up of {x} and let E = p−1 ({x}); since Y is normal and codim(E, Y ) = 1, there exists a smooth closed point y ∈ Y such that y ∈ E. −1 (sn,Y (y)) is a closed constructible subset of p−1 Then D = θn,Y ∗ (C) which is not not contained in L∞ (Ysing ). By proposition 6/2.4.6, it is not thin. Consequently, p∗ (D) is not thin neither. Since p∗ (D) ⊂ C, this implies that C is not thin. (3.4.4). — Let k be a field. Let X be a k-variety, and let Y be a closed subscheme of X. For every integer q  0, the contact locus Contq (X, Y ) is the set of arcs γ ∈ L (X) such that ordγ (f )  q for every element f ∈ OX,γ(0) which belongs to the ideal of Y . One defines analogously the contact locus Contq (X, Y ). For any ideal I ⊂ OX , one defines the contact loci Contq (I) and Contq (I) as Contq (V (I)) and Contq (V (I)), respectively. Lemma 3.4.5. — Let X be a k-variety, and let Y be a closed subscheme of X. Let q  0. Then Contq (X, Y ) is a closed constructible subset of L (X), and Contq (X, Y ) is a constructible open subset of Contq (X, Y ). Proof. — The set Contq (X, Y ) is closed in L (X) by proposition 4/4.4.7; thus Contq (X, Y ) = Contq (X, Y ) Contq+1 (X, Y ) is open in Contq (X, Y ). Proposition 3.4.6. — Let X be a smooth integral k-variety. Let r be a positive integer; let Y1 , . . . , Yr be smooth integral subschemes of X which meet transversally; for every i, let ci = codim(Yi , X). Let Y be an irreducible component of Y1 ∩ · · · ∩ Yr . Let (q1 , . . . , qr ) be a family of integers  1. Let C and C  be the subsets of L (X) consisting of arcs γ such that γ(0) ∈ Y and such that ordγ (Yi ) = qi and ordγ (Yi )  qi , respectively.

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a) The sets C and C  are irreducible homogeneous constructible subsets of L (X). Moreover, C is open and dense in C  , C  is closed, and r codim(C, L (X)) = i=1 qi ci . −1  b) For every y ∈ Y , C ∩ θ0−1 (y) ris open and dense in C ∩ θ0 (y), and −1  codim(C ∩ θ0 (y), L (X)) = n + i=1 (qi − 1)ci . Proof. — First of all, it follows from the definition of the order valuation defined by an arc that C = θq−1 (θq (C)) for every integer q such that q > max(qi ) and similarly for C  . This shows that C and C  are constructible. It also follows from their definition in terms of the valuations associated with arcs that these sets are homogeneous. Moreover, the conditions defining θq (C  ) are closed, so that C  is closed; similarly, the conditions defining θq (C) within θq (C  ) are open, so that C is an open subset of C  . We then consider the case where X = An and where, for every i, there exists a subset Ji of {1, . . . , n} such that the subscheme Yi is defined by the vanishing of the coordinates xj , for j ∈ Ji . Since Y1 , . . . , Yr meet transversally, the subsets J1 , . . . , Jr are pairwise disjoint. Moreover, the intersection Y1 ∩ · · · ∩ Yr is irreducible and hence is equal to Y . In this case, we identify L (X) with L (A1 )n , and one has ordγ (Yi ) = min ord(γj ) j∈Ji

for every γ = (γ1 , . . . , γn ) ∈ L (X). Furthermore, through the canonical isomorphism L (A1 )  AN ,

γ = γ0 + γ1 t + . . . → (γ0 , γ1 , . . . )



the subspace C is defined by the conditions γj,m = 0 for every i, every j ∈ Ji and every m < qi . Fix an integer q such that q > max(q1 , . . . , qn ), one has C  = θq−1 (θq (C  )), and θq (C  ) is  an irreducible closed subset of Lq (X)  Lq (A1 )n  A(q+1)n of r  codimension i=1 qi ci . Since X is smooth, r C is irreducible, constructible,  and closed, and codim(C , L (X)) = i=1 qi ci . Moreover, one has C = θq−1 (θq (C)), and θq (C) is the dense open subset of θq (C  ) defined by the conditions for every i, there exists j ∈ Ji such that γj,qi = 0. Consequently, C is a dense open subset of C  , and codim(C, L (X)) = r i=1 qi ci as well. Let y = (y1 , . . . , yn ) ∈ Y , and let Wq be the closed subset of Lq (X) defined by the conditions γj,m = 0 for every i, every j ∈ Ji , every m such that 0 < m < qi , as well as the conditions γj,0 = yj for j ∈ {1, . . . , n} such that j ∈ Ji . Then Wq is irreducible and closed in Lq (X), and its codimension is equal to r r r    qi ci + (n − ci ) = (qi − 1)ci + n. i=1

i=1

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Let also Wq = θq (C) ∩ Wq ; this is a dense open subset of Wq . Moreover, C  ∩ θ0−1 (y) = θq−1 (Wq ) and C ∩ θ0−1 (y) = θq−1 (Wq ); in particular, these constructible sets are irreducible. Let us now prove the general case. Let n = dim(X) and let y ∈ Y . Since X is smooth, and the subschemes Yi are smooth and meet transversally, there exist an open neighborhood U of y in X and an étale morphism f : U → An , smooth closed subschemes Z1 , . . . Zr of An , where Zi is defined by the vanishing of  some coordinates xj , for j ∈ Ji , such that U ∩ Yi = f −1 (Zi ). Letting Z = i Zi , we may moreover assume that U ∩ Y = f −1 (Z). Let D and D be the subsets of L (An ) defined by the subschemes Zi and the integers qi . Since f is smooth, it induces a commutative diagram L (U )

L (f )

L (An )

θq

Lq (U )

θq L q (f )

Lq (An ) θ0q

θ0q f

U

An

with Cartesian squares. The horizontal arrows are étale, and the vertical arrows θ0q are compositions of q locally trivial fibrations, with fiber an affine space of dimension n. Moreover, C  ∩ L (U ) and C ∩ L (U ) are the inverse images of D and D in L (An ). As a consequence, θq (C  ∩ L (U )) = θq (C  ) ∩ of its projection to Y Lq (U ) is a closed subset of Lq (U ), and its fibers are closed irreducible subsets of codimension n + (qi − 1)ci ; moreover, θq (C ∩ L (U )) is a dense open subset of θq (C  ∩ L (U )). This implies the second assertion. As a composition of locally trivial fibrations, the morphism θ0q is surjective and open. Applying (ÉGA Isv , Chap. 0, prop. 2.1.14), it follows from this that θq (C  ) is irreducible. Moreover, its dimension is equal to

dim(θq (C  )) = dim(Y ) + min dim θq (C  ∩ (θ0q )−1 (y)) y∈Y

= dim(Y ) + n(q + 1) −

r 

(qi − 1)ci

i=1

= n(q + 1) −

r 

qi ci ,

i=1

since dim(Y ) = n − codim(Y ) = n −

r  i=1

This concludes the proof of the proposition.

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(0.0.1). — In this chapter we develop the theory of motivic integration on formal schemes X over a complete discrete valuation ring R, introduced by Sebag (2004a) and generalizing the constructions of Kontsevich (1995), Denef and Loeser (1999), and Looijenga (2002). We define a measure on the class of constructible subsets in the Greenberg schemes Gr(X), which takes its values in the dimensional completion of the Grothendieck ring K0 (VarX0 ) of varieties over the special fiber X0 of X. This measure is constructed in close analogy with the theory of p-adic integration. In section 1, we first present the case where X is smooth. This case is significantly simpler than the general case, but already sufficient for several interesting applications. In section 2, we then define the measure of constructible subsets in the Greenberg scheme without any smoothness assumption. In order to obtain a broader class of measurable sets, we then construct a Carathéodory-type extension of the measure in section 3. In section 5, we define semi-algebraic subsets of Gr(X) and show their measurability in this sense. Finally, in section 4 we define the motivic integral of integrable functions. We then prove one of the most important results in this book, the change of variables formula for motivic integrals (theorem 4.3.1), which is a crucial tool in almost all the applications of motivic integration. (0.0.2). — In this chapter, R is a complete discrete valuation ring with residue field k, maximal ideal m, and quotient field K. We assume that k is perfect when R has mixed characteristic; if R has equal characteristic, we choose a section of the projection morphism R → k. We define Rn to be the ring R/mn+1 ; in particular, R0 = k. We consider adic formal schemes over Spf(R), denoted by fraktur letters, e.g., X, Y, Z. . . An admissible formal Rscheme is a flat formal R-scheme of finite type. For every formal R-scheme of finite type X, we denote by X0 = X ⊗R k its special fiber (a scheme of finite type over k) and by Xη its generic fiber (a compact K-analytic space). © Springer Science+Business Media, LLC, part of Springer Nature 2018 A. Chambert-Loir et al., Motivic Integration, Progress in Mathematics 325, https://doi.org/10.1007/978-1-4939-7887-8_6

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We refer to section A/3 for an overview of the theory of formal schemes and analytic spaces. (0.0.3). — Let us recall that, for every formal R-scheme of finite type X, we have defined its Greenberg schemes Grn (X), n  0, and Gr∞ (X) = lim Grn (X) in chapter 4. These are related by affine truncation morphisms ← − n ∞ θm,X : Grn (X) → Grm (X) and θm,X : Gr∞ (X) → Grm (X). In particular, the Greenberg schemes are naturally endowed with a scheme structure over Gr0 (X) = X0 . If no confusion is possible, we omit the index X in the nota∞ n , θm,X . To further simplify the notation, we will tion of the morphisms θm,X sometimes denote Gr∞ (X), Gr∞ (f ) and θn∞ by Gr(X), Gr(f ) and θn for any formal R-scheme X and any morphism f : X → Y of formal R-schemes. We extend the usual order  from N to N ∪ {+∞} by setting a  ∞ for all a in N ∪ {+∞}. (0.0.4). — This chapter is written in the general framework of Greenberg schemes of formal schemes, but we emphasize that the results also apply to the following two important special cases. – (Arc schemes). Let X be a k-variety. Let R = k[[t]] and define X = ˆ k R. Then the Greenberg schemes Grn (X) and Gr∞ (X) are canonically X⊗ isomorphic to the jet schemes Ln (X/k) and the arc scheme L∞ (X/k). This is the classical setting of motivic integration, as introduced by Kontsevich (1995) in the smooth case and developed by Denef and Loeser (1999) in general. – (Greenberg schemes of R-varieties, equal characteristic). We still consider R = k[[t]] but start from an R-variety X and define X as its formal t-adic completion. One has canonical identifications Grn (X)  Grn (X ) for all n. This setting has been proposed by Looijenga (2002). (0.0.5). — We recall that we denote by MX0 the localization of the Grothendieck ring K0 (VarX0 ) obtained by adjoining the inverse of the class L = LX0 of the affine line. This ring admits a filtration by dimension, where F  MX0 is the set of classes x/Ln such that dim(x/X0 ) − n  , where the dimension dim(x/X0 ) is defined as in §2/4.1.1. We also fix a real number r > 1, and we define a norm · on MX0 by setting x/Ln = rdim(x/X0 )−n for every nonzero element x ∈ MX0 . This induces a non-Archimedean ring norm on MX0 . We X the associated separate and complete ring. We still denote denote by M 0 X the induced norm and the induced filtration on M X . by · and F • M 0 0 We also recall that we defined modified Grothendieck rings in section 2/4.4, denoted with an exponent uh. R : To make the notation uniform, we shall write MXR0 and M X0 X , if R has equal characteristic; – for MX0 and M 0 uh , if R has mixed characteristic. – for M uh and M X0

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(0.0.6). — Let X be a formal R-scheme of finite type. Recall that we denote by ConsGr(X) the Boolean algebra formed by the constructible subsets of −1 (Cn ) Gr∞ (X) and that these are precisely the subsets of the form C = θn,X for some integer n  0 and some constructible subset Cn of Grn (X). For such a subset Cn , we have defined its class e(Cn /X0 ) in the Grothendieck ring of X0 -varieties K0 (VarX0 ) by writing Cn as a disjoint union of subvarieties of Grn (X). (0.0.7). — To unburden the notation, we adopt the following conventions throughout this chapter: when the formal R-scheme of finite type X is fixed, we will write L and e(Y ) instead of LX0 and e(Y /X0 ), where Y is a variety over X0 . It will always be clearly indicated in which ring the class e(Y ) is considered.

§ 1. MOTIVIC INTEGRATION IN THE SMOOTH CASE In this section, we present a version of the geometric theory of motivic integration for smooth formal schemes of finite type over Spf(R) (in fact, we will impose a slightly weaker condition). The smoothness assumption substantially simplifies the constructions; we will study the general case in the following sections. For every smooth formal R-scheme X of finite type of pure relative dimension d  0, we will construct a map μX : ConsGr(X) → MX0 that is additive with respect to disjoint unions: for every pair of disjoint constructible subsets (A, B) in Gr(X), one has μX (A ∪ B) = μX (A) + μX (B). This map will play the role of the measure in the theory of motivic integration on smooth formal schemes. 1.1. Working with Constructible Sets Lemma 1.1.1. — Let X be a formal R-scheme of finite type of pure relative dimension d, and let c be its Elkik-Jacobi constant. Let e ∈ N and let A be a constructible subset of Gr(e) (X) of level  . Then for all integers m, n with m  n  max( , ce), we have the equality e(θn,X (A))L−(n+1)d = e(θm,X (A))L−(m+1)d in MXR0 . If X is smooth over R, then the equality e(θn,X (A))L−(n+1)d = e(θm,X (A))L−(m+1)d holds already in MX0 , for all m, n  .

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Proof. — By induction, we may assume that m = n + 1. We then consider the constructible sets θn+1,X (A) and θn,X (A) and the map induced by the n+1 truncation morphism θn,X . If R has equal characteristic, then all of its fibers are affine spaces of dimension d, by theorem 5/2.3.11. This implies the equality e(θn+1,X (A)) = e(θn,X (A))Ld in MX0 . If R has mixed characteristic, then theorem 5/2.3.11 asserts that the fibers are affine spaces of dimension d as well, provided one makes a base change to a perfection of the ground field. In that case, the preceding equality holds in the modified Grothendieck ring of varieties MXuh0 . The lemma thus follows from the definition of MXR0 . If X is smooth over R, then the truncation morphism (m−n)d

m is a locally trivial fibration with fiber Ak for all m, n  0, by θn,X proposition 5/1.2.2. Thus the equality holds already in MX0 .

Proposition 1.1.2. — Let X be a formal R-scheme of finite type of pure relative dimension d. Let A be a constructible subset of Gr(X) such that A ∩ Gr(Xsing ) = ∅. Then the sequence   e(θn,X (A))L−(n+1)d n∈N

in

MXR0

is stationary. In particular, it converges to an element of MXR0 . If X is smooth over R, then the sequence is stationary already in MX0 .

Proof. — By the quasi-compactness of the constructible topology, there exists a positive integer e such that A is contained in Gr(e) (X). Thus the result follows from lemma 1.1.1. Definition 1.1.3. — Let X be a formal R-scheme of finite type of pure relative dimension d. For every constructible subset A of Gr(X) that does not intersect Gr(Xsing ), we define μX (A) := lim e(θn,X (A))L−(n+1)d n→+∞

in the ring MXR0 . This element is called the motivic volume of A. If X is smooth over R, then μX (A) is well-defined already in MX0 . Example 1.1.4. — Let X be a smooth formal R-scheme of finite type of pure relative dimension d and let A be a constructible subset of level . Then we have μX (A) = e(θ,X (A))L−(+1)d . In particular, μX (Gr(X)) = e(X0 )L−d . This element is called the motivic volume of X.

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309

Corollary 1.1.5. — Let X be a formal R-scheme of finite type of pure relative dimension d. The map μX : ConsGr(X)

Gr(Xsing )

→ MXR0

is additive. If X is smooth over R, then the same holds for the map μX : ConsGr(X) → MX0 . Proof. — This follows from proposition 1.1.2 and the scissor relations in the Grothendieck ring. 1.2. The Change of Variables Formula in the Smooth Case (1.2.1). — Let X be a formal R-scheme of finite type of pure relative dimension d. Let A be a constructible subset of Gr(X) that does not meet Gr(Xsing ). Let α: A → Z be a constructible function defined on A (i.e., α−1 (n) is a constructible subset of A, for every n ∈ Z). The map α takes only a finite number of values, by the quasi-compactness of the constructible topology (see theorem A/1.2.4). Consequently, the series  μX (α−1 (n))L−n n∈Z

is a finite sum and defines an element of the ring MXR0 . Definition 1.2.2. — The motivic integral of the function α is defined as   L−α dμX := μX (α−1 (n))L−n A

MXR0 .

in the ring already in MX0 .

n∈Z

If X is smooth over R, then we can consider this integral

Example 1.2.3. — Let α : A → Z be the constructible function on A defined by α(x) = 0 for every x ∈ A. Then  L−α dμX = μX (A). A

(1.2.4). — The main technical result on motivic integrals is the change of variables formula. We now give a very simple formulation of this important result in our particular case. Theorem 1.2.5. — Let X, Y be formal R-schemes of finite type of pure relative dimension d. Let f : Y → X be a morphism of formal R-schemes. Let A be a constructible subset of Gr(X) Gr(Xsing ), and let B be a constructible subset of Gr(Y) Gr(Ysing ). Assume that f induces a Gr-bijection from B to A. Assume also that B ∩ ordjac−1 f (+∞) = ∅. Let α : A → Z be a constructible function on A. Then the following properties hold:

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a) The function β : B → Z, defined by y → (α ◦ Gr(f ))(y) + ordjacf (y), is constructible. b) The transformation rule   L−α dμX = (f0 )! L−β dμY A

B

holds in the ring MXR0 . Recall from definition 5/3.2.1 that we say that f induces a Gr-bijection from B to A if it induces a bijection B(k  ) → A(k  ) for every extension (resp. every perfect extension) k  of k, if R has equal (resp. mixed) characteristic. Proof. — a) Note that ordjacf is a constructible function on B and thus takes only a finite number of values. Since θm,X ◦ Gr(f ) = Grm (f ) ◦ θm,Y , for every m ∈ N, and the fibers of the map α are constructible subsets of Gr(X), we deduce from theorem A/1.2.4 that (α ◦ Gr(f ))−1 (m) is a constructible subset of Gr(Y). It follows that β is constructible and takes only finitely many values, as well. b) For every e ∈ N, we denote by Be the constructible subset of B consisting of the points where ordjacf = e. The image of Be in Gr(X) is constructible, by theorem 5/3.2.2. We denote this image by Ae . Theorem 5/3.2.2 also guarantees that for every integer n and every sufficiently large integer m  0, e(θm,Y (Be ∩ β −1 (n + e))) = e(θm,X (Ae ∩ α−1 (n)))Le R in MX0 . This implies that μX (Ae ∩ α−1 (n))L−n = μY (Be ∩ β −1 (n + e))L−n−e for all integers n and e. Taking the sum over all n and e, we obtain the desired equality of motivic integrals. In the statement of theorem 1.2.5, the assumption B ∩ ordjac−1 f (+∞) = ∅ is valid, for instance, when f induces a quasi-étale morphism fη : Yη → Xη of K-analytic spaces, then ordjacf is finite everywhere on Gr(Y). We emphasize that, even when X and Y are smooth over R so that both sides of the equality in the transformation rule are well defined over MX0 , our proof only produces an equality in MXR0 .

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§ 2. THE VOLUME OF A CONSTRUCTIBLE SUBSET OF A GREENBERG SCHEME In this section, we generalize the constructions of section 1 by allowing the space to be singular and the functions to take infinitely many values. The basic question here is to understand when and where the limit of the sequence in proposition 1.1.2 exists. As we will see, these problems are solved R . by considering the completion MXR0 → M X0 2.1. What Is a Motivic Volume? (2.1.1). — Let X be a formal scheme of finite type over Spf(R) whose generic fiber has dimension d  0. We have seen in corollary 5/1.5.7 that, for every integer n ∈ N, and for every constructible subset A of Gr(X), the subset θn,X (A) is a constructible subset of Grn (X). Hence, it is still meaningful R . We have seen in proposito consider its class e(θn,X (A)) in the ring M X0 tion 1.1.2 that, when X has pure relative dimension and A does not meet Gr(Xsing ), the sequence (e(θn,X (A))L−(n+1)d )n∈N is stationary. This is not true without the smoothness assumption, but we can still try to consider its limit (2.1.1.1)

lim e(θn,X (A))L−(n+1)d .

n→+∞

The main result of this section is theorem 2.5.1, which asserts that this limit R if X satisfies a mild tameness condition (definition 2.4.2). always exists in M X0 This limit will be taken as the definition of the motivic volume μX (A) of A. We will then prove in theorem 2.5.5 that the map R μX : ConsGr(X) → M X0 is additive. This will be the basic measure of motivic integration. 2.2. Reduction to the Reduced Flat Case (2.2.1). — Let X be a formal R-scheme of finite type and let d  0 be the dimension of its generic fiber. Let X and Xflat denote the maximal reduced and flat closed formal subschemes of X, respectively. By lemma 4/4.1.2, the closed immersions X → X and Xflat → X induce homeomorphisms of Greenberg schemes that preserve constructible subsets. Moreover, the generic fibers Xη and (Xflat )η still have dimension d. Lemma 2.2.2. — Let A be a constructible subset of Gr(X). The three sequences (e(θn,X (A))L−(n+1)d )n∈N , (e(θn,X (A))L−(n+1)d )n∈N , and R coincide. In particular, if any of them (e(θn,Xflat (A))L−(n+1)d )n∈N in M X0 R . converges, then all of them converge and they have the same limit in M X0

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Proof. — The Greenberg schemes Grn (X ) and Grn (Xflat ) are closed subschemes of Grn (X). By lemma 4/4.1.2, one has θn,X (A) = θn,Xflat (A) = θn,X (A) in Grn (X), hence the lemma.

2.3. A Dimensional Lemma The following proposition is a key statement in the construction of the measure μX , which will allow us to neglect thin constructible subsets in the proof of existence of motivic volumes. Proposition 2.3.1. — Let X be a formal R-scheme of finite type and let d  0 be the dimension of its generic fiber. Let Z be a closed formal subscheme of X such that Zη is of dimension at most d − 1. Denote by γZ the Greenberg function of Z (definition 5/1.5.6). Let A be a constructible subset of Gr(X). Let ∈ N. Then, for every pair of integers (m, n) ∈ N2 , with n  m  γZ ( ), −1 (Grm (Z)))) · L−(n+1)d ∈ F +1 MXR0 . e(θn,X (A ∩ θm,X

Proof. — By definition of the dimensional filtration on MX0 , it suffices to prove that −1 (Grm (Z))))  (n + 1)d − − 1. dim(θn,X (A ∩ θm,X

It is sufficient to prove this inequality when A = Gr(X). Let c = γZ ( ); one has c  . By proposition 5/2.3.2, we have the following inequality: −1 −1 (Grc (Z))))  dim(θ,X (θc,X (Grc (Z)))) + (n − )d. dim(θn,X (θc,X

On the other hand, proposition 5/1.5.9 asserts that −1 (Grc (Z))) = θ,Z (Gr(Z)), θ,X (θc,X

so that −1 (Grc (Z))))  dim(θ,Z (Gr(Z))) + (n − )d. dim(θn,X (θc,X

Applying proposition 5/2.3.2 to Z, we also have dim(θ,X (Gr(Z)))  ( + 1)(d − 1). Consequently, we have −1 (Grc (Z))))  ( + 1)(d − 1) + (n − )d dim(θn,X (θc,X

= (n + 1)d − − 1, which proves the desired inequality when m = c. The general case follows readily. Indeed, we first observe the inclusion −1 −1 (Grm (Z))) ⊂ θn,X (θc,X (Grc (Z))), θn,X (θm,X

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313

m −1 deduced from the obvious inclusion Grm (Z) ⊂ (θc,X ) (Grc (Z)). We thus have −1 −1 dim(θn,X (θm,X (Grm (Z))))  dim(θn,X (θc,X (Grc (Z))))

 (n + 1)d − − 1, as was to be shown. Corollary 2.3.2. — Let X be a formal R-scheme of finite type and let d  0 be the dimension of its generic fiber. Let Z be a closed formal subscheme of X such that dim(Zη )  d − 1. Then, for every constructible subset A of Gr(X), one has −1 (Grn (Z)))) · L−(n+1)d = 0 lim e(θn,X (A ∩ θn,X n→+∞

R . in M X0 Proof. — This is an immediate consequence of proposition 2.3.1. Corollary 2.3.3. — Let X be a formal R-scheme of finite type and let d  0 be the dimension of its generic fiber. Let Z be a closed formal subscheme of X such that dim(Zη )  d − 1. Then, for every constructible subset A of Gr(X), one has lim e(θn,X (A ∩ Gr(Z)))) · L−(n+1)d = 0 n→+∞

R . in M X0 Proof. — The set θn,X (A ∩ Gr(Z)) is a constructible subset of Grn (Z) by R is well-defined. Moreover, proposition 5/1.5.9, so that its class in M X0 −1 Gr(Z) ⊂ θn,X (Grn (Z)), so that the result follows from corollary 2.3.2. Corollary 2.3.4. — Let X be a formal R-scheme of finite type and let d  0 be the dimension of its generic fiber. Let (Xi )I be the family of rig-irreducible components of X. Let I0 be the set of indices i ∈ I such that Gr(Xi ) is fat in Gr(X). Then, for every constructible subset A of Gr(X), the sequence in R with general term: M X0    e(θn,X (A)) − e(θn,X (A ∩ Gr(Xi )) · L−(n+1)d i∈I0

R . converges to zero in M X0 Proof. — Let B be the subset of Gr(X) consisting of the points that lie in Gr(Xi ) for at least two different indices i ∈ I. The generic fiber of an intersection of two different rig-irreducible components of X has dimension at most d − 1. Thus, it follows from corollary 2.3.3 that e(θn,X (B))L−(n+1)d R . This immediately implies the desired result. converges to zero in M X0

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2.4. Volume of Thin Constructible Subsets In this subsection, we show that the volume of a thin constructible subset always exists and equals 0. In particular, proposition 2.4.6 will allow us to neglect thin constructible subsets in the construction of the measure μX . (2.4.1). — Let X be a formal scheme of finite type over Spf(R) and let d  0 be the dimension of its generic fiber. Recall from (4/4.3.1) that a subset A of Gr(X) is said to be thin if there exist a finite affine open covering (Ui )i∈I of X, and, for every i ∈ I, a closed formal subscheme Zi of Ui , such that: a) For every i ∈ I, A ∩ Gr(Ui ) ⊂ Gr(Zi ); b) For every i ∈ I, the generic fiber (Zi )η is an analytic space of dimension at most d − 1. Recall also that a subset of Gr(X) which is not thin is said to be fat. Finally, recall (definition 5/1.3.1) that the singular locus Xsing of a formal R-scheme of finite type X is its closed formal subscheme defined by its Jacobian ideal JacX . Definition 2.4.2. — Let X be a formal R-scheme of finite type, and denote by X its maximal reduced closed formal subscheme. We say that X is tame if Gr(Xsing ) is thin in Gr(X). Proposition 2.4.3. — Assume that k is perfect. Then every formal scheme X of finite type over R of pure relative dimension d is tame. Proof. — By the definition of a tame formal scheme, we may assume that X is reduced and affine. The set of regular points in Spec(O(X)) is open because O(X) is excellent. We denote by Z its complement, endowed with its induced reduced structure, and by Z the formal R-adic completion of Z. The generic fiber of Z has dimension at most d − 1, because X is reduced. We claim that Gr(Xsing ) is contained in Gr(Z). It suffices to check this inclusion on points defined over finite extensions of k, because these points form a dense subset of Gr(Xsing ) by the Greenberg approximation theorem (theorem 5/1.5.1). Let L be a finite extension of K of ramification index one. Then L is separable over K because k is perfect. Thus Xη is smooth at every L-point that does not lie in Zη . It follows that Gr(Ysing ) ⊂ Gr(Z). Example 2.4.4. — Let X be a formal R-scheme of finite type and let X be its maximal reduced closed formal subscheme. If every irreducible component of maximal dimension of Xη has a nonempty quasi-smooth locus, then X is tame. (2.4.5). — Let e ∈ N. Recall from (5/1.4.2) that we denote by Gr(e) (X) the open subscheme of Gr(X) defined by Gr(e) (X) = Gr(X)

−1 θe,X (Gre (Xsing )).

This is a constructible open subset of Gr(X).

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315

Proposition 2.4.6. — Let X be a tame formal scheme of finite type over Spf(R) and let d  0 be the dimension of its generic fiber. Let A be a constructible subset of Gr(X). Then the following properties are equivalent: (i) The subset A is thin; (ii) The sequence of integers with general term dim(e (θn,X (A)) − (n + 1)d converges to −∞; R with general term (iii) The sequence in M X0 (e (θn,X (A))) · L−(n+1)d R . converges to zero in M X0 If Xη is equidimensional, then the previous properties are also equivalent with: (iv) the set A is contained in Gr(Xsing ), where X denotes the maximal reduced closed formal subscheme of X. Proof. — (i)⇒(ii). Let us assume that A is thin. By assumption, there exist an open affine covering (Ui )i∈I of X and, for every i ∈ I, a closed formal subscheme Zi of Ui , with generic fiber of dimension at most d − 1, such that A ∩ Gr(Ui ) ⊂ Gr(Zi ) for every i ∈ I. Since dim (θn,X (A)) = sup (dim (θn,Ui (A ∩ Gr(Ui )))) i∈I

we may assume that X is affine and that there exists a closed formal subscheme Z of X such that dim(Zη )  d − 1 and A ⊂ Gr(Z). Then proposition 5/2.3.2 applied to Z implies that: dim(θn,Z (Gr(Z)))  (n + 1)(d − 1), for every integer n ∈ N. Consequently, dim θn,X (A) − (n + 1)d  (n + 1)(d − 1) − (n + 1)d = −n − 1 converges to −∞.

R . (ii)⇔(iii). This follows from the definition of the convergence in M X0 (ii)⇒(iv). Assume that X has pure relative dimension d and that A ⊂ Gr(Xsing ). Then there exists an integer e ∈ N such that A ∩ Gr(e) (X) = ∅. Let us set A(e) = A ∩ Gr(e) (X). By theorem 5/2.3.11, there exists an integer N  0 such that for every n  N ,   e(θn (A(e) )) · L−(n+1)d = e(θN (A(e) )) · L(n−N )d · L−(n+1)d = e(θN (A(e) )) · L−(N +1)d . In particular, for every integer n  N , we have dim(e(θn (A)) − (n + 1)d)  dim(e(θn (A(e) )) − (n + 1)d) = dim(θN (A(e) )) − (N + 1)d,

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contradicting (iii). (iv)⇒(i). This follows immediately from the fact that Gr(Xsing ) is thin, by proposition 2.4.3, since X is purely of relative dimension d. (iii)⇒(i). Let (Xi )i∈I be the rig-irreducible components of X. Since A = i∈I A ∩ Gr(Xi ) by lemma 4/4.1.3, we may assume that X is rigirreducible. Then Xη has pure dimension d, and then the result follows from the implications that we have already proven. 2.5. Existence of the Volume of a Constructible Subset Let us now prove the existence of a volume for the constructible subsets of Greenberg schemes. Theorem 2.5.1 is the main statement of this section. Theorem 2.5.1. — Let X be a tame formal scheme of finite type over R whose generic fiber has dimension d  0. Let A be a constructible subset of Gr(X). Then for every integer m, there exists a positive integer N , depending only on m and X but not on A, such that 

e(θn,X (A))L−(n+1)d − e(θn ,X (A))L−(n +1)d  * lies in F m+1 M X0 for all n, n  N . R with general term In particular, the sequence in M X0

e(θn,X (A)) · L−(n+1)d R . has a limit in M X0 Proof. — By corollary 2.3.4, we may assume that X is rig-irreducible and reduced. Then we may also assume that X is flat over R, since otherwise, Gr(X) is empty. Finally, we may assume that (Xsing )η has dimension at most n − 1, since otherwise, Gr(X) = Gr(Xsing ) is thin and the result follows from proposition 2.4.6 and its proof. Let m be an integer. Let γ be the Greenberg function associated with Xsing . By lemma 1.1.1, there exists an integer N  γ(m) such that, for every integer n  N , (2.5.1.1)     e θn,X (A ∩ Gr(γ(m)) (X)) = e θN,X (A ∩ Gr(γ(m)) (X)) · Ld(n−N ) .

Let n  N . We set B = θn,X (A)

θn,X (A ∩ Gr(γ(m)) (X)).

This is a constructible subset of Grn (X), contained in −1 θn,X (A ∩ θγ(m),X (Grγ(m) (Xsing )))

and satisfying

  e (θn,X (A)) − e θn,X (A ∩ Gr(γ(m)) (X)) = e (B) .

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317

From proposition 2.3.1, we deduce that e(B) · L−(n+1)d ∈ F m+1 MXR0 , since n  γ(m). Then it follows from (2.5.1.1) that   (2.5.1.2) e (θn,X (A)) · L−(n+1)d − e θN,X (A ∩ Gr(γ(m)) (X)) · L−(N +1)d R . The same holds when we replace n by n . Subalso belongs to F m+1 M X0 tracting these expressions for n and n , we find:  R . e (θn,X (A)) · L−(n+1)d − e (θn ,X (A)) · L−(n +1)d ∈ F m+1 M X0

This implies that the sequence (e (θn,X (A))) · L−(n+1)d satisfies the Cauchy R . criterion; thus it has a limit in M X0 Definition 2.5.2. — Let X be a tame formal R-scheme of finite type and let d be the dimension of its generic fiber. For every constructible subset A of Gr(X), we define the motivic volume of A by R . μX (A) = lim e(θn,X (A)) · L−(n+1)d ∈ M X0 n→+∞

R is called This limit exists by theorem 2.5.1. The map μX : ConsGr(X) → M X0 the measure on Gr(X). Remark 2.5.3. — This function is sometimes called the “motivic measure” in the literature. However, it should not be confused with the motivic measures in the sense of definition 2/2.1.1, which are ring morphisms with source K0 (VarX0 ). Example 2.5.4. — if A is disjoint from Gr(Xsing ),

By proposition 1.1.2, then the sequence e(θn (A)) · L−(n+1)d n∈N is stationary and has a limit in MXR0 (i.e., we do not need to complete the ring MXR0 in that case). The volume provided by definition 2.5.2 is then the image of that limit in the ring R by the completion morphism M R → M R . M X0 X0 X0 Theorem 2.5.5. — Let X be a tame formal R-scheme of finite type and let d  0 be the dimension of its generic fiber. R is additive, i.e., if A, B are two disa) The map μX : ConsGr(X) → M X0 joint constructible subsets of Gr(X), then μX (A ∪ B) = μX (A) + μX (B); b) If A1 , . . . , As are constructible subsets of Gr(X), we have μX (

s 

i=1

with AJ =

 j∈J

Ai ) =



(−1)Card(J)−1 μX (AJ ),

J=∅ J⊂{1,...,s}

Aj , for every nonempty subset J of {1, . . . , s};

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c) For all constructible subsets A, B of Gr(X), we have (2.5.5.1)

μX (A ∪ B) = max ( μX (A) , μX (B) ) .

In particular, if A ⊂ B, then μX (A)  μX (B) . d) Let (Xi )i∈I be the family of rig-irreducible components of X whose Greenberg schemes are fat in Gr(X). For every constructible subset A of Gr(X), we have  μX (A) = μXi (A ∩ Gr(Xi )) i∈I

R ; in particular, if I = ∅, then μX (A) = 0; in M X0 Proof. — a) For every sufficiently large integer n  0, the constructible subsets θn,X (A) and θn,X (B) are disjoint. The scissor relations in the Grothendieck ring then imply the following equality: e(θn,X (A)∪θn,X (B))·L−(n+1)d = e(θn,X (A))·L−(n+1)d +e(θn,X (B))·L−(n+1)d . By passing to the limit, we obtain the required relation. b) We only need to prove the case s = 2; the result then follows by induction. The subsets A2 and A1 A2 form a partition of A1 ∪ A2 . From a), we deduce that (2.5.5.2)

μX (A1 ∪ A2 ) = μX (A2 ) + μX (A1

Besides, the subsets A1 ∩ A2 and A1 deduce from a) that (2.5.5.3)

A2 ).

A2 form a partition of A1 . We also

μX (A1 ) = μX (A1 ∩ A2 ) + μX (A1

A2 ).

The property then follows from (2.5.5.2) and (2.5.5.3). c) The second assertion follows from the first one, so it suffices to prove (2.5.5.1). For every constructible subset C of Gr(X), the value θn,X (C)L−d(n+1) converges to μX (C) . Thus (2.5.5.1) follows from the fact that dim(S ∪ T /X0 ) = max{dim(S/X0 ), dim(T /X0 )} when S and T are constructible subsets of a X0 -variety. The second assertion follows from this. d) This assertion follows directly from corollary 2.3.4.

§ 3. MEASURABLE SUBSETS OF GREENBERG SCHEMES Let X be a tame formal R-scheme of finite type. The aim of this section is to extend the measure μX to a larger class of measurable subsets, which includes countable unions of constructible subsets whose volumes tend to zero or countable intersections of constructible subsets such as Gr(Z), with Z a closed formal subscheme of X.

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319

R 3.1. Summable Families in M X0 In this subsection, we review some elementary results of (non-Archimedean) analysis that we will use without specific mention in the following sections. The experienced reader can drop this section. R indexed (3.1.1). — Let I be a set and (ai )i∈I be a family of elements of M X0 by I. We write limi→∞ ai = 0 if, for every real number ε > 0, there exists a finite subset J of I such that ai < ε for every j ∈ I J. This condition is void when I is finite. When I is infinite, it means that ai converges to 0 with respect to the Fréchet filter on I (whose members are the complements of finite subsets of I). If limi→∞ ai = 0, then for every integer n, the set of indices i ∈ I such that ai > 1/n is finite, by assumption, so that the set of indices i ∈ I such that ai = 0 is countable. Consequently, it often suffices to consider families indexed by N, i.e., sequences. Definition 3.1.2. — Let I be a set. We say that a family (ai )i∈I of eleR indexed by I is summable if there exists an element a ∈ M R ments of M X0 X0 such that, for every real number ε such that ε > 0, there exists a finite subset J0 of I, such that, for every finite set J such that J0 ⊂ J ⊂ I, we have:  -a − aj - < ε. j∈J If the family (ai )i∈I is summable, then there exists exactly one  such element a; it is called the sum of the family (ai ) and denoted by i∈I ai . It follows from the ultrametric inequality that - ai -  sup ai . - i∈I i∈I

Every finite family (ai )i∈I is summable, and its sum is its sum in the usual sense. The sum (ai + bi )i∈I of two summable families (ai )i∈I and (bi )i∈I is summable, and one has    (ai + bi ) = ai + bi . i∈I

i∈I

i∈I

R )I . The Proposition 3.1.3. — Let I be an infinite set; let (ai )i∈I ∈ (M X0 following assertions are equivalent: (i) The family (ai )i∈I is summable; (i ) One has limi→∞ ai = 0;  (ii) There exists a bijection σ : N → I such that the series n∈N aσ(n) R  converges in MX0 ; (ii ) There exists a bijection σ : N → I such that limn→+∞ aσ(n) = 0 in R ; M X0

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(iii) For every bijection σ : N → I, the series  MXR0 ;

 n∈N

aσ(n) converges in

R . (iii ) For every bijection σ : N → I, we have limn→+∞ aσ(n) = 0 in M X0 If they hold, then for any bijection σ : N → I, the series with general term R to the sum of the family (ai )i∈I . aσ(n) converges in M X0 Proof. — We may assume that I = N. It is clear that (i ) is equivalent to the fact that an → 0 when n → +∞, that is, to (ii ). Since every bijection σ : N → N satisfies limn→+∞ σ(n) = +∞, we also have the implication (ii )⇒(iii ), and the implication (iii )⇒(ii ) is obvious. The implications (i)⇒(i ), (ii)⇒(ii ), and (iii)⇒(iii ) follow classically from the triangular inequality. Moreover, it follows from the definition of a summable family that (i)⇒(iii)⇒(ii). R implies that the ultrametric inequality in M If (ii ) holds, then the  X0 partial sums of the series an form a Cauchy sequence; since the normed  R is complete, the series ring M an converges, hence (ii). One proves X0 similarly that (iii )⇒(iii)  It remains to prove that (ii)⇒(i). To that aim, we let a = n∈N an and with sum a. Let ε > 0 and let prove that the family n ) is summable, - (a n N ∈ N such that -a − j=0 aj -  ε for every integer n  N . Writing n−1 n an = (a − j=0 aj ) − (a − j=0 aj ), we first observe that an  ε for every n > N . Let J0 = {0, . . . , N } and let J be any finite subset of N containing J0 . Then the relation a−



N  

aj = a − aj − aj

j∈J

j=0

j∈J J0

 and the ultrametric inequality imply that -a − j∈J aj -  ε.

3.2. Definition of Measurable Subsets In this paragraph, we generalize the notion of constructible subsets to a general setting suitable for applications. We introduce the notion of measurable subsets of Greenberg schemes and extend the measure μX to this bigger class of subsets. If A and B are subsets of a set X, we denote by A B their symmetric difference, defined by A

B = (A ∪ B)

(A ∩ B) = (A ∩ B) ∪ (A ∩ B).

Definition 3.2.1. — Let X be a tame formal scheme of finite type over Spf(R). Let A be a subset of Gr(X) and let ε > 0 be a real number. We say that a family of constructible subsets (A0 , (Ai )i∈I ) of Gr(X) is a constructible

§ 3. MEASURABLE SUBSETS OF GREENBERG SCHEMES

321

ε-approximation of A, if the two following conditions are satisfied:  Ai . (3.2.1.1) A A0 ⊂ i∈I

(3.2.1.2)

∀i ∈ I,

μX (Ai ) < ε.

The set A0 is called the principal part of the constructible ε-approximation (A0 , (Ai )i∈I ). Such a constructible ε-approximation is said to be strong if A0 ⊂ A. Remark 3.2.2. — In the literature, the corresponding terminology for constructible approximation is cylindrical approximation. Definition 3.2.3. — Let X be a tame formal scheme of finite type over Spf(R). One says that a subset A of Gr(X) is measurable (resp. strongly measurable) if, for every real number ε > 0, the set A admits a constructible ε-approximation (resp. whose principal part is contained in A). The set of all measurable subsets of Gr(X) is denoted by Cons∗Gr(X) . Remark 3.2.4. — We can refine definition 3.2.3 by specifying the cardinal of the index set in a constructible approximation. Let κ be a cardinal. We say that a measurable subset A of Gr(X) is κ-measurable if, for every real number ε > 0, there exists a constructible ε-approximation (A0 , (Ai )i∈I ) where I a set of cardinal < κ. Let us note that, in the literature, e.g., Sebag (2004a), only ℵ1 -measurable subsets are considered. Example 3.2.5. — Every constructible subset of Gr(X) is strongly measurable. Indeed, for every real number ε > 0, and every constructible subset A of Gr(X), the family (A, ∅) is a strong constructible ε-approximation of A. Example 3.2.6. — Let X be a tame formal scheme of finite type over Spf(R) and let d  0 be the dimension of its generic fiber. Let Z be a closed formal subscheme of X with generic fiber of dimension at most d − 1. Let ε > 0. We have the following relation:  −1 θn,X (Grn (Z)). Gr(Z) = n∈N

From corollary 2.3.2, we deduce that there exists an integer N such that, for every integer n  N , −1 μX (θn,X (Grn (Z))) < ε. In particular, the family of constructible subsets of Gr(X) −1 (∅, θN (ε),X (GrN (Z)))

is a strong constructible ε-approximation of Gr(Z). This shows that Gr(Z) is strongly measurable.

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Observe, however, that Gr(Z) is not constructible in general. Otherwise, by quasi-compactness of the constructible topology on Gr(X) (see theorem A/1.2.4) implies that there would exist an integer n such that Gr(Z) = −1 (Grn (Z)). However, that property does not hold in general: consider, for θn,X instance, the closed formal subscheme of the formal affine line defined by the datum of the ideal (X) in the ring of restricted power series R{X}. Example 3.2.7. — Let (An )n∈N be a sequence of constructible subsets of R . Let A = Gr(X) such that limn→+∞ μX (An ) = 0 in M X0 n∈N An . Let us show that A is strongly measurable. Let indeed ε > 0. Let N ∈ N be such that μX (An )  ε for n  N . Then ( n 0; for every integer i ∈ {1, . . . , n}, let (Ai,0 , (Ai,j )j∈Ji ) be a constructible ε-approximation of Ai . Let J = {(i, j) ; j ∈ Ji }. a) The family (A1,0 A2,0 , (Ai,j )(i,j)∈J ) is a constructible ε-approximation of A1 A2 . In particular, the family (Gr(X) A1,0 , (A1,j )j∈J1 ) is a constructible ε-approximation of Gr(X) A1 . n b) The family ( i=1 Ai,0 , (Ai,j )(i,j)∈J ) is a constructible ε-approximation n of i=1 Ai . n c) The family ( i=1 Ai,0 , (Ai,j )(i,j)∈J ) is a constructible ε-approximation n of i=1 Ai . d) Assume that A1 ⊂ A2 ⊂ . . . ⊂ An . Then for every integer s ∈ {1, . . . , n}, the family n  ( Ai,0 , (Ai,j )(i,j)∈J ) i=s

is a constructible ε-approximation of As . In particular, for every integer i ∈ {1, . . . , n}, there exists a constructible ε-approximation of Ai with principal part A˜i,0 , such that A˜1,0 ⊂ A˜2,0 ⊂ . . . ⊂ A˜n,0 . e) Assume that the Ai are mutually disjoint. Then for every integer s ∈ {1, . . . , n}, the family  Ai,0 , (Ai,j )(i,j)∈J ) (As,0 1in,i=s

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323

is a constructible ε-approximation of As . In particular, there exist constructible ε-approximations of A1 , . . . , An whose principal parts are mutually disjoint. Proof. — The proof of the lemma consists in straightforward set-theoretic calculations. The first assertion of a) follows from  (A1 A2 ) (A1,0 A2,0 ) ⊂ (A1 A1,0 ) ∪ (A2 A2,0 ) ⊂ Ai,j , 1i2 j∈Ji

and the second follows from that, using that (Gr(X), ∅) is a constructible ε-approximation of Gr(X). For b), we observe that n n n     Ai Ai,0 ⊂ Ai,j , i=1

i=1

i=1 j∈Ji

while the proof of c) follows from n n n     Ai Ai,0 ⊂ Ai,j . i=1

i=1

i=1 j∈Ji

Th proof of d) follows from c) and the remark that As = every s ∈ {1, . . . , n}. The proof of e) follows from a) by remarking that As = As for every s ∈ {1, . . . , n}.

n i=s

Ai , for

1i 0 and let (A0 , (Ai )i∈I ) and (A0 , (Ai )i∈I  ) be constructible ε-approximations of A. Then μX (A0 ) − μX (A0 ) < ε.

(3.3.1.1)

Proof. — We observe the following inclusion A0

A0 ⊂ (A

A0 ) ∪ (A

A0 ).

Indeed, if x ∈ A0 A0 but x ∈ A A0 , then x ∈ A, hence x ∈ A symmetry, A0 A0 ⊂ A A0 . Consequently,         (3.3.1.2) A0 A0 ⊂ Ai ∪ Ai . i∈I

A0 ; by

i∈I 

Since ConsGr(X) is a Boolean algebra, A0 A0 is constructible. By quasicompactness of the constructible topology (see theorem A/1.2.4), we conclude

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from (3.3.1.2) that there exist two finite sets J, J  , such that J ⊂ I and J  ⊂ I  , and:         (3.3.1.3) A0 A0 ⊂ Ai ∪ Ai . i∈J 

i∈J

By theorem 2.5.5 and by (3.3.1.3), we deduce the following inequality: μX (A0

(3.3.1.4)

A0 ) < ε.

On the other hand, one has μX (A0 ) − μX (A0 ) = μX (A0

A0 ) − μX (A0

A0 ),

so that μX (A0 ) − μX (A0 )  μX (A0

A0 ) < ε.

This concludes the proof. Theorem 3.3.2. — Let X be a tame formal scheme of finite type over Spf(R). For every measurable subset A of Gr(X) there exists a unique element R such that μ∗X (A) ∈ M X0 μ∗X (A) − μX (A0 ) < ε and every constructible ε-approximation of A with principal part A0 . The map μ∗X : Cons∗Gr(X) is additive and coincides with μX on ConsGr(X) Definition 3.3.3. — Let X be a tame formal scheme of finite type over Spf(R). Let A be a measurable subset of Gr(X). The element μ∗X (A) defined by theorem 3.3.2 is called the motivic volume of the measurable subset A. Proof. — Let A be a measurable subset of Gr(X). Let us first prove the uniqueness of the element μ∗X (A). Let μ(A) and μ (A) be two elements R such that μ(A) − μX (A0 ) < ε and μ (A) − μX (A0 ) < ε for every of M X0 constructible ε-approximation of A with principal part A0 . This implies μ(A) − μ (A)  max( μ(A) − μX (A0 ) , μX (A0 ) − μ (A) ) < ε, since the norm · is non-Archimedean. Passing to the limit ε → 0, it follows that μ(A) = μ (A). For every integer n > 0, let us choose a constructible (1/n)-approximation of A with principal part An . By proposition 3.3.1, one has μX (An ) − μX (Am ) < max(1/n, 1/m). Consequently, the sequence (μX (An )) satisfies the Cauchy criterion. Since R is complete, this implies that it converges. Let μ∗ (A) be its limit; it M X0 X satisfies μ∗X (A) − μX (An )  1/n for every integer n.

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325

Let us then consider a constructible ε-approximation of A with principal part A0 . For n large enough so that 1/n < ε, one has

μ∗X (A) − μX (A0 )  max μ∗X (A) − μX (An ) , μX (An ) − μX (A0 ) < ε. This proves that μ∗X (A) satisfies the condition of the theorem. By construction and example 3.2.5, the map μ∗X extends the map μX . Let us finally prove now that the map μ∗X is additive. Let A, B be two disjoint measurable subsets of Gr(X) and let C = A ∪ B; we have to prove that μ∗X (C) = μ∗X (A) + μ∗X (B). Let ε > 0. Let A0 and B0 be the principal parts of constructible ε-approximations of A and B, respectively, such that A0 ∩ B0 = ∅ (lemma 3.2.9). Then C0 = A0 ∪ B0 is the principal part of a constructible ε-approximation of A ∪ B (loc. cit.), and one has μX (C0 ) = μX (A0 ) + μX (B0 ), since μX is additive (theorem 2.5.5). Consequently, μ∗X (A) + μ∗X (B) − μX (C0 ) = (μ∗X (A) − μX (A0 )) + (μ∗X (B) − μX (B0 )) . By definition of μ∗X (A) and μ∗X (B), the two terms have norms < ε. Since R is Archimedean, this implies that M X0 μ∗X (A) + μ∗X (B) − μX (C0 ) < ε. By definition of μ∗X (C), this implies the desired equality. Corollary 3.3.4. — Let X be a tame formal scheme of finite type over Spf(R). Let A1 , . . . , As be measurable subsets of Gr(X). We have μ∗X (

s 

 j∈J

(−1)Card(J)−1 μ∗X (AJ ),

J=∅ J⊂{1,...,s}

i=1

with AJ =



Ai ) =

Aj for every finite set J ⊂ {1, . . . , s} and J = ∅.

Proof. — By induction on s, it suffices to prove the formula when s = 2. Let A, B be measurable subsets of Gr(X). Obviously, we have: A ∪ B = (A

A ∩ B)  (B

A ∩ B)  (A ∩ B),

A = (A

A ∩ B)  (A ∩ B),

B = (B

A ∩ B)  (A ∩ B).

By additivity of μ∗X (theorem 3.3.2), we thus have μ∗X (A ∪ B) = μ∗X (A

A ∩ B) + μ∗X (B

A ∩ B) + μ∗X (A ∩ B)

= μ∗X (A) − μ∗X (A ∩ B) + μ∗X (B) − μ∗X (A ∩ B) + μ∗X (A ∩ B) = μ∗X (A) + μ∗X (B) − μ∗X (A ∩ B), as claimed.

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Corollary 3.3.5. — Let A, B be measurable subsets of Gr(X). We have μ∗X (A ∪ B) = max ( μ∗X (A) , μ∗X (B) ) . In particular, if A ⊂ B then μ∗X (A)  μ∗X (B) . Proof. — First assume that A ⊂ B. Let ε > 0. By lemma 3.2.9, there exist a constructible ε-approximation of A with principal part A0 and a constructible ε-approximation of B with principal part B0 such that A0 ⊂ B0 . Consequently, μX (A0 )  μX (B0 ) . This inequality implies that μ∗X (A) < max( μ∗X (A0 ) , ε)  max( μ∗X (B0 ) , ε)  max( μ∗X (B) , ε), hence the inequality μ∗X (A)  μ∗X (B) when ε → 0. In general, since μ∗X is additive, we have μ∗X (A ∪ B) = μ∗X (A) + μ∗X (B) − μ∗X (A ∩ B). Consequently, μ∗X (A ∪ B)  max( μ∗X (A) , μ∗X (B) , μ∗X (A ∩ B) ). By the first assertion, this implies the desired statement. Remark 3.3.6. — Corollary 3.3.5 is a kind of “positivity” property for the measure μ∗X . In the same spirit, notice that motivic volumes are limits of elements of the form e(S/X0 )L−n , where S is an X0 -variety, i.e., are effective R . elements of M X0 3.4. Countable Additivity of the Measure μ∗X In this paragraph, we establish some infinite additivity properties of the measure μ∗X . Note that Cons∗Gr(X) fails to be a σ-algebra in general, see proposition 3.4.3 below, so that μ∗X has no chance to be σ-additive. Lemma 3.4.1. — Let X be a tame formal scheme of finite type over Spf(R). Let (Bi )i∈I be a family of measurable subsets of Gr(X) and let A = i∈I Bi . If A is measurable, then μ∗X (A) = sup μ∗X (Bi ) . i∈I

Proof. — For every i ∈ I, one has μ∗X (Bi )  μ∗X (A) because Bi ⊂ A (corollary 3.3.5). Consequently, sup μ∗X (Bi )  μ∗X (A) , i∈I

and it remains to prove the reverse inequality.

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327

Let ε be a positive real number. Let (A0 ; (Aj )j∈J ) be a constructible εapproximation of A; for every i ∈ I, let (Bi0 ; (Bij ) j∈Ji ) be a constructible ε-approximation of Bi . The inclusions A A0 ⊂ j∈J Aj and Bi Bi0 ⊂ j j∈Ji Bi imply that     j Bi0 ∪ Aj ∪ Bi . A0 ⊂ i∈I

j∈J

i∈I j∈Ji

By quasi-compactness of the constructible topology on Gr(X) (theorem A/1.2.4), there exist a finite subset I 0 of I, a finite subset J 0 of J, and, for every i ∈ I, a finite subset Ji0 of Ji such that     j A0 ⊂ Bi0 ∪ Aj ∪ Bi . i∈I 0

Consequently, -μX (A0 )- ⊂ sup



j∈J 0

i∈I 0 j∈Ji0

 j 0 j sup -μX (Bi )- , sup -μX (A )- , sup sup -μX (Bi )- .

i∈I 0

j∈J 0

i∈I 0 j∈Ji0

By definition of the measure of a measurable set, one has μ∗X (A)  sup(-μX (A0 )- , ε) and

-μX (Bi0 )-  sup( μ∗X (Bi ) , ε), for every i ∈ I. Consequently, μ∗X (A) ⊂ sup(sup μ∗X (Bi ) , ε), i∈I

and the desired inequality follows by letting ε → 0. Proposition 3.4.2. — Let X be a tame formal scheme of finite type over measurable) Spf(R). Let (Ai )i∈I be a family of measurable (resp. strongly subsets of Gr(X) such that limi→∞ μ∗X (Ai ) = 0. Let A = i∈I Ai . Then A is measurable (resp. strongly measurable) and (3.4.2.1)

μ∗X (A) = sup μ∗X (Ai ) . i∈I

Proof. — Let ε > 0. We have to find a constructible ε-approximation of A. For every i ∈ I, let (Ai,0 , (Ai,j )j∈Ij ) be a constructible ε-approximation of Ai . Let us fix i ∈ I. By theorem 3.3.2, we have: (3.4.2.2)

μX (Ai,0 ) − μ∗X (Ai ) < ε.

Since μ∗X (Ai ) → 0, there exists a finite set I0 ⊂ I such that for every i ∈ I I0 , (3.4.2.3)

μ∗X (Ai ) < ε

R is non-Archimedean, for every i ∈ I I0 , By (3.4.2.2) and (3.4.2.3), since M X0 we conclude that: (3.4.2.4)

μX (Ai,0 ) < ε.

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Let L = (I I0 ) ∪ ( i∈I ({i} × Ji )). Let us show that the following family (A0 , (A )∈L ) of subsets of Gr(X), defined by  A0 = Ai,0 i∈I0

 Ai,0 B = Ai,j

if = i ∈ I I0 , if = (i, j) with i ∈ I and j ∈ Ji

is a constructible ε-approximation of A. By construction, it is clear that μX (A ) < ε for every ∈ L. Let us then note that ⎞  ⎛     A ∩ A0 ⊂ ⎝ Ai,j ⎠ ∪ Ai ⎛

i∈I0 j∈Ji

 

⊂⎝



Ai,j ⎠ ∪ ⎝

i∈I0 j∈Ji





i∈I I0











Ai,j ⎠ ∪

i∈I I0 j∈Ji



 Ai,0

i∈I I0

A .

∈L

On the other hand, we have



A0 ∩ A ⊂ ⎝

 

⎞ Ai,j ⎠ ⊂

i∈I0 j∈Ji



A .

∈L

This shows that A A0 ⊂ ∈L A and concludes the proof that (A0 , (B )∈L ) is a constructible ε-approximation of A. Consequently, A is measurable. Let us moreover assume that Ai is strongly measurable, for every i ∈ I, and let us choose strong approximations, so that Ai,0 ⊂ Ai . Then A0 ⊂ A, hence (A0 , (B )∈L is a strong approximation of A. This implies that A is strongly measurable. The remaining assertion follows from lemma 3.4.1. Proposition 3.4.3. — Let X be a tame formal scheme of finite type over Spf(R) and let d  0 be the dimension of its generic fiber. Let (Ai )i∈I be a family of mutually disjoint measurable subsets of Gr(X). Then the following assertions are equivalent:  (i) The union Ai is measurable; i∈I

R ; (ii) The family (μ∗X (Ai ))i∈I is summable in M X0 ∗ (iii) One has lim μX (Ai ) = 0. i→+∞

Moreover, in this case, we have   μ∗X (Ai ). μ∗X ( Ai ) = i∈I

i∈I

Proof. — By proposition 3.1.3, the assertions (ii) and (iii) are equivalent. Moreover, proposition 3.4.3 establishes the implication (iii)⇒(i).

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329

Let us then assume that A = i∈I Ai is measurable and let us prove that the family (μ∗X (Ai )) is summable, with sum μ∗X (A). Let ε > 0. Fix a constructible ε-approximation (A0 , (Bj )j∈J ) of A, as well as, for every i ∈ I, a constructible ε-approximation (A0i , (Bi,j )j∈Ji ) of Ai . Applying lemma 3.2.9, we note that (A0i ∩ A0 , (Bi,j )j∈J ∪ (Bj )j∈J ) is a constructible ε-approximation of Ai whose principal part is contained in A0 . Consequently, we assume in what follows that A0i ⊂ A0 for every i ∈ I. We then observe that     A0 ⊂ A0i ∪ Bi,j ∪ Bj . i∈I

i∈I j∈Ji

j∈J

By quasi-compactness of the constructible topology, A0 is covered by a finite union of the constructible sets appearing in the right hand side. Using the fact that A0i ⊂ A0 for every i, we deduce that there exists a finite subset N0 of I such that, for every finite subset N of I containing N0 , there exists a constructible subset CN of Gr(X), disjoint from A0i for i ∈ N such that  A0 = A0i ∪ CN i∈N

and μX (CN ) < ε. By additivity of μX , we thus have  A0i ) < ε. (3.4.3.1) μX (A0 ) − μX ( i∈N



0

0 On the other hand, A and i∈N Ai are the principal parts of εapproximations of A and i∈N Ai , respectively, so that   (3.4.3.2) Ai ) − μX ( A0i ) < ε. μ∗X (A) − μX (A0 ) < ε, μ∗X ( i∈N

Consequently, we have (3.4.3.3)

μX (A) − μ∗X (



i∈N

Ai ) < ε.

i∈N

Since the Ai are mutually disjoint, the additivity of μ∗X implies that  (3.4.3.4) μX (A) − μ∗X (Ai ) < ε. i∈N

We thus have proved that the family (μ∗X (Ai ))i∈I is summable and that its sum equals μ∗X (A). That concludes the proof of the proposition. Corollary 3.4.4. — Let X be a tame formal scheme of finite type over Spf(R). Let (An )n∈N be an increasing sequence of measurable subsets of Gr(X). The following assertions are equivalent: (i) The union A = n∈N An is measurable; R ; (ii) The sequence (μ∗X (An )) converges in M X0 R . (iii) One has limn→+∞ μ∗ (An An−1 ) = 0 in M X

X0

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Moreover, in this case, we have  An ) = lim μ∗X (An ). μ∗X ( n∈N

n→+∞

Proof. — This follows from proposition 3.4.3 applied to the sequence (Bn ) defined by B0 = A0 and Bn = An An−1 for n  1. Corollary 3.4.5. — Let X be a tame formal scheme of finite type over Spf(R). Let (An )n∈N be a decreasing sequence of measurable subsets of Gr(X). The following assertions are equivalent:  (i) The intersection n∈N An is measurable; R ; (ii) The sequence limn→+∞ μ∗X (An ) converges in M X0 ∗ R  . (iii) One has limn→+∞ μX (An An+1 ) = 0 in M X0 Moreover, in this case, we have  An ) = lim μ∗X (An ). μ∗X ( n∈N

n→+∞

Proof. — This follows from the preceding corollary by taking complements.

3.5. Negligible Sets (3.5.1). — Let X be a tame formal scheme of finite type over Spf(R) and let d  0 be the dimension of its generic fiber. Let us denote by NGr(X) the set formed by the measurable subsets of Gr(X) whose volume is zero. An element of the set NGr(X) is called a negligible subset of Gr(X). Let us note that a negligible subset is in fact strongly measurable. We also say that a subset A of Gr(X) is strongly negligible if, for every real number ε > 0, there exists a constructible subset B of Gr(X) such that μX (B) < ε. Example 3.5.2. — Let X be a tame formal scheme of finite type over Spf(R) and let d  0 be the dimension of its generic fiber. Let Z be a closed formal subscheme of X whose generic fiber has dimension at most d − 1. By example 3.2.6, Gr(Z) is measurable, and the definition of the measure μ∗X implies that μ∗X (Gr(Z)) = 0. Consequently, Gr(Z) is negligible. It is in fact strongly negligible. Example 3.5.3. — Let X be a tame formal scheme of finite type over Spf(R). Let X be the maximal reduced closed formal subscheme of X and assume that Xη is equidimensional. Proposition 2.4.6 and the tameness assumption on X imply that the following properties of a constructible subset A of Gr(X) are equivalent: 1) It is contained in Gr(Xsing ); 2) it is thin; 3) it is negligible; and 4) it is strongly negligible. Lemma 3.5.4. — Let X be a tame formal scheme of finite type over Spf(R). Let A be a subset of Gr(X). The following assertions are equivalent: (i) The set A is negligible;

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331

(ii) For every real number ε > 0, the subset A admits a constructible εapproximation with principal part ∅; (iii) For every real number ε > 0, there exists a family (Ai )i∈I of constructible subsets of Gr(X) such that A ⊂ i∈I Ai and μX (Ai ) < ε for every i ∈ I. Proof. — The equivalence (ii)⇔(iii) and the implication (ii)⇒(i) follow from the definitions of a constructible approximation and the measure μ∗X . (i)⇒(iii). Let ε > 0. By the definition of a negligible subset, A is measurable; hence, there exists a constructible ε-approximation (A0 , (Ai )i∈I ). Moreover, the definition of the measure μ∗X implies that μX (A0 ) = μ∗X (A) − μX (A0 )  ε. The assertion thus follows from the inclusion A ⊂ A0 ∪ (A

A0 ) ⊂ A0 ∪



Ai .

i∈I

Corollary 3.5.5. — Let X be a tame formal scheme of finite type over Spf(R). a) Any subset of a negligible subset of Gr(X) is negligible. b) The union of a family of negligible subsets of Gr(X) is negligible.

3.6. C-Measurable Subsets of Gr(X) In this section, we describe a class of measurable subsets that admit a Carathéodory-type property and characterize the restriction of the motivic measure to this class. Definition 3.6.1. — Let X be a tame formal scheme of finite type over Spf(R) and let A be a subset of Gr(X). Let ε be a positive real number. A (strong) C-constructible ε-approximation of A is a (strong) constructible ε-approximation (A0 , (Ai )i∈I ) of A such that I is countable and limi→∞ μX (Ai ) = 0. We say that A is C-measurable (resp. strongly C-measurable) if, for every real number ε > 0, it admits a (resp. strong) C-constructible εapproximation. In particular, a C-measurable subset is measurable, and a strongly Cmeasurable subset is strongly measurable. However, we do not know whether the corresponding inclusion Cons∗Gr(X),C ⊂ Cons∗Gr(X) is strict. Proposition 3.6.2. — The set Cons∗Gr(X),C of all C-measurable subsets of Gr(X) is a Boolean algebra.

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Proof. — As for the proof of proposition 3.2.8, this follows by observing that the constructible approximations furnished by lemma 3.2.9 satisfy the required condition if the initial approximations satisfy them as well. Example 3.6.3. — Let X be a tame formal scheme of finite type over Spf(R) and let d  0 be the dimension of its generic fiber. Let Z be a closed formal subscheme of X whose generic fiber has dimension at most d − 1. Let us show that Gr(Z) is a strongly C-measurable subset of Gr(X). Let ε > 0. We have the following relation:  −1 θn,X (Grn (Z)). Gr(Z) = n∈N

From corollary 2.3.2, we deduce that there exists an integer N such that, for every integer n  N , −1 (Grn (Z)) < ε. μX (θn,X −1 Then the pair (∅; θn,X (Grn (Z)) is a strong C-constructible ε-approximation of Gr(Z).

Example 3.6.4. — Let X be a tame formal scheme of finite type over Spf(R). Let (An )n∈N be a sequence of constructible subsets of Gr(X) such R that lim n→+∞ μX (An ) = 0 in MX0 . We have seen in example 3.2.7 that A = n∈N An is measurable. In fact, the constructible ε-approximation exhibited there is a strong C-constructible ε-approximation, thus implying that A is strongly C-measurable. Theorem 3.6.5. — Let X be a tame formal scheme of finite type over Spf(R). There exists a unique additive map R μ∗C,X : Cons∗Gr(X),C → M X0 R and satisfies the two following properwhich extends μX : ConsGr(X) → M X0 ties: a) For every countable family (Ai )i∈I of constructible subsets of Gr(X) R ; such that limi→∞ μX (Ai ) = 0, then μ∗X,C ( i∈I Ai ) = μ∗X ( i∈I Ai ) in M X0 ∗ ∗ ∗ b) For every A, B ∈ ConsGr(X),C , if A ⊂ B, then μX,C (A)  μX,C (B) . Besides, the map μ∗X,C coincides with the restriction of μ∗X to Cons∗Gr(X),C . Proof. — By theorem 3.3.2, the restriction of μ∗X is additive and extends μX ; by corollary 3.3.4 and proposition 3.4.3, it satisfies the required properties. Let us denote by μ∗X,C the restriction of the measure μ∗X to Cons∗Gr(X),C . Let us prove that this restriction is characterized by these properties. Let R such a map. Let A ∈ Cons∗ μ : Cons∗Gr(X),C → M Gr(X),C . Let ε > 0, and X0 let (A0 , (Ai )i∈I ) be a C-constructible ε-approximation of A. Let us first prove that (3.6.5.1)

μ(A

A0 )  ε.

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333

Indeed, for every i ∈ I, we have μ(Ai ) = μX (Ai )  ε. Because of the assumptions on the map μ, the relation  A A0 ⊂ Ai i∈I

implies that (3.6.5.2)

A0 )  μ(

μ(A



Ai ) = μ∗X (

i∈I



Ai ) .

i∈I

Moreover, (3.6.5.3)

μ∗X (



Ai )  sup μ∗X (Ai ) < ε. i∈I

i∈I

This proves (3.6.5.1). By additivity of μ, we then have: (3.6.5.4)

μ(A) = μ(A

(A ∩ A0 )) + μ(A ∩ A0 ),

and (3.6.5.5)

μ(A0 ) = μ(A0

(A ∩ A0 )) + μ(A ∩ A0 ),

Since A0 (A∩A0 ), A (A∩A0 ) ⊂ A A0 , we deduce from (3.6.5.1), (3.6.5.4), and (3.6.5.5), thanks to the assumptions on μ that: (3.6.5.6)

μ(A) − μ(A ∩ A0 )  ε,

and (3.6.5.7)

μX (A0 ) − μ(A ∩ A0 )  ε.

In the end, the triangular inequality gives by (3.6.5.6) and (3.6.5.7) that: (3.6.5.8)

μ(A) − μX (A0 )  ε.

By theorem 3.3.2, this concludes the proof of the theorem.

§ 4. MOTIVIC INTEGRALS In this section, we define the motivic integrals and establish the change of variables formula for motivic integrals, which is a crucial result for applications.

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4.1. Integrable Functions (4.1.1). — We let Z = Z ∪ {+∞}, endowed with the natural extension of addition for which m + ∞ = +∞ + m = +∞ for every m ∈ Z. Definition 4.1.2. — Let X be a tame formal scheme of finite type over Spf(R), let A be a measurable subset of Gr(X), and let f : A → Z be a map. We say that f is measurable if each finite fiber of f is a measurable subset of Gr(X). R , then If, moreover, the family (L−n · μ∗X (f −1 (n)))n∈Z is summable in M X0 −f is integrable. In that case, we denote by we say that L   (4.1.2.1) L−f dμ∗X := μ∗X (f −1 (n))L−n A

n∈Z

R and call it the motivic integral of L−f . the sum of this family in M X0 Example 4.1.3. — Let f : Gr(X) → Z be a constructible function, i.e., a map with constructible fibers. Then f is measurable. Moreover, it follows from the quasi-compactness of Gr(X) (theorem A/1.2.4) that f takes only finitely many values. Consequently, L−f is integrable. Proposition 4.1.4. — Let X be a tame formal scheme of finite type over Spf(R); let A and B be measurable subsets of Gr(X). a) Assume that B ⊂ A and let f : A → Z be a measurable function on A. Then f |B is measurable. If L−f is integrable on A, then L−f |B is integrable on B. b) Let f : A ∪ B → Z be a function on A ∪ B. Then f is measurable if and only if f |A and f |B are measurable, and L−f is integrable if and only if L−f |A and L−f |B are integrable. In this case, we have:     −f ∗ −f |A ∗ −f |B ∗ L dμX = L dμX + L dμX − L−f |A∩B dμ∗X . A∪B

A

B

A∩B

c) Assume that A is negligible and let f : A → Z be a function on A. Then L−f is integrable on A and  L−f dμ∗X = 0. A

Proof. — a) The measurability of f|B follows from the definitions, because Cons∗Gr(X) is stable by intersection. Since B ⊂ A, we conclude from corollary 3.3.4 that L−n μ∗X (f −1 (n) ∩ B)  L−n μ∗X (f −1 (n)) for every n ∈ Z. Since L−f is integrable, this implies the result. b) It follows from a) that if f is measurable, then f |A and f |B are measurable, and if L−f is integrable, then L−f |A and L−f |B are integrable. Conversely, if f |A and f |B are measurable, then f is measurable, because

§ 4. MOTIVIC INTEGRALS

335

Cons∗Gr(X) is stable by union. It then follows from the additivity of μ∗X (corollary 3.3.4) that −1 −1 ∗ ∗ μ∗X (f −1 (n)) = μ∗X (f |−1 A (n)) + μX (f |B (n)) − μX (f |A∩B (n)).

Assertion b) then follows from the basic properties of summable families. c) Let us finally assume that A is negligible. Then, every subset of A is negligible, in particular measurable, so that f is measurable; moreover, for every n ∈ Z, μ∗X (f −1 (n)) = 0, so that A L−f dμ∗X = 0. That concludes the proof. Remark 4.1.5. — Let A be a measurable subset of Gr(X) and let f : A → Z be a measurable function. Assume that L−f is integrable on A. For n ∈ Z, let An = f −1 (n).  Let ε > 0. By definition, A L−f dμ∗X is the sum of the summable fam−n ily (L−n μ∗X (An ))n∈Z , so that L μ∗X (An ) converges to 0 when n → ±∞. Consequently, there exists a (nonempty) finite subset I of Z such that L−n μ∗X (Aj ) < ε for n ∈ Z I, and -  −f ∗ −n ∗ L μX (An )- < ε. - L dμX − - A n∈I

By the definition of the motivic volume (see remark 3.3.6), for every n ∈ I, there exist an X0 -variety and an integer mn such that - −n ∗ -L μX (An ) − L−mn e(Cn /X0 )- < ε. Let m = supn∈I mn and let C be the disjoint union of the X0 -varieties Lm−mn Cn , for n ∈ I; one then has - - L−f dμ∗X − L−m e(C/X0 )- < ε. 

A

R . This shows that A L−f dμ∗X is a limit of effective elements of M X0 Moreover, one has dim(C/X0 ) = supn∈I (m − mn ) + dim(Cn /X0 ). Passing to the limit ε → 0, this implies that - - L−f dμ∗X - = sup L −n μ∗X (An ) . (4.1.5.1) n∈Z

A

When f = 0, one recovers lemma 3.4.1. Let g : A → Z be another measurable function such that L−g is integrable on A as well. It then follows from equation (4.1.5.1) that - − sup(g−f ) - L−f dμ∗X (4.1.5.2) L -A - − inf(g−f ) −g ∗−f ∗ L L dμ L dμ XX- . A

A

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CHAPTER 6. MOTIVIC INTEGRATION

In particular, one has (4.1.5.3)

− sup(f )

L

- − inf(f ) −f ∗ μ∗X (A)  L dμ μ∗X (A) . X -  L A

4.2. Direct and Inverse Images of Measurable Subsets Definition 4.2.1. — Let X, Y be formal schemes of finite type over Spf(R). Let f : Y → X be a morphism of formal schemes over Spf(R). We denote by X the maximal reduced closed formal subscheme of X. Let us define the wild locus of Gr(f ) to be (4.2.1.1)

Σf := Gr(f )−1 (Gr(Xsing )) ∪ (ordjacf )−1 (∞)

It is a closed subset of Gr(Y), countable intersection of constructible closed subsets. Its complement Gr(Y) Σf is called the tame locus of Gr(f ). (4.2.2). — Assume that Y is tame. We say that f is tame if Σf is a negligible subset of Gr(Y). Otherwise, we say that f is wild. More generally, if A is a subset of Gr(Y), we say that f is tame on A if the subset A ∩ Σf of Gr(Y) is strongly negligible. Remark 4.2.3. — If f is tame, then Σf is strongly negligible. Let indeed subsets of Gr(f ) such ε > 0 and let (Bi )i∈I be a family of constructible that μY (Bi ) < ε for every i and Σf ⊂ i∈I Bi . Since Σf is a countable intersection of constructible closed subsets, the quasi-compactness of the constructible topology implies that there exists a finite subset J of I such that Σf ⊂ i∈J Bi , hence the claim since the latter set is constructible. Example 4.2.4. — Let X be a smooth formal R-scheme of finite type. Let f : Y → X be an admissible blowing-up. Then the morphism f is tame. More generally, if the induced morphism fη : Yη → Xη is an analytic domain immersion, then f is tame. Proposition 4.2.5. — Let X, Y be tame formal R-schemes of finite type whose generic fibers have pure dimension d  0, let f : Y → X be a morphism of formal R-schemes and let B be a constructible subset of Gr(Y). Let us assume, either that Y is smooth or that k is perfect and Yη is quasi-smooth. a) If B ∩ Σf = ∅, then Gr(f )(B) is a constructible subset of Gr(X). b) In any case, there exist a real number α, independent of B, and a constructible subset A of Gr(X) such that Gr(f )(B) ⊂ A and μX (A)  α μY (B) . When Y is smooth, one can even take α = 1. c) If f is Gr-injective on B, then  μX (Gr(f )(B)) = L− ordjacf dμ∗Y . B

We note that assertion c) is a particular case of the general change of variables formula that we will prove below (theorem 4.3.1).

§ 4. MOTIVIC INTEGRALS

337

Proof. — We first prove the proposition under the assumption that Y is smooth. In that case, Y is reduced and flat, and the morphism f factors through the maximal reduced and flat closed subscheme of X. By lemma 2.2.2, we may assume that X is reduced and flat. First assume that μY (B) = 0. Since B is constructible and Y is smooth, proposition 2.4.6 implies that B = ∅. The lemma obviously holds in this case, with A = ∅ in c). For the rest of the proof, we assume that μY (B) = 0. Let us first treat the particular case where there exists an integer e ∈ N such that B ⊂ Gr(f )−1 (Gr(e) (X)). Since B is constructible, there exists an −1 (Bq ). integer q and a constructible subset Bq of Grq (Y) such that B = θq,Y Let then c be the Elkik-Jacobi constant of X and let n be an integer such that n  max(q, ce); set (4.2.5.1)

−1 A = θn,X (θn,X (Gr(f )(B))).

Obviously, one has Gr(f )(B) ⊂ A. By theorem 5/2.3.11, one has −1 A = θn,X (Grn (f )(θn,Y (B))).

Observe that θn,Y (B) is a constructible subset of Grn (Y) (corollary 5/1.5.7). By Chevalley’s theorem (theorem A/1.2.4), Grn (f )(θn,Y (B)) is a constructible subset of Grn (X); hence, A is a constructible subset of Gr(X). By lemma 1.1.1, one has (4.2.5.2)

μX (A) = e(Grn (f )(θn,Y (B)))L−(n+1)d .

On the other hand, the smoothness of Y implies that (4.2.5.3)

μX (B) = e(θn,Y (B))L−(n+1)d .

Since dim(Grn (f )(θn,Y (B)))  dim(θn,Y (B)), we conclude that μX (A)  μY (B) . To treat the general case, we recall that the sequence (Gr(X) Gr(e) (Xsing ))e∈N of constructible subsets of Gr(X) is decreasing and its intersection is equal to Gr(Xsing ). Since X is tame, then Gr(Xsing ) is negligible and μ∗X (Gr(Xsing )) = 0. Consequently, for all integers e large enough, one has -μX (Gr(X) Gr(e) (X))- < μY (B) ; let then B  = B ∩ Gr(f )−1 (Gr(e) (X)). If, moreover, B ∩ Σf = ∅, it follows from the quasi-compactness of the constructible topology that B ⊂ Gr(f )−1 (Gr(e) (X)) for all integers e large enough; we may thus assume that B  = B in that case. By what precedes, A = Gr(f )(B  ) is a constructible subset of Gr(X) and μX (A )  μY (B  )  μY (B) . Let A = Gr(X) Gr(e) (X) and let A = A ∪ A . By construction, A is a constructible subset of Gr(X)

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containing Gr(f )(B) and μX (A)  μY (B) . When B ∩ Σf = ∅, we have Gr(f )(B) = A . This concludes the proof of a) and b). Let us now deduce c) from theorem 5/3.2.2. Let A = Gr(f )(B). Since B ∩ Σf = ∅, it follows from the quasi-compactness of the constructible topol ogy that there exists e ∈ N such that A ⊂ Gr(e ) (X), and the function ordjacf takes only finitely many values on B; by additivity, we may thus assume that ordjacf is constant on B, with value e ∈ N. As above, let n  max(2e, e , ce) be a large enough integer; we thus have μY (B) = e(θn,Y (B))L−(n+1)d and μX (A) = e(θn,X (A))L−(n+1)d . Let x ∈ θn,X (A); theorem 5/3.2.2 describes the fibers Bx = θn,Y (B) ∩ Grn (f )−1 (x). When R has equal characteristic, one has (Bx )red  Aeκ(x) for every such x, where κ(x) is its residue field. Consequently, e(θn,Y (B)) = e(θn,X (A))Le in MX0 . When R has mixed characteristic, theorem 5/3.2.2 asserts that (Bx ⊗ F )red  AeF for every perfect extension F of κ(x). Consequently, the equality e(θn,Y (B)) = e(θn,X (A))Le holds in in MXR0 in both cases. This implies that μY (B) = e(θn,Y (B))L−(n+1)d = e(θn,X (A))Le L−(n+1)d = Le μX (A), as was to be shown. This concludes the proof of the proposition in the case where Y is smooth. Let us now prove the second case, assuming that k is perfect and Y is quasi-smooth. Let h : Z → Y be a weak Néron model of Y (theorem 4/3.4.5), obtained by the composition of admissible blow-ups Z → Z and of the immersion of the smooth locus Z = Sm(Z ) → Z . Since k is perfect, the morphism h is Gr-bijective (proposition 4/3.5.1). Moreover, the function ordjach sup(ordjach ) is bounded on Gr(Z); let α = L . Let C = h−1 (B); this is a constructible subset of Gr(Z); since h is Gr-bijective, one has B = Gr(h)(C). Applying c) of the first case to C and h, we thus have  μZ (B) = L− ordjach dμ∗Z . C

By remark 4.1.5, b), it follows that μZ (C)  α μY (B) . Let us now apply the first case of the proposition to C and the composition: h

f

g: Z − →Y− → X, observing that Σg = h−1 (Σf ) and ordjach = ordjacf ◦h+ordjacg . If B∩Σf = ∅, then C ∩ Σg = ∅, so that A = Gr(f )(B) = Gr(h)(C) is a constructible subset of Gr(X). In general, there exists a constructible subset A of Gr(X) such that μX (A )  μZ (C)  α μY (B) . Assume finally that f is Gr-injective on B and let us prove that μX (A) = B L− ordjacf dμ∗Y . By quasicompactness of the constructible topology, we may assume that ordjacg and ordjach are constant on C. Then g is Gr-injective on C; hence, μX (A) = L− ordjacg μZ (C) = L− ordjach L− ordjacf μY (B) = L− ordjacg μY (B), as was to be shown.

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339

Corollary 4.2.6. — Let X, Y be tame formal R-schemes of finite type and assume that their generic fibers have pure dimension d  0; let f : Y → X be a morphism of formal R-schemes. Let us moreover assume that Y is smooth or that Yη is quasi-smooth and k is perfect. Let B be a negligible (resp. strongly negligible) subset of Gr(Y). Then Gr(f )(B) is a negligible (resp. strongly negligible) subset of Gr(X). Proof. — Let α be as in proposition 4.2.5. Let ε > 0 and let (∅; (Bi )i∈I ) be a constructible ε/α-approximation of B. By proposition 4.2.5, there exists, for every i ∈ I, a constructible subset Ai of Gr(X) such that Gr(f )(Bi ) ⊂ Ai and μX (Ai )  α μY (Bi ) . Consequently, (∅; (Ai )i∈I ) is a constructible ε-approximation of Gr(f )(B). This proves that Gr(f )(B) is negligible, as claimed. Let us now assume that B is strongly negligible. Let ε > 0 and let B  be a constructible subset of Gr(Y) such that B ⊂ B  and μY (B  ) < ε/α. By proposition 4.2.5, there exists a constructible subset A of Gr(X) such that Gr(f )(B  ) ⊂ A and μX (A )  α μY (B  ) < ε. Since Gr(f )(B) ⊂ A , this proves that Gr(f )(B) is strongly negligible, as was to be shown. Corollary 4.2.7. — Let X, Y be tame formal R-schemes of finite type and assume that their generic fibers have pure dimension d  0; let f : Y → X be a morphism of formal R-schemes. Let us assume that Y is smooth or that Yη is quasi-smooth and k is perfect. Let α be as in proposition 4.2.5. Let B be a measurable (resp. strongly measurable) subset of Gr(Y). If f is tame on B, then Gr(f )(B) is a measurable (resp. strongly measurable) subset of Gr(X); moreover, μ∗X (Gr(f )(B))  α -μ∗Y (B)-. Proof. — Let ε > 0. By lemma 3.2.9, there exists a constructible ε/αapproximation (B0 , (Bi )i∈I ) of B. If, moreover, B is strongly measurable, then we can insure that B0 ⊂ B. By the definition of f being tame on B, the intersection B ∩ Σf is a strongly negligible subset of Gr(Y); by corollary 4.2.6, the set Gr(f )(B∩Σf ) is therefore a strongly negligible subset of Gr(X). Let then B  be a constructible subset of Gr(Y) containing B ∩ Σf such that μY (B  ) < ε/α. Replacing B0 by B0 B  and adding B  to the family (Bi )i∈I , we are thus reduced to the case where B0 ∩ Σf = ∅. By proposition 4.2.5, A0 = Gr(f )(B0 ) is a constructible subset of Gr(X), and μX (A0 )  α μY (B0 ) . Furthermore, we have  Gr(f )(Bi ) (4.2.7.1) Gr(f )(B) Gr(f )(B0 ) ⊂ Gr(f )(B B0 ) ⊂ i∈I

For each i ∈ I, proposition 4.2.5 asserts the existence of a constructible subset Ai of Gr(X) such that Gr(f )(Bi ) ⊂ Ai and μX (Ai )  α μY (Bi )  ε. We observe that (A0 , (Ai )i∈I ) is a constructible ε-approximation of Gr(f )(B). This proves that Gr(f )(B) is a measurable subset of Gr(X) and that μX (Gr(f )(B))  α max( μY (A0 ) , ε)  α max(-μ∗Y (A)- , ε).

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If B is a strongly measurable subset of Gr(Y), we have chosen B0 so that B0 ⊂ B. In particular, A0 ⊂ Gr(f )(B). This implies that Gr(f )(B) is strongly measurable, as claimed. Proposition 4.2.8. — Let X, Y be tame formal R-schemes of finite type, and assume that their generic fibers have pure dimension d  0; assume either that Y is smooth or that k is perfect and Yη is quasi-smooth. Let f : Y → X be a Gr-injective tame morphism of formal R-schemes. Let A be a measurable (resp. strongly measurable, resp. negligible) subset of Gr(X). Then Gr(f )−1 (A) is a measurable (resp. strongly measurable, resp. negligible) subset of Gr(Y). Proof. — By lemma 2.2.2, we may assume that the formal scheme X is reduced and flat. For every integer m, let Cm = f −1 (Gr(m) (X)) ∩ ordjac−1 f ([0; m]); this is a closed constructible subset of Gr(Y). m Let α = L . Let ε > 0 and let (A0 ; (Ai )i∈I ) be a constructible ε/αapproximation of A; we moreover assume that A0 ⊂ A if A is strongly measurable and that A0 = ∅ if A is negligible. For every i ∈ {0} ∪ I, let Bi = Cm ∩ Gr(f )−1 (Ai ) and let Ai = Gr(f )(Bi ), so that  (Gr(f )−1 (Ai ) ∩ Cm ). (Gr(f )−1 (A) ∩ Cm ) B0 ⊂ i∈I

By theorem A/1.2.4, Bi is a constructible subset of Gr(Y), contained in Cm . By proposition 4.2.5, Ai is a constructible subset of Ai and  m  μX (Ai ) = L− ordjacf dμ∗Y = L−n μY (Bi,n ), Bi

where Bi,n = Bi ∩

n=0

ordjac−1 f (n). −m

μY (Bi )  L

For i ∈ I, this implies μX (Ai )  α μX (Ai )  ε,

so that (B0 , (Bi )) is a constructible ε-approximation of Gr(f )−1 (A) ∩ Cm ; in the case A is strongly measurable, we moreover have B0 ⊂ Gr(f )−1 (A) ∩ Cm , since A0 ⊂ A; in the case A is negligible, one has B0 = ∅. We thus have shown that Gr(f )−1 (A) ∩ Cm is measurable (resp. strongly measurable, resp. negligible). By definition of Σf , one has m∈N Cm = Gr(Y) Σf . By assumption, Gr(f ) is tame; hence, Σf is negligible. By corollary 3.4.4, it follows that Gr(f )−1 (A) is measurable (resp. strongly measurable, resp. negligible). This concludes the proof of the proposition. 4.3. The Change of Variables Formula This is certainly the most important result in the theory of motivic integration. We shall see in the next chapter a sample of its remarkable applications.

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341

Theorem 4.3.1. — Let X, Y be tame formal R-schemes of finite type, whose generic fibers have pure dimension d  0. Assume that Y is smooth or that k is perfect and that Yη is quasi-smooth. Let f : Y → X be a tame morphism of formal R-schemes. Let A, B be measurable subsets of Gr(X) and Gr(Y), respectively, such that A = Gr(f )(B). We assume that f is Gr-injective on B. Let α : A → Z be a function and define β : B → Z by β = α ◦ Gr(f ) + ordjacf . a) The function α is measurable on A if and only if the function β is measurable on B. b) The function L−α is integrable on A if and only if the function L−β is integrable on B. If this holds, then the transformation rule for motivic R : integral holds, ı.e., we have the following equality in M X0  

L−α dμ∗X = (f0 )! L−β dμ∗Y . A

B

Proof. — By lemma 2.2.2, we may assume that the formal scheme X is reduced and flat. a) For every m ∈ Z, let Am = α−1 (m) and Bm = β −1 (m); for every m ∈ Z and every n ∈ N, let Bm,n = β −1 (m) ∩ ordjac−1 f (n). We thus have Bm,n = Gr(f )−1 (Am−n ) ∩ ordjac−1 f (n)  Am = Gr(f )(Bm+n,n ). ∗

Let A =

m∈Z

n∈N



Am and B = A

B



m,n

Bm,n , so that



A = α−1 (∞)

B ∗ = β −1 (∞) = Gr(f )−1 (α−1 (∞)) ∪ (ordjac−1 f (∞) ∩ B).

Let us first assume that α is measurable, so that Am is measurable for every m ∈ Z, and let us prove that Bm is measurable. By proposition 4.2.8, Bm,n is measurable, for every pair (m, n). Since f is tame, Σf is negligible; −1 hence, ordjac−1 f (∞) is negligible as well, so that μY (ordjacf (n)) converges to 0 when n → +∞, by corollary 3.4.4. Moreover, Bm is the disjoint union of Bm ∩ Σf and of the family (Bm,n )n∈N . It then follows from corollary 3.4.4 that Bm is measurable. Conversely, let us assume that β is measurable and let us prove that α is measurable. For every pair (m, n), Bm,n = Bm ∩ ordjac−1 f (n) is then measurable. Since Bm is the disjoint union of the negligible set Bm ∩ ordjac−1 f (∞) and of the family (Bm,n ) of measurable sets, it follows as above from corollary 3.4.4 that μ∗Y (Bm,n ) → 0 when n → ∞. Since f is Gr-injective, we see that Am is the disjoint union of a negligible set and of the family (Gr(f )(Bm,n )). By corollary 4.2.7, Gr(f )(Bm,n ) is measurable, for every n ∈ N, and μ∗X (Bm,n ) → 0 when n → +∞. Consequently, Am is measurable.

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b) Let us now prove that L−α is integrable on A if and only if L−β is integrable on B and that in this case, their integrals agree. By definition, L−α is integrable  on A if and only if the family −m ∗ (L μX (Am ))m is summable, and A L−α is then the sum of this family. For every pair (m, n), let Am−n,n = Gr(f )(Bm,n ), so that Am is the disjoint union of the family (Am−n,n )n∈N and of a negligible set. Consequently, L−α is integrable A if and only if the family (L−m+n μ∗X (Am−n,n ))m,n is  on−α is then the sum of this family. summable, and A L Similarly, L−β is integrable on B if and only if the family (L−m μ∗Y (Bm ))m  is summable, and B L−β is the sum of this family. Since Bm is the disjoint union of the family (Bm,n )n (whose volumes converge to 0) and of a negligible −β is integrable on B if and only if the set Bm ∩ ordjac−1 f (∞), we see that L −m ∗ family (L μY (Bm,n ))m,n is summable, and then   L−β = L−m μ∗Y (Bm,n ). B

m,n

It thus suffices to prove the following formula L−n μ∗Y (Bm,n ) = μ∗X (Am−n,n ). In other words, we are reduced to the case where B is a measurable subset of ordjac−1 f (n), for some n ∈ N. Since X and f are tame, one has μ∗X (A) = lim μ∗X (A ∩ Gr(e) (X)) e→∞

and μ∗Y (B) = lim μ∗Y (B ∩ Gr(f )−1 (Gr(e) (X))). e→∞

We may thus assume that A ⊂ Gr(e) (X). −n Let ε > 0 and let (B0 ; (Bi )i∈I ) be a constructible ε L -approximation (e) −1 of B. Since Gr (X) ∩ ordjacf (n) is constructible, we may assume that Bi ⊂ Gr(e) (X) ∩ ordjac−1 f (n) for every i ∈ {0} ∪ I, by lemma 3.2.9. For every i ∈ {0} ∪ I, let us then set Ai = Gr(f )(Bi ). By proposition 4.2.5, subset of A and μX (Ai ) = L−n μX (Bi ). Moreover, Ai is a constructible A A0 ⊂ i∈I Ai . This shows that the family (A0 ; (Ai )i∈I ) is a constructible ε-approximation of A and that μ∗X (A) − μX (A0 ) < ε. Consequently, - ∗ -μX (A) − L−n μ∗X (B)- < ε, hence the theorem by letting ε → 0.

4.4. An Example: The Blow-Up In this subsection, we treat an example of computation with the change of variables formula.

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343

(4.4.1). — Let X be a connected smooth k-variety of dimension d; let C be a connected smooth closed subvariety of X of codimension c  2. Let f : Y → X be the blowing-up of X along C and let E be its exceptional divisor. By example 2/2.4.3, we know that (4.4.1.1)

e(X) = e(Y ) − e(E) + e(C)

in K0 (Vark ). Moreover, we know that f |E : E → C is a locally trivial projective bundle of dimension c − 1, so that e(E) = e(Pc−1 k ) e(C).

(4.4.1.2)

(4.4.2). — We will now show how the change of variables formula (theorem 4.3.1) allows to recover this formula, or at least a slight weakening of it ˆ k R; recall that Gr(X) = L (X) is the arc (see remark 4.4.7). We let X = X ⊗ space of X. We will write μ∗X for the measure on L (X). Since R has equal X . characteristic and X0 = X, it takes its values in the ring M We define similarly Y, E, and C and write ϕ : Y → X for the morphism of formal schemes induced by f : Y → X. Since C is a smooth closed subvariety of X, we observe that Y and E are smooth as well, so that these formal schemes are smooth. (4.4.3). — Since X is smooth over k, it follows directly from the definition of the motivic measure that (4.4.3.1)

μX (L (X)) = L−d e(X)

X . in the ring M (4.4.4). — On the other hand, the morphism f : Y → X being proper, and an isomorphism above X C, the morphism ϕ induces a Gr-bijective map L (Y )

L (E) → L (X)

L (C).

Since E and C are strict closed subvarieties of the smooth varieties Y and X, their arc schemes L (E) and L (C) are negligible in L (Y ) and L (X), respectively. Applying theorem 4.3.1, we get  μ∗X (L (X)) = L−0 dμ∗X L (X)  = L− ordjacϕ dμ∗Y L (Y )

=



L−n μ∗Y (ordjac−1 ϕ (n)).

n∈N

Since the Jacobian ideal of f is the (c − 1)-power of the ideal of E, one has ordjacϕ = (c − 1) ordI , where I is the ideal of E in Y. Consequently,  (4.4.4.1) μ∗X (L (X)) = L−n(c−1) μ∗Y (ord−1 I (n)). n∈N

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(4.4.5). — For n = 0, one has ord−1 I (0) = L (Y )

−1 θ0,Y (L0 (E)),

so that ∗ μ∗Y (ord−1 I (0)) = μY (L (Y ))

−1 μ∗Y (θ0,Y (L0 (E)))

= L−d e(Y ) − L−d e(E), by definition of the motivic measure of a constructible subset of the smooth formal scheme Y. Similarly, for every integer n  1, one has −1 −1 ord−1 I (n) = θn−1,Y (Ln−1 (E)) ∩ L (Y ) θn,Y (Ln (E));

hence, −1 −1 ∗ ∗ μ∗Y (ord−1 I (n)) = μY (θn−1,Y (Ln−1 (E))) − μY (θn,Y (Ln (E)))

= L−dn e(Ln−1 (E)) − L−d(n+1) e(Ln (E)). Since E is a smooth R-formal scheme of relative dimension d−1, the morphism Ln (E) → E is a locally trivial fibration for the Zariski topology, with fiber n(d−1) Ak . Consequently, e(Ln (E)) = Ln(d−1) e(E). for every integer n  1. This implies −dn (n−1)(d−1) L e(E) − L−d(n+1) Ln(d−1) e(E) μ∗Y (ord−1 I (n)) = L

= (L−n−d+1 − L−n−d ) e(E) = L−n−d (L − 1) e(E). Finally, we obtain μ∗X (L (X)) = L−d (e(Y ) − e(E)) +

∞ 

L−n(c−1) L−n−d (L − 1) e(E)

n=1 −d

=L

−d

(e(Y ) − e(E)) − L

(L − 1) e(E)

∞ 

L−nc .

n=1

∞ R , and one has Since -L−1 - < 1, the series n=1 L−nc converges in M X0 ∞  n=1

Therefore, (4.4.5.1)

L−nc =

L−c 1 . = c 1 − L−c L −1

 L−1 μ∗X (L (X)) = L−d e(Y ) − e(E) − c e(E) . L −1

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345

c−1 (4.4.6). — One has e(Pc−1 ), so that k ) = (1 + L + . . . L c (L − 1) e(Pc−1 k ) = L − 1.

By equation (4.4.3.1) and equation (4.4.1.2), equation (4.4.4.1) thus rewrites as L−d e(X) = L−d e(Y ) − L−d e(E) − L−d e(C). compatibly with the blow-up relation (4.4.1.1). Remark 4.4.7. — Let us emphasize the fact that recovering the blow-up relation by using the (general) change of variables formula is pretty “unnatX , and not in the ural,” since it only gives rise to an equality in the ring M ring K0 (Vark ). That is weaker, since the localization map K0 (Vark ) → Mk is not injective. Also recall that we do not know whether the further process of completion, necessary for the theory of motivic integration which is presented in this book, is faithful or not.

§ 5. SEMI-ALGEBRAIC SUBSETS OF GREENBERG SCHEMES In this section, we assume that R = k[[t]], where k is a field of characteristic zero. This implies, in particular, that MSR = MS for every k-variety S. We will introduce a large class of well-behaved measurable subsets of Greenberg schemes of R-varieties, namely, the semi-algebraic sets. They appear naturally in various applications in algebraic geometry and give rise to interesting motivic generating series with strong rationality properties. 5.1. Semi-algebraic Subsets (5.1.1). — We first recall the notion of a semi-algebraic condition. Let F be an extension of k. The angular component ac(x) of an element x ∈ F ((t)) is defined to be 0 if x = 0, and to be the coefficient of lowest degree of x otherwise. Let x1 , . . . , xm be variables running over F ((t)), let y1 , . . . , yn be variables running over F , and let 1 , . . . , r be variables running over Z. A semi-algebraic condition θ(x1 , . . . , xm , y1 , . . . , yn , 1 , . . . , r ) (over k) of type (m, n, r) is a (finite) Boolean combination of conditions of the form (5.1.1.1)

ordt (f1 (x1 , . . . , xm ))  ordt (f2 (x1 , . . . , xm )) + L( 1 , . . . , r ),

(5.1.1.2)

ordt (f1 (x1 , . . . , xm )) ≡ L( 1 , . . . , r ) mod d,

(5.1.1.3)

h(ac(f1 (x1 , . . . , xm )), . . . , ac(fp (x1 , . . . , xm )), y1 , . . . , yn ) = 0,

where f1 , f2 , . . . , fp are polynomials with coefficients in k((t)), L is a polynomial with coefficients in Z of degree at most 1, d is a positive integer, and

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h ∈ k[T1 , . . . , Tp+n ] is a polynomial with coefficients in k. We use here the conventions ordt 0 = +∞, (+∞) + = +∞ and +∞ ≡ mod d, for every integer and every positive integer d. Definition 5.1.2. — For every semi-algebraic condition θ(x1 , . . . , xm ; y1 , . . . , yn ; 1 , . . . , r ) of type (m, n, r) over k, one denotes by VF (θ) the set of points m

(x1 , . . . , xm ; y1 , . . . , yn ; 1 , . . . , r ) ∈ F ((t)) ×F n × Zr satisfying the condition θ. m A subset A of F ((t)) ×F n × Zr is said to be semi-algebraic (over k) if there exists a semi-algebraic condition θ of type (m, n, r) over k such that A = VF (θ). m

The set of all semi-algebraic subsets of F ((t)) ×F n × Zr is a Boolean algebra. m

(5.1.3). — Let A be a semi-algebraic subset of F ((t)) ×F n × Zr . A map p F : A → F ((t)) ×F q × Zs is said to be semi-algebraic if its graph is a semim+p algebraic subset of F ((t)) ×F n+q × Zr+s . For example, a polynomial in m variables with coefficients in k((t)) induces m × a semi-algebraic maps from F ((t)) to F ((t)); the function ordt : F ((t)) → Z is semi-algebraic; the function ac : F ((t))→ F is semi-algebraic. Restrictions and compositions of semi-algebraic maps are semi-algebraic maps, as well as the inverse of a bijective semi-algebraic map between semialgebraic subsets. Example 5.1.4. — Semi-algebraic subsets of F n (of type (0, n, 0)) exactly coincide with constructible subsets defined over k: for every semi-algebraic subset V of F n , there exists a unique constructible subset C ⊂ Ank such that V = C(F ). Similarly, semi-algebraic maps F m → F n are the maps with k-constructible graph. By Chevalley’s theorem (theorem A/1.2.4), such semi-algebraic sets admit elimination of quantifiers if F is algebraically closed: for every semi-algebraic map f : F m → F n , the image f (A) of a semi-algebraic subset A of F m is a semi-algebraic subset of F n . Example 5.1.5. — A semi-algebraic subset of Zr (of type (0, 0, r)) is also called a Presburger subset. Explicitly, Presburger sets are defined by Boolean combinations of the form: (5.1.5.1)

L( 1 , . . . , r )  0

(5.1.5.2)

L( 1 , . . . , r ) ≡ 0 (mod d)

in which L is a polynomial of degree  1 with coefficients in Z and d is a positive integer. Let A be a Presburger subset of Zr and let f : A → Zs be a map; one says that f is a Presburger map if its graph is a Presburger subset of Zr+s .

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347

(Thus, this corresponds to a semi-algebraic map.) For example, affine maps are Presburger. Presburger sets admit elimination of quantifiers (see Presburger 1930): with the previous notation, f (A) is a Presburger subset of Zs . Theorem 5.1.6 (Pas 1989). — Assume that F is an algebraically closed m field extension of k. Let A be a semi-algebraic subset of F ((t)) ×F n × Zr p and let f : A → F ((t)) ×F q × Zs be a semi-algebraic function. Then f (A) is n a semi-algebraic subset of F ((t)) ×Zs . More precisely, for every semi-algebraic condition θ over k defining the graph of f , there exists a semi-algebraic condition ψ(x , y  ,  ) over k such that for every algebraically closed extension F of k and every (x , y  ,  ) ∈ p F ((t)) ×F q × Zs , the conditions ψ(x , y  ,  ) and (∃x ∈ F ((t)) )(∃y ∈ F n )(∃ ∈ Zr )θ((x, x ), (y, y  ), ( ,  )) m

are equivalent. Remark 5.1.7. — This last formulation indicates that this theorem is a “quantifier elimination” result for semi-algebraic sets, generalizing the theorems of Chevalley and Presburger. More generally, theorem 5.1.6 implies that every “semi-algebraic formula with quantifiers” is equivalent to a semialgebraic formula. The proof of this fact runs by induction on the length of the formula, removing one quantifier at a time. The case of an existential quantifier is explicitly theorem 5.1.6, and the case of a universal quantifier follows by passing to complementary subsets. Since it is not stated in the same way in Pas (1989), let us give some additional comments about its proof. By assumption, f (VF (θ)) is defined p in F ((t)) ×F q × Zs by the formula (∃x ∈ F ((t)) )(∃y ∈ F n )(∃ ∈ Zr )θ((x, x ), (y, y  ), ( ,  )). m

By theorem 4.1 of Pas (1989) (whose proof is outside of the scope of this book), there exists an equivalent formula that has no F ((t))-quantifiers. Then quantifiers on F can be eliminated by Chevalley’s theorem (theorem A/1.2.4), since F is assumed to be algebraically closed. Finally, the quantifiers over Z can be removed by applying Presburger’s elimination theorem. 5.2. Semi-algebraic Subsets of Greenberg Schemes Let X be an R-variety. Following Denef and Loeser (1999), we define the class of semi-algebraic subsets of Gr(X ) and prove that they are C-measurable subsets of Gr(X ). If F is a field extension of k, and if x ∈ Gr(X )(F ), we denote by ϕx the corresponding point of X (F [[t]]). (5.2.1). — Let A be a subset of Gr(X ). One says that A is elementary semi-algebraic if there exist a semi-algebraic condition θ of type (m, 0, 0) and functions g1 , . . . , gm ∈ OX (X ) such that (5.2.1.1)

A = {x ∈ Gr(X ) ; θ(g1 (ϕx ), . . . , gm (ϕx ))}.

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In other words, A is defined by a finite Boolean combination of conditions of the following form: (5.2.1.2)

ordt (f1 (ϕx ))  ordt (f2 (ϕx )),

(5.2.1.3)

ordt (f1 (ϕx )) ≡ 0 mod d,

(5.2.1.4)

h(ac(f1 (ϕx )), . . . , ac(fn (ϕx ))) = 0,

in which f1 , f2 , . . . , fn are elements of OX (X ), and h ∈ k[T1 , . . . , Tn ] is a polynomial. Also observe that ordt (f1 (ϕx )) = ordx (f1 ), with the notation of (4/4.4.2.1). (5.2.2). — More generally, a family (A )∈Zr of subsets of Gr(X ) is called an elementary semi-algebraic family if there exist a semi-algebraic condition θ of type (m, 0, r) and functions g1 , . . . , gm ∈ OX (X ) such that for every ∈ Zr , A = {x ∈ Gr(X ) ; θ(g1 (ϕx ), . . . , gm (ϕx ), )}. Definition 5.2.3. — One says that A (resp. a family (A )∈Zr ) is semialgebraic if, for every affine open subscheme U of X , the trace A ∩ Gr(U ) (resp. the family (A ∩ Gr(U ))∈Zr ) is elementary semi-algebraic (resp. an elementary semi-algebraic family) in Gr(U ). Let us denote by SGr(X ) the set of all semi-algebraic subsets of Gr(X ). Lemma 5.2.4. — Let X be an R-variety. The set of all semi-algebraic subsets (resp. of all elementary semi-algebraic subsets) of Gr(X ) is a Boolean algebra. More generally, semi-algebraic families indexed by Zr form a Boolean algebra. Proof. — It suffices to treat the case of elementary semi-algebraic subset. The empty set is an elementary semi-algebraic subset of Gr(X ). Let A, B be elementary semi-algebraic subsets of Gr(X ). If A and B are defined by elementary semi-algebraic conditions θ and ψ, then A ∩ B is defined by the semi-algebraic condition θ ∧ ψ, A ∪ B by the condition θ ∨ ψ, and Gr(X ) A by the negation ¬θ of θ. Consequently, A ∩ B, A ∪ B, and Gr(X ) A are elementary semi-algebraic. Proposition 5.2.5. — Let A be a subset of Gr(X ). a) Assume that X is affine. Then A is semi-algebraic if and only if it is elementary semi-algebraic. b) Assume that X is affine and let U be an affine open subscheme of X . If A is contained in Gr(U ), then A is semi-algebraic if and only if it is semi-algebraic as a subset of Gr(U ). c) Assume that there exists a finite covering (Ui ) of X by affine open subschemes such that for every i, A∩Gr(Ui ) is a semi-algebraic subset of Gr(Ui ). Then A is semi-algebraic.

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349

We leave it to the reader to formulate and prove the analogous result for semi-algebraic families. Proof. — a) It follows from the definition of a semi-algebraic subset that if A is semi-algebraic, then it is elementary semi-algebraic. Conversely, if A is elementary semi-algebraic, as witnessed by a semi-algebraic condition θ and functions g1 , . . . , gm in Γ(X , OX ), then the condition θ and the restrictions to U of g1 , . . . , gm prove that A ∩ Gr(U ) is elementary semi-algebraic, as a subset of Gr(U ). This proves a). b) It follows from the definition of a semi-algebraic subset that if A is a semi-algebraic subset of Gr(X ), then A ∩ Gr(U ) is a semi-algebraic subset of Gr(U ). In particular, if a subset A of Gr(U ) is semi-algebraic in Gr(X ), then it is semi-algebraic in Gr(U ). Conversely, let us assume that A is a semi-algebraic subset of Gr(U ), and let us prove that A is semi-algebraic in Gr(X ). of OX (X ) such that U = D(vi ). Let (vi ) be a finite family of elements Let Ui = D(vi ); then Gr(U ) = Gr(Ui ), and by a), applied to Ui , we see that A ∩ Gr(Ui ) is an elementary semi-algebraic subset of Gr(Ui ). It suffices to prove that A ∩ Gr(Ui ) is semi-algebraic as a subset of Gr(X ), since, then, A = (A ∩ Gr(Ui )) will be a finite union of a semi-algebraic subsets of Gr(X ), hence is semi-algebraic in Gr(X ). Consequently, we may assume that U = D(v), for some regular function v on X . In this case, a point x ∈ Gr(X ), with associated arc ϕx ∈ F [[t]], belongs to Gr(U ) if and only if v(ϕx ) is invertible in F [[t]], that is, if ordt (v(ϕx )) = 0. This is a semi-algebraic condition. Let g ∈ Γ(U , OU ); since X is affine, there exists g  ∈ Γ(X , OX ) such that g = g  /v n . We then remark that for x ∈ Gr(U ), one has ordt (g(ϕx )) = ordt (g  (ϕx )), ordt (v(ϕx )) = 0; hence, ac(g(ϕx )) = ac(g  (ϕx ))/ ac(v(ϕx ))n . These two observations imply that A can be defined by an elementary semialgebraic condition in Gr(X ), as claimed. Consequently, A is elementary semi-algebraic in Gr(X ), hence semi-algebraic. c) Let U be an affine open subscheme of X , and let us show that A ∩ Gr(U ) is an elementary semi-algebraic subset of Gr(U ). For every i, let (Vi,j ) be a finite affine covering of U ∩ Ui . Since Vi,j is contained in Ui , the intersection A ∩ Gr(Vi,j ) is semi-algebraic in Gr(Vi,j ). By b), applied to U and Vi,j , we deduce that A ∩ Gr(Vi,j ) is semi-algebraic in Gr(U ). Since   A ∩ Gr(U ) = A ∩ Gr(U ∩ Ui ) = A ∩ Gr(Vi,j ) i

i,j

it is semi-algebraic in Gr(U ), as was to be shown. Definition 5.2.6. — Let A be a semi-algebraic subset of Gr(X ). A function α : A × Zr → Z ∪ {∞} is said to be simple if the family (A )∈Zr+1 defined by A = {x ∈ Gr(X ) ; α(x, 1 , . . . , r ) = r+1 },

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for ∈ Zr+1 , is a semi-algebraic family of semi-algebraic subsets of Gr(X ). Example 5.2.7. — Let I be a coherent sheaf of ideals on X , let n ∈ N, and let A be the constructible subset of points x ∈ Gr(X ) such that ordI (x)  n. Then A is a semi-algebraic subset of Gr(X ). Indeed, when X is affine and I is generated by global sections f1 , . . . , fm , then A is defined by the conjunction of the semi-algebraic conditions ordt (fi (ϕx ))  n, for i ∈ {1, . . . , m}. As a consequence, the function Gr(X ) → Z ∪ {∞} given by x → ordI (x) is simple. Example 5.2.8. — Let C be a constructible subset of Gr0 (X ) and let −1 (C) be its inverse image in Gr(X ). Then A is semi-algebraic. A = θ0,X We may assume that X is affine and C is closed in Gr0 (X ). We fix a closed embedding h = (h1 , . . . , hm ) : X → Am R . Let (f1 , . . . , fn ) be a family −1 of polynomials of k[T1 , . . . , Tm ] generating the ideal of C. Then θ0,X (C) is defined by the semi-algebraic condition (ordt (f1 (ϕx ))  1) ∧ · · · ∧ (ordt (fn (ϕx ))  1). Remark 5.2.9. — Let us give some examples that show that the classes of constructible and of semi-algebraic sets are different. We set X = A1R . Then the singleton A = {t} is semi-algebraic in Gr(X ) (defined by the condition ordt (x)  ordt (0)), but not constructible. This example is somewhat artificial, since A is semi-algebraic in the Greenberg scheme of the closed subscheme of X defined by x − t. So let us give a more refined example. Let B be the subset of Gr(X ) consisting of points x of the form x0 + x1 t + x2 t2 + . . . with x1 = 0. Then B is constructible, but it is not semi-algebraic. To see this, let ξ0 be the generic point of Gr0 (X ) and let ξ be any element in (θ0,X )−1 (ξ0 ). Then for every polynomial f in one variable over k((t)), we have ordt (f (ϕξ )) = ordt (f (ξ0 )) and ac(f (ϕξ )) = ac(f (ξ0 )). This means that any semi-algebraic subset of Gr(X ) that contains ξ must contain the entire set (θ0,X )−1 (ξ0 ). In particular, B is not semi-algebraic. Conversely, let C be the semi-algebraic subset of Gr(X ) consisting of arcs γ such that ac(γ) = 1. If A were constructible, there would exist an integer n such that for any two arcs γ, γ  such that γ ≡ γ  (mod tn ), then γ ∈ A if and only if γ  ∈ A. However, the arcs tn+1 and 0 are congruent modulo tn , the first one belongs to A, but the second one doesn’t. Proposition 5.2.10. — Let X be an R-variety. Let A be a semi-algebraic subset of Gr(X ). Then, for every integer n ∈ N, the set θn,X (A) is a constructible subset of Grn (X ). Proof. — We may assume that X is affine, presented as a closed subscheme −1 N of AN R . Let B = θn,AN (θn,X (A)). By construction, B is defined in Gr(AR ) R by the formula ∃x1 . . . ∃xN (ordt (x1 − y1 )  n + 1) ∧ · · · ∧ (ordt (xN − yN )  n + 1) ∧ θA (x),

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351

where θA is a semi-algebraic condition defining A in AN R . It is thus the image, under the second projection, of the the semi-algebraic set C ⊂ Gr(X ×R AN R) defined by the formula (ordt (x1 − y1 )  n + 1) ∧ · · · ∧ (ordt (xN − yN )  n + 1) ∧ θA (x). It therefore follows from Pas’s theorem (theorem 5.1.6) that B is a semialgebraic subset of Gr(AN (B) = R ). By construction of B, one has θn,AN R −1 θn,X (A) and B = θn,AN (θn,X (A)). Recall the identifications Gr(AN R) = R

N N N L (AN k ) and Grn (AR ) = Ln (Ak ). A jet of order n on Ak can be extended N (by 0) to an arc on Ak . Arguing as in §3/3.5, this implies the existence N ∞ of a morphism of schemes s : Ln (AN k ) → L∞ (Ak ) such that θn ◦ s = id; −1 (B) = s (B). Let us now consider a semi-algebraic hence, θn,X (A) = θn,AN R condition θB defining B in Gr(AN R ). By inspection of the conditions given in (5.2.1.2, 5.2.1.3, 5.2.1.3), we conclude that s−1 (B) is a constructible subset in Ln (AN k ). This concludes the proof.

5.3. Measurability of Semi-algebraic Subsets Proposition 5.3.1. — Let X be an R-variety of dimension d. Let A be a semi-algebraic subset of Gr(X ). There exist a thin negligible semi-algebraic subset E of A and a semi-algebraic family (Am )m∈Np of mutually disjoint subsets of Gr(X ) that verify the following properties: a) The Am and E form a partition of A; b) For every m ∈ Np , the set Am is constructible in Gr(X ); c) We have lim μX (Am ) = 0. m→∞

Proof. — We may argue locally on X , which allows us to assume that X is affine. Considering the traces of A on Gr(Y ), for each rig-irreducible component Y of X , we may also assume that X is R-flat and integral. Let θ be a semi-algebraic condition defining A and let f1 , . . . , fp be the nonzero functions on X that appear in θ. For every m ∈ Np , let ψm be the semi-algebraic condition (ordt f1 (ϕx ) = m1 ) ∧ · · · ∧ (ordt fp (ϕx ) = mp ) and let Am be the semi-algebraic subset of A defined by the conjunction θ ∧ ψm . By construction, the family (Am )m∈Np is a semi-algebraic family of mutually disjoint subsets of Gr(X ). We observe moreover that for every m ∈ Np , the set Am is constructible, by proposition 4/4.4.7. For every i ∈ {1, . . . , p}, Am is contained in the constructible subset of Gr(X ) defined by the condition ordt (fi,x )  mi . It thus follows from example 3.2.6 that μX (Am ) → 0 when m → ∞. Let E = A m∈Np Am . By construction, one has E = A ∩ Gr(V (f )), where f = f1 . . . fp , so that E is semi-algebraic. Since X is integral, one has f = 0. Consequently, E is thin.

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Corollary 5.3.2. — Let X be an R-variety. Every semi-algebraic subset of Gr(X ) is strongly C-measurable. In particular, SGr(X ) ⊂ Cons∗Gr(X ),C . Proof. — Let A be a semi-algebraic subset of Gr(X ). Let E be a negligible subset and let (An )n∈Np be a semi-algebraic family of constructible subsets of Gr(X ) satisfying the properties of proposition 5.3.1. Let ε > 0. There exists a finite subset N of Np such that for every n ∈ Np N , μ∗X (An ) < ε. Besides, there exists a constructible subset B of Gr(X ) such that E ⊂ B and μX (B) < ε (example 3.2.6). Then the family ( n∈N An ; (An )n∈N ∪ B) is a strong C-constructible ε-approximation of A. Consequently, A is strongly C-measurable. Theorem 5.3.3. — Let X be an R-variety purely of relative dimension d. The restriction of the measure μ∗X to SGr(X ) is the unique map * μ∗sa,X : SGr(X ) → M X0 that satisfies the following properties: a) If A is semi-algebraic and constructible, then μ∗sa,X (A) = μX (A); b) If A is a thin semi-algebraic subset of Gr(X ), then μ∗sa,X (A) = 0; c) For every family (Ai )i∈N of mutually disjoint semi-algebraic  subsets of Gr(X ) such that i∈N Ai is semi-algebraic, then μ∗sa,X (A) = i∈N μX (Ai ). Proof. — Since semi-algebraic subsets are measurable, proposition 3.4.3 and example 3.5.2 show that the restriction of μ∗X to semi-algebraic subsets satisfies the required properties. Let us thus show that it is the only such map. Let μ be such a map and let A ∈ SGr(X ) ; we have to prove that μ(A) = μ∗X (A). By proposition 5.3.1, there exist a thin negligible semi-algebraic subset E, an integer p, and a semiof Gr(X ), algebraic family (Am )m∈Np of semi-algebraic constructible subsets all these sets being pairwise disjoint, such that A = E ∪ ( m∈Np Am ) and limm→∞ μX (Am ) = 0. Property a) ensures that μ(Am ) = μX (Am ). By proposition 3.4.3, the set A = m∈Np Am is measurable, and its measure is equal to   μ∗X (A ) = μX (Am ) = μ(Am ). m∈Np 

μ∗X

m∈Np 

By property c), one thus has μ(A ) = (A ). Moreover, one has μ(E) = 0 = μ∗X (E) = 0, by b). Consequently, applying c) to the family (E, A ), we have μ(A) = μ(E) + μ(A ) = μ∗X (E) + μ∗X (A ) = μ∗X (A), as was to be shown. The following result is a motivic analog of a result for p-adic volumes due to Oesterlé (1982). It provides a concrete interpretation of the motivic volume of a semi-algebraic set.

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353

Proposition 5.3.4. — Let X be an R-variety of pure relative dimension d, and let A be a semi-algebraic subset of Gr(X ). Then the sequence (e(θn,X (A))L−(n+1)d )n0 * converges to the motivic volume μ∗X (A) in M X0 . Proof. — Note that the sets θn,X (A) are constructible by proposition 5.2.10, so that it is meaningful to consider their classes in the Grothendieck ring = (θn,X )−1 θn,X (A). This is a of varieties. For every n  0, we set An  constructible subset of Gr(X ), and A = n An . By corollary 3.4.3, the * sequence μX (An ) converges to μ∗X (A) in M X0 . By theorem 2.5.1, we know that for every n  0, the sequence (e(θm,X (An ))L−(m+1)d )n0 * converges to μX (An ) in M X0 , and this convergence is uniform in n. It follows that (e(θn,X (An ))L−(n+1)d )n0 * converges to μ∗X (A) in M X0 . But, by construction, θn,X (An ) = θn,X (A). This concludes the proof.

5.4. Rationality of Motivic Power Series Let X be an R-variety. The goal of this section is to show that power series with coefficients in the Grothendieck ring MX0 (or its completion) which are defined “naturally,” by semi-algebraic formulas, have nice rationality properties. We will revisit these results in chapter 7 (particularly §7/3) and make them more precise in some specific instances. The results of this section were proved by Denef and Loeser (1999) for varieties over a field and extended to the relative case by Sebag (2004b). Theorem 5.4.1. — Let k be a field of characteristic 0, let R = k[[t]], and let X be a flat R-variety of pure relative dimension d. Let (An )n∈Zr be a semi-algebraic family of Gr(X ), and let α : Gr(X ) × Zr → N be a simple function (see definition 5.2.6). Then the power series   −α(·,n) ∗ L dμX T n n∈Nr

An

* belongs to the subring of M X0 [[T1 , . . . , Tr ]] generated by MX0 [T1 , . . . , Tr ] and −a b the power series 1/(1 − L T ), for a ∈ N and b ∈ Nr , not both zero. The proof of this rationality result relies on the following proposition.

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Proposition 5.4.2. — Let P be a Presburger subset of Zd ; let a = (a1 , . . . , an ) : P → Nn be a (finite) Presburger function with finite fibers and consider the power series   a (p) T a(p) = T1 1 . . . Tnan (p) ∈ Z[[T1 , . . . , Tn ]]. ZP,a (T ) = p∈P

p∈P

Then ZP,a belongs to the subring of Z[[T ]] generated by Z[T ] and the power series 1/(1 − T c ), for c ∈ Nn {0}. More precisely, let m be an integer such that the map (a1 , . . . , am ) : P → Nm has finite fibers. Then ZP,a belongs to the subring of Z[[T ]] generated by Z[T ] and the power series 1/(1 − T c ), for c ∈ (Nm {0}) × Nn−m . Proof. — We first treat the case where n = d and a is the identity map. Let N  1 be a common multiple of all moduli appearing in congruences defining P . By Euclidean division, we write P as a finite union of subsets of Zn of the form N Pa + a, for Presburger sets Pa defined by a formula without congruence condition. Therefore, we may assume that P itself is defined by a formula without congruence condition. In this case, P is a finite disjoint union of subsets of the form (a + σ) ∩ Zn , where a ∈ Qn and σ is an open rational polyhedral cone in Rn , that is, the open cone generated by finitely many rational vectors. Thus it suffices to prove the result for P = σ. We can subdivide σ into a disjoint union of simplicial open rational polyhedral cones of Rn , generated by Q-linearly independent vectors; thus we may assume that σ is itself simplicial. Let σ = u1 , . . . , ur , where u1 , . . . , ur are Q-linearly independent vectors in Zn . Let P be the finite set of lattice points in the fundamental parallelepiped {λ1 u1 + . . . + λr ur , λi ∈ Q ∩ [0, 1) for all i}. Then we have Zσ,id (T ) = (



p∈P

T p )(

n  

u1,i q1 +···+ur,i qr

Ti

q∈Nr>0 i=1

)=(



p∈P

T p)

r 

T uj . 1 − T uj j=1

If m is a positive integer such that m  n and the projection of σ onto Nm has finite fibers, then the vectors (uj,1 , . . . , uj,m ) are nonzero for all j, hence the claim. We now prove the general case. We add variables and consider the power series  T m U p ∈ Z[[T, U ]]. GP,a (T, U ) = (p,m)∈P ×Nn a(p)=m

By the first case, it belongs to the subring of Z[[T, U ]] generated by Z[T, U ] and the power series 1/(1 − T b U c ), for b ∈ Nn and c ∈ Nd , with (b, c) = 0. In fact, the exponents b that appear satisfy (b1 , . . . , bm ) = 0, by the refinement of the rationality property. In particular, we may make the substitution U = 1, and we get that FP,a (T ) = GP,a (T, 1) belongs to the subring of Z[T ]

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355

generated by Z[T ] and the power series 1/(1 − T b ), for b ∈ Nn such that (b1 , . . . , bm ) = 0. (5.4.3). — We now begin the proof of theorem 5.4.1 with a particular case. Let X be a smooth affine R-variety of pure relative dimension d, and let u = (u1 , . . . , ud ) : X → AdR be an étale morphism. We say that a regular function f ∈ O(X ) is monomial in u if there exist g ∈ O(X )× and m = md 1 (m1 , . . . , md ) ∈ Nd such that f = gum 1 . . . ud ; then m is called the exponent of f and g its coefficient. We also let ω : Gr(X ) → (N ∪ {∞})d be the map x → (ordx (u1 ), . . . , ordx (ud )). Let (A )∈Zr be a family of semi-algebraic subsets of Gr(X ). Assume that there exists a semi-algebraic condition θ defining this family on X such that all nonzero regular functions on X that appear in θ are monomial with respect to u. We also assume that the simple function α in the statement of the theorem is equal to 0. We then have to study the power series  μX (A )T11 . . . Trr FA (T ) = ∈Nr

X [[T1 , . . . , Tr ]]. in M 0 Lemma 5.4.4. — Under the assumptions of 5.4.3, there exist a finite set Q, a family (Cq )q∈Q of X0 -varieties, and a family (Pq )q∈Q of Presburger subsets of Zd+r such that  μX (A ∩ ω −1 (n)) = L−d e(Cq /X0 )L−n1 −···−nd q∈Q (n,)∈Pq

for every n = (n1 , . . . , nd ) ∈ Nd and every ∈ Zr . If, moreover, A is constructible and ω is finite on A , for every ∈ Zr , then we can moreover assume that the projection (n, ) → from Pq to Zr has finite fibers, for every q ∈ Q. Proof. — We choose a closed embedding of X into an affine space AN R. ˜1 , . . . , u ˜d We lift arbitrarily the regular functions u1 , . . . , ud to polynomials u N d N on AN . Let us then consider the map a from A to A × A given by k R R k k ud (x)), θAN a(x) = (ac(˜ u1 (x)), . . . , ac(˜ ,0 (x)). R r d N d+r Let us also consider the map h from AN given R × Z to (Ak ×k Ak ) × Z by h(x, ) = (a(x), ω(x), ). The maps a and h are semi-algebraic in the sense that there exist semialgebraic conditions (of type (d, d + N ) and (N, d + N, 2r + d), respectively) that define their graphs over any algebraically closed extension F of k. Let A ⊂ Gr(X ) × Zr be the union of the subsets A × { }, for ∈ Zr ; by assumption, it is a semi-algebraic subset of Gr(X ) × Zr . The given semi-algebraic definition ψ of A is of type (N, 0, r) with variables

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(x1 , . . . , xN ; ; 1 , . . . , r ). We modify it to a a semi-algebraic formula ψ  of type (0, d + N, d + r), with variables (; y1 , . . . , yd , x1 , . . . , xN ; n1 , . . . , nd , 1 , . . . , r ) as follows. By the assumption on the family (A ), each nonzero function f ∈ O(X ) that appears in ψ is a monomial in u, with some exponent m and some coefficient h ∈ O(X )× . We then replace the expression ordx (f ) by the linear combination n1 m1 + · · · + nd md , and the expression ac(f ) ˜ 0 (x)y m1 . . . y md , where h ˜ ∈ k[X1 , . . . , Xr ] is any lift by the polynomial h 1 d × of h0 ∈ O(X0 ) . By construction, the condition ψ(x, ) is thus equivalent to the semi-algebraic formula “∃(y, n, ) ψ  (y, m, ) ∧ (y = a(x)) ∧ (n = ω(x)).” Observe that the formula ψ  is of type (0, d + N, d + r). However, the definition of semi-algebraic subsets does not allow for any interplay between the variables over the residue field k and the variables over the value group Z, so that semi-algebraic subsets of F d+N × Zd+r are Boolean combinations of the form C × P , where C ⊂ F d+N is semi-algebraic (i.e., constructible) and P ⊂ Zd+r is Presburger. We thus obtain a finite family (Cq )q∈Q of constructible subsets in Adk ×k X0 and a finite family (Pq )q∈Q of Presburger sets in Nd+r such that VF (ψ  ) is the disjoint union of Cq (k a ) × Pq for every algebraically closed extension F of k. The condition for a point x ∈ Gr(X ) to belong to A is then transformed to the conjunction a(x) ∈ Cq ∧(ω(x), ) ∈ Pq . Since u : X → AdR is étale, the canonical morphism (Gr(u), θX0 ) : Gr(X ) → Gr(AdR ) ×Ad X0 k

is an isomorphism of X0 -schemes. Let n ∈ N be such that (n, ) ∈ Pq . Let p be any integer such that p  sup(n1 , . . . , nd ). If we view a point x ∈ Gr(X ) as power series, the condition n = ω(x) means that u1 (x), . . . , ud (x) are of the form uj (x) = aj tnj + . . . , and the condition a(x) = y is equivalent to (a1 , . . . , ad , θ0 (x)) = y. This description implies that A ∩ ω −1 (n) is constructible of level  p and that  μX (A ∩ ω −1 (n)) = L−(p+1)d e(Cq /X0 )L(p−n1 )+···+(p−nd ) d

q∈Q (n,)∈Pq

= L−d



e(Cq /X0 )L−(n1 +···+nd ) .

q∈Q (n,)∈Pq

This concludes the proof of the first part of the lemma. Let us now assume that for every ∈ Zr , the set A is constructible and that ω is finite on A . In this case, the quasi-compactness of the constructible topology implies that ω is bounded from above on A . Consequently, the set of n ∈ Nd such that (n, ) ∈ Pq is finite, for every ∈ Zr . (5.4.5). — With the same hypotheses and notation as in §5.4.3, one can thus write   e(Cq /X0 ) L−n1 −···−nd T11 . . . Trr . FA (T ) = L−d q∈Q

(n,)∈Pq

§ 5. SEMI-ALGEBRAIC SUBSETS OF GREENBERG SCHEMES

For every q ∈ Q, set GA,q (T ) =



357

U n1 +···+nd T11 . . . Trr ∈ Z[[U, T1 , . . . , Tr ]].

(n,)∈Pq

By proposition 5.4.2, this is an element of the subring of Z[[T, U ]] generated by Z[T, U ] and the power series 1/(1 − U a T b ), for a ∈ N and b ∈ Nr {0}. Substituting U = L−1 , we thus obtain that FA (T ) belongs to the subring X [[T ]] generated by MX [T ] and the power series 1/(1 − L−a T b ), for of M 0 0 a ∈ N and b ∈ Nr , not both zero. (5.4.6). — We now prove theorem 5.4.1. By an additivity argument, we may assume that X is affine and integral. For each m ∈ Z and each n ∈ Zr , let Am,n = {x ∈ An ; α(x, n) = m}. We thus can rewrite the power series of the theorem as   −α(·,n) ∗ F (T ) = L dμX T n n∈Nr



=

An

L−m μX (Am,n )T n . r

(m,n)∈Z×N

Let us then set G(U, T ) =



X [[U, T ]]. μX (Am,n )U m T n ∈ M 0 r

(m,n)∈Z×N

Since α is a simple function, the family (Am,n )(m,n) is a semi-algebraic family of subsets of Gr(X ). Let Z be the reduced closed subscheme of X supported on the union of Xsing , the special fiber X0 , and the hypersurfaces defined by all nonzero regular functions appearing in a semi-algebraic description of this family. Let h : X  → X be a log resolution of (X , Z ); in particular, the generic fiber XK is smooth. By the change of variables formula (theorem 4.3.1), we compute the measure of Am,n as an integral on Gr(X  ).   be the smooth locus of X  ; since Gr(Xsm ) = Gr(X  ), we are thus Let Xsm reduced to the case where X is smooth and the closed subscheme Z has strict normal crossings relatively over R. By a further affine covering of X , we are then reduced to the situation of §5.4.3, so that the power series G(U, T ) belongs to the subring * X [[U, T ]] generated by M of M X0 [U, T ] and the power series of the form 0 −a b c 1/(1 − L U T ), with a, b ∈ N and c ∈ Nr , not all zero. This implies the theorem. (5.4.7). — In fact, the proof gives a finer result, for power series with coefficients in MX0 . Before we state and prove this result (theorem 5.4.9), we need to make some further preparations. Let X be an affine R-variety. We say that a semi-algebraic family (A ) in Gr(X) is bounded if it can be defined by a semi-algebraic formula θ such that all the nonzero regular functions f on X that occur in θ have bounded order on each set A . We then call θ a bounded semi-algebraic formula for (A ) . Note that every member A of a

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bounded family (A ) is automatically constructible: if N is an upper bound for the orders of the functions f on A , then the condition defining A only depends on ϕx modulo tN +1 , so that A is constructible of level N . Lemma 5.4.8. — Let X be an affine R-variety. Let (A )∈Zp be a semialgebraic family in Gr(X ) such that each member A is constructible. Then we can write (A ) as a finite union of semi-algebraic families of the form

B (C1 ∪ · · · ∪ Cm ) )∈Zp where m is a positive integer, (B ) and (Ci ) are bounded semi-algebraic families, and Ci ⊂ B for all i and . If A is disjoint from Gr(Xsing ) for every , then we can arrange that each set B is disjoint from Gr(Xsing ). Proof. — We fix a semi-algebraic formula θ that defines the family (A ) . Let f1 , . . . , fr , fr+1 , . . . , fs be the nonzero regular functions on X that appear in θ, where fr+1 , . . . , fs have bounded order on each member A , and each of the functions f1 , . . . , fr has unbounded order on some member A . We will argue by induction on r. For every , we denote by α( ) the smallest nonnegative integer such that A is constructible of level  and the functions fr+1 , . . . , fs have order at most α( ) on A . The graph of the function α : Zp → Z can be defined by a semi-algebraic formula with quantifiers. Quantifier elimination (theorem 5.1.6) then implies that this graph can also be defined by a formula without quantifiers; in other words, α is a Presburger function. Let Y be the closed subscheme of X defined by the functions f1 , . . . , fr , and denote by γY its Greenberg function. By corollary 5/1.5.4, we can bound γY from above by an affine function. Thus we can choose a Presburger function β : Zr → Z such that γY (α( ))  β( ) for every in Zp . Now we consider the semi-algebraic families defined by A1 = {x ∈ A , ordt f1 (ϕx )  β( )}, ... Ar = {x ∈ A , ordt fr (ϕx )  β( )}, A = {x ∈ A , ordt fi (ϕx ) > β( ) for i = 1, . . . , r}, for every in Zp . By the induction hypothesis, we can write (Ai ) in the required form, for every i in {1, . . . , r}. Thus it suffices to prove the result for the family (A ) . We define a semi-algebraic family (B ) by modifying the formula defining the family (A ) in the following way: we replace the functions f1 , . . . , fr by 0, and we add the conditions ordt fi (ϕx )  α( ) for i ∈ {r + 1, . . . , s}. Then (B ) is a bounded semi-algebraic family, and B is constructible of level  α( ) for every in Zp . If x is a point of Gr(X ) that satisfies ordt fi (ϕx ) > β( ) for every i in {1, . . . , r}, then by the definition of the Greenberg function γY , we can find a point x in Gr(Y ) such that θα(),Y (x ) = θα(),X (x).

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359

Since A and B are stable of level  α( ), we know that x lies in A (resp. B ) if and only if the same holds for x ; but for x , the memberships to A and B are equivalent. It follows that A = {x ∈ B , ordt fi (ϕx ) > β( ) for i = 1, . . . , r}. Thus if we define bounded semi-algebraic families (Ci ) by Ci = {x ∈ B , ordt fi (ϕx )  β( )} for every i in {1, . . . , r}, then A = B

(C1 ∪ · · · ∪ Cm )

is an expression for A of the required form. Now assume that A is disjoint from Gr(Xsing ) for every in Zp . Let (g1 , . . . , gq ) be a family of generators for the Jacobian ideal JacX of X . For every in Zp , let us denote by δ( ) the smallest nonnegative integer such that A is contained in Gr(δ()) (X ). Quantifier elimination for semi-algebraic formulas again implies that δ is a Presburger function. Then we can ensure that all the semi-algebraic families that appear in the bounded expression for (A ) are disjoint from Gr(Xsing ) by adding the conditions ordt gj (ϕx )  δ( ) for j ∈ {1, . . . , q}. Theorem 5.4.9. — Let k be a field of characteristic 0, let R = k[[t]], and let X be an R-variety of pure relative dimension d. Let (An )n∈Zr be a semialgebraic family of Gr(X ) and let α : Gr(X )×Zr → N be a simple function. We assume that for every n ∈ Nr , we have An ∩ Gr(Xsing ) = ∅. We also assume that An and An ∩ {α(·, n) = m} are constructible, for every n in Nr and every m ∈ N. Then the power series     L−α(·,n) dμX T n n∈Nr

An

is well defined in MX0 [[T1 , . . . , Tr ]] and belongs to the subring generated by MX0 [T1 , . . . , Tr ] and the power series 1/(1 − L−a T b ), for a ∈ N and b ∈ Nr {0}. Proof. — We will explain how the proof of theorem 5.4.1 in §5.4.6 needs to be refined. We again may assume that X is affine and integral. With the same notation as above, the sets Am,n are constructible and disjoint from Gr(Xsing ). Consequently, their measure exists in MX0 and not only in X . the completed ring M 0 By lemma 5.4.8 and the additivity of the motivic volume, we may assume that the semi-algebraic family (Am,n )m,n is bounded. Then, taking a log resolution h : X  → X for (X , Z ) as in §5.4.6 such that h is an isomorphism over X Z , the function ordjach will only take finitely many values on the inverse image of each set Am,n , so that we can apply the change of variables formula in MX0 stated in theorem 1.2.5. Continuing as in §5.4.6, we can further reduce to the case where X is smooth over R, and all the nonzero regular functions appearing in some

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bounded semi-algebraic formula θ for (Am,n )m,n are monomial with respect to some fixed étale coordinate system u = (u1 , . . . , ud ) : X → AdR . Then, at the end of §5.4.6, we need to make sure that the exponents c that appear in the rational expression for G(U, T ) can all be taken in Nr {0}. After a permutation of the coordinates (u1 , . . . , ud ), we may assume that u1 , . . . , ue are precisely the coordinates that appear with a positive exponent in our bounded semi-algebraic formula θ, for some e  d. Let X  be the R-smooth closed subscheme of X defined by ue+1 = . . . = ud = 0. Then we have μX (Am,n ) = μX  (Am,n ∩ Gr(X  ))Ld−e in MX0 . Thus, replacing X by X  , we may assume that d = e. Then the function ω = (ord(u1 ), . . . , ord(ud )) is bounded on Am,n , for all (m, n) ∈ Zr+1 , because θ is a bounded formula. This means that, for all the sets Pq that appear in lemma 5.4.4, the projection Pq → Zr+1 has finite fibers. Proposition 5.4.2 now provides the guarantee that the series G(U, T ) in §5.4.6 belongs to the subring of MX0 [[U, T ]] generated by MX0 [U, T ] and the power series of the form 1/(1 − L−a U b T c ), with a, b ∈ N and c ∈ Nr {0}. Evaluating at U = L−1 yields the desired result. Corollary 5.4.10. — Let X be an R-variety of pure relative dimension d. Let A be a semi-algebraic subset of Gr(X ). Then the motivic volume μ∗X (A) * belongs to the subring of M X0 generated by the image of MX0 and the in−a verses 1/(1 − L ), for a ∈ N>0 . Proof. — For every n  0, we set An = A ∩ ordjac−1 X (n), the set of elements in A that have contact order n with the singular locus Xsing . The sets An are constructible and form a semi-algebraic family, and their union is equal to A Gr(Xsing ). Since Gr(Xsing ) is thin, we have  μ∗X (A) = μX (An ) n0

by corollary 3.4.3. Setting α = 0 in theorem 5.4.9, we find that the series  μX (An )T n n0

lies in the subring of MX0 [[T ]] generated by MX0 [T ] and the power series 1/(1 − L−a T b ), for a ∈ N and b ∈ N {0}. Evaluating at T = 1 yields the desired result. Remark 5.4.11. — The proof of corollary 5.4.10 produces a canonical el∗ * ement in MX0 [(1 − L−a )−1 ]a>0 whose image in M X0 is μX (A). Thus the motivic volume of semi-algebraic sets can be defined in this finer ring, and one can correspondingly refine the change of variables formula by means of a similar argument. The ring MX0 [(1 − L−a )−1 ]a>0 is the value ring of the

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361

theory of motivic integration developed in Cluckers and Loeser (2008) by means of cell decomposition techniques for semi-algebraic sets. Corollary 5.4.12. — Let X be an R-variety and let A be a semi-algebraic subset of Gr(X ). The series ∞  PA (T ) = e(θX ,n (A))T n ∈ MX0 [[T ]] n=0

belongs to the subring of MX0 [[T ]] generated by MX0 [T ] and the power series 1/(1 − La T b ), for a ∈ Z and b ∈ N {0}. Series as in this corollary are motivic analogues of the p-adic Poincaré series studied in §1/3.5. To make sense of the statement, we recall that for every n ∈ N, the set θX ,n (A) is a constructible subset of Grn (X ) (proposition 5.2.10). Proof. — We may assume that X = AN R . For every n ∈ N, let An = −1 θn,X (θn,X (A)). For every n ∈ N, the set θX ,n (A) is a constructible subset of Grn (X ) (proposition 5.2.10), so that the set An is a constructible subset of Gr(X ). In fact, if θ(x) is a semi-algebraic formula defining A, then the semi-algebraic formula with quantifiers ψ(y, n) given by ∃x1 . . . ∃xN (ordt (x1 − y1 )  n + 1) ∧ · · · ∧ (ordt (xN − yN )  n + 1) ∧ θ(x) defines the family (An ). For every n, one has μX (An ) = L−N (n+1) e(θX ,n (A)), so that the power series PA (T ) can be rewritten as  LN (n+1) μX (An )T n . PA (T ) = n∈N

The assertion then follows from theorem 5.4.9.

CHAPTER 7 APPLICATIONS

This final chapter is devoted to a selection of notable applications of motivic integration. In §1, we present the construction of the motivic zeta function of a variety, as introduced by Kapranov (2000). While it does not use motivic integration per se, it fits perfectly within the circle of ideas of this book. We go on in §2 with applications to singularity theory, especially an expression for the log canonical threshold of a variety in terms of the dimension of constructible subspaces of jet spaces or arc spaces, after Mustaţă (2001); Ishii (2008); Zhu (2013). We also discuss the Nash problem in §2.5. In §3 we study applications to the construction of birational invariants of algebraic varieties. We give the general construction of the “Gorenstein volume” of integral varieties with log terminal singularities (definition 3.4.3) and of its invariance under K-equivalence (theorem 3.5.4). The theorem of Batyrev–Kontsevich appears here as a particular case of corollary 3.5.5. We also discuss the definition of stringy Hodge numbers introduced by Batyrev (1999a). In §4, we explain the construction by Denef and Loeser (1998) of the motivic Milnor fiber of a function f as a formal limit of a variant of the motivic Igusa zeta functions defined in §3.3. In §5, we define motivic integrals on non-Archimedean analytic spaces and the motivic Serre invariant of a compact analytic space, after Loeser and Sebag (2003). We use these to give a non-Archimedean interpretation of the motivic Igusa zeta function in §6, following Nicaise and Sebag (2007a); Nicaise (2009).

© Springer Science+Business Media, LLC, part of Springer Nature 2018 A. Chambert-Loir et al., Motivic Integration, Progress in Mathematics 325, https://doi.org/10.1007/978-1-4939-7887-8_7

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§ 1. KAPRANOV’S MOTIVIC ZETA FUNCTION 1.1. Symmetric Products of Varieties (1.1.1). — Let k be a field. Let X be a quasi-projective k-variety and let n be nonnegative integer. The symmetric group Sn acts on the n-fold product X n of X; by definition, the nth symmetric product X (n) of X is the quotient X n /Sn . Since X is quasi-projective, every finite set of points of X is contained in an affine open subscheme of X. It follows that every orbit of the Sn -action on X n is contained in an affine open subscheme of X n , so that the quotient X (n) exists as a scheme, by (SGA I, V, 1.8). The quotient morphism pn : X n → X (n) satisfies the following properties: – It is finite and commutes with the action of Sn , and its geometric fibers are orbits of Sn . – Under the morphism OX (n) → (pn )∗ OX n , the sheaf OX (n) is identified with the ring subsheaf of (pn )∗ OX n consisting of G-invariant elements. When X is a smooth curve, X (n) is smooth. However, this is no longer true in higher dimensions; when X is a smooth surface, the punctual Hilbert schemes furnish beautiful resolutions of singularities. Let f : X → Y be a morphism of quasi-projective k-varieties. It induces a morphism fn : X n → Y n which is equivariant with respect to the action of Sn on both sides. Passing to the quotient, we get a morphism f (n) : X (n) → Y (n) . (1.1.2). — When K is an algebraically closed field extension of k, it follows from the definition of the symmetric power that X (n) (K) is the set of unordered n-tuples of K-points on X, or, equivalently, effective zero-cycles of degree n on X ⊗k K. More generally, when K is a perfect field, it follows by Galois descent that X (n) (K) identifies with the set of effective zero-cycles of degree n on X ⊗k K. The situation is more delicate when K is imperfect, as will be illustrated in example 1.1.4. If X is a smooth projective curve over k, then X (n) represents the functor that sends a k-scheme T to the set of effective relative Cartier divisors of degree n on the T -scheme X ×k T , by (SGA IV3 , XVII, 6.3.9). In particular, we can identify X (n) (K) with the set of effective zero-cycles of degree n on X ⊗k K, for every field extension K of k. We will use this description below to compute the classes of symmetric powers of curves in the Grothendieck ring of varieties. Example 1.1.3. — Let σ1 , . . . , σn be the elementary symmetric polynomials, defined by  σm (T1 , . . . , Tn ) = Ti1 . . . Tim . i1 2g − 2. Then for every x ∈ X, the dimension of H 0 (X × {x}, Lx ) equals n + 1 − g by the Riemann–Roch theorem. In particular, it is independent of x. Since Picn (X) is reduced, this implies that E is locally free (Mumford 1974, II, §5). We consider the projective bundle π : P(E ∨ ) → Picn (X) associated with the dual of E . We will show that X (n) is isomorphic to P(E ∨ ) over X. Giving a morphism of X-schemes X n → P(E ∨ ) is equivalent to giving a subline bundle M of p∗ E such that the quotient p∗ E /M is locally free. Let M be the subline bundle whose fiber at a point x = (x1 , . . . , xn ) of X n is given by H 0 (X × {p(x)}, Lp(x) (−[x1 ] − . . . − [xn ])) ⊂ H 0 (X × {p(x)}, Lp(x) ). The corresponding morphism X n → P(E ∨ ) factors through a morphism of kschemes X (n) → P(E ∨ ). This is an isomorphism: its inverse is the morphism P(E ∨ ) → X (n) that corresponds to the relative effective Cartier divisor on

§ 1. KAPRANOV’S MOTIVIC ZETA FUNCTION

369

P(E ∨ )×X whose fiber over a point y of P(E ∨ ) is the divisor of any generator of the line in E |π(y) corresponding to y. As a consequence, we have (1.1.10.1)

e(X (n) ) = e(Pn−g ) e(Picn (X)) = e(Pn−g ) e(J)

for all integers n such that n > 2g − 2. Now, assume that n = 2g − 2. Let x be a point of Picn (X), and let Lx be the corresponding degree n line bundle on X × {x}. By the Riemann–Roch theorem, one has dim H 0 (X × {x}, Lx ) = n + 1 − g + dim H 1 (X × {x}, Lx ). Let ωX be the canonical line bundle of X; we will slightly abuse notation by using the same symbol for its pullback to X × {x}. Applying Serre duality, H 1 (X × {x}, Lx )  H0 (X × {x}, ωX ⊗ Lx∨ ). The degree of the line bundle ωX ⊗ Lx∨ is equal to 0; consequently, this line bundle has no nonzero section if Lx is not isomorphic to ωX . In that case, we get dim H 0 (X × {x}, Lx ) = n + 1 − g = g − 1. ∼ ωX , we find For Lx = dim H 0 (X × {x}, Lx ) = n + 2 − g = g. This shows that E is locally free of rank g − 1 over Picn (X) {ωX } and that its fiber over ωX has dimension g. The same argument as above proves that the restriction of X (n) over Picn (X) {ωX } is a projective bundle of . In rank g − 2 and that the fiber of X (n) over ωX is isomorphic to Pg−1 k particular, one has (1.1.10.2)

e(X (2g−2) ) = e(Pg−2 )(e(J) − 1) + e(Pg−1 ) = e(Pg−2 ) e(J) + Lg−1 .

A similar computation yields some interesting relations between the classes e(X (n) ) for small values of n. Denote by F the pushforward to Picn (X) of the line bundle L ∨ ⊗ pr∗X ωX on X × Picn (X). We choose a partition Π of Picn (X) into integral subvarieties V such that the restriction of E and F to V are locally free, and we denote their ranks by r(V ) and s(V ), respectively. Thus for every point x in V , we have r(V ) = h0 (X × {x}, Lx ) and s(V ) = h0 (X × {x}, ωX ⊗ Lx∨ ). Then by the same argument as above, we deduce the following formulas in K0 (Vark ):  e(X (n) ) = e(Pr(V )−1 ) e(V ), V ∈Π

e(X

(2g−2−n)

)=



e(Ps(V )−1 ) e(V ).

V ∈Π

By the Riemann–Roch theorem and Serre duality, one has, for every stratum V in Π and every x ∈ V , the relation s(V ) = h0 (X × {x}, ωX ⊗ Lx∨ )

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= h1 (X × {x}, Lx ) = h0 (X × {x}, ωX ⊗ Lx∨ ) − deg(Lx ) + g − 1 = r(V ) + g − 1 − n. Therefore, e(X (2g−2−n) ) =



(1 + L + · · · + Lr(V )+g−2−n ) e(V )

V ∈Π



= Lg−1−n

(1 + L + · · · + Lr(V )−1 ) e(V )

V ∈Π

+



(1 + · · · + Lg−2−n ) e(V )

V ∈Π g−1−n

=L

e(X (n) ) + e(Pg−2−n )



e(V )

V ∈Π

(1.1.10.3)

= Lg−1−n e(X (n) ) + e(Pg−2−n ) e(J),

since the members of Π form a partition of Picn (X)  J into subvarieties. Proposition 1.1.11 (Totaro, see Göttsche 2001, lemma 4.4) Let X be a quasi-projective k-variety and let m, n  1 be positive integers. Then e+,uh ((X × Am )(n) ) = e+,uh (X (n) )Lmn in the Grothendieck semiring K0+,uh (Vark ) of varieties up to universal homeomorphisms. Proof. — By induction, it suffices to prove this formula when m = 1. We may assume that X is reduced, because (X (n) )red  ((Xred )(n) )red . Let p : (X ×A1 )(n) → X (n) be the obvious morphism. Let α = (n1 , . . . , nr ) be a partition of the integer n, that is, a finite non-decreasing sequence of positive integers such that n1 + n2 + · · · + nr = n. With this partition, we can associate a closed immersion X r → X n that maps each r-tuple (x1 , . . . , xr ) to the n-tuple (x1 , . . . , x1 , . . . , xr , . . . , xr ). . /0 1 . /0 1 n1 times

nr times (n)

We denote the image of this closed immersion by Δα , and we denote by Xα the image of Δα in X (n) minus the images of the closed subsets Δα for all partitions α = α of n. Since the quotient morphism X n → X (n) is closed, (n) Xα is a locally closed subset of X (n) . We endow it with its induced reduced structure. (n) The geometric points on Xα correspond to zero-cycles of the form n1 x1 + · · · + nr xr , where x1 , . . . , xr are distinct geometric points of X. When α runs along all partitions of the integer n, these locally closed subsets are pairwise disjoint, and their union is X (n) . We shall show that for every partition α,

§ 1. KAPRANOV’S MOTIVIC ZETA FUNCTION

371

one has the formula e+,uh (p−1 (Xα(n) )) = e+,uh (Xα(n) )Ln . The required result then follows from summing both sides over all the partitions α of n. Fix a partition α = (n1 , . . . , nr ) of n. Let X∗r be the open subscheme of (X r )red consisting of r-tuples with pairwise distinct entries. The action on X r of the symmetric group Sr induces a free action on X∗r . Let Sα be the subgroup of Sr consisting of permutations σ such that nσ(i) = ni for every i ∈ {1, . . . , r}; it acts freely on X∗r , so that the quotient morphism (n) q : X∗r → X∗r /Sα is finite, surjective, and étale. The morphism X∗r → Xα  defined by (x1 , . . . , xr ) → ni xi factors through a finite surjective mor(n) phism X∗r /Sα → Xα , which is a bijection on geometric points, and thus a universal homeomorphism. Consequently, if we write Yα = (X ×A1 )(n) ×X (n) (X∗r /Sα ), then it suffices to show that e+,uh (Yα ) = e+,uh (X∗r /Sα )Ln in K0+,uh (Vark ). r Let Sα act on i=1 (A1 )(ni ) by permutation of the factors. We endow  r X∗r × i=1 (A1 )(ni ) with the diagonal action. Then the projection morphism X∗r ×

r 

(A1 )(ni ) → Xr∗

i=1

onto the first factor is Sα -equivariant. Recall that by example 1.1.3, we m every integer m  1. These furnish have isomorphisms (A1 )(m) r  A1 ,(nfor r an isomorphism of X∗ × i=1 (A ) i ) with X∗r × An , which transforms the action of the group Sα into a linear action on X∗r × An over X∗r . Passing to the quotient modulo Sα , we obtain a rank n vector bundle (X∗r

×

r 

(A1 )(ni ) )/Sα → Xr∗ /Sα .

i=1

A priori, this vector bundle is only locally trivial for the étale topology; however, Hilbert’s theorem 90 then asserts that it is locally trivial already for the Zariski topology. In particular, e((X∗r ×

r 

(A1 )(ni ) )/Sα ) = e(X∗r /Sα )Ln .

i=1

Now consider the natural morphism of Xr∗ /Sα -schemes (X∗r ×

r 

(A1 )(ni ) )/Sα → (X∗r /Sα ) ×X (n) (X × A1 )(n) = Yα

i=1

This is a finite morphism that induces a bijection on geometric points, and thus a universal homeomorphism, so that e+,uh (Yα ) = e+,uh (X∗r /Sα )Ln in K0+,uh (Vark ).

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Example 1.1.12 (Galkin and Shinder 2014). — Assume that the characteristic of k is different from 2 (otherwise, the identities below need to be considered in the Grothendieck ring of varieties modulo universal homeomorphisms). Let V be a k-vector space of dimension d+2, and let f ∈ Sym3 (V ∨ ) be a cubic form on V which is not divisible by the square of a linear form. Let X be the cubic hypersurface of P(V )  Pd+1 defined by the equation f = 0; it is reduced. Let FX be the variety of lines in X: this is the reduced closed subscheme of the Grassmann variety G(2, V ) parameterizing lines ⊂ P(V ) such that ⊂ X. Let Xsing be the complement of the ksmooth locus of X. We will prove that the following relation holds in the Grothendieck ring K0 (Vark ): (1.1.12.1)

e(X (2) ) = (1 + Ld ) e(X) + L2 e(FX ) − Ld e(Xsing ).

Let W ⊂ X × G(2, V ) be the closed subscheme parameterizing pairs (x, ) such that x ∈ , and let p : W → X be the first projection. For every x ∈ X, the fiber p−1 (x) is the space of lines in Pd+1 κ(x) containing x, hence is d isomorphic to the projective space Pκ(x) . This implies that p is a piecewise trivial fibration with fiber Pd . In particular, one has e(W ) = e(Pd ) e(X).

(1.1.12.2)

Let Z be the closed subscheme of W consisting of the points (x, ) ∈ W such that ∈ FX , and let q : Z → FX be the second projection. For every ∈ FX , the fiber q −1 ( ) is nothing but the line , hence is isomorphic to P1κ() . Consequently, q is a piecewise trivial fibration with fiber P1 , so that e(Z) = e(P1 ) e(FX ).

(1.1.12.3)

Let T ⊂ W be the closed subscheme of pairs (x, ) ∈ W such that either x ∈ Xsing or is tangent to X at x. The fiber at a point x ∈ X of the first projection T → X is thus a projective space of dimension d if x is singular, and a projective space of dimension d − 1 if x is smooth. Consequently, e(T ) = e(Pd−1 ) e(X (1.1.12.4)

Xsing ) + e(Pd ) e(Xsing )

= e(Pd−1 ) e(X) + Ld e(Xsing ).

Observe that if a line belongs to FX , then (x, ) ∈ T for every point x ∈ ; in other words, Z ⊂ T . Let then W  = W Z and T  = W  ∩ T = T Z. Let Z  be the closed subset of W  consisting of pairs (x, ) such that the intersection X ∩ is a divisor on of the form [x] + 2[y], for some y ∈ , and let W  = W  Z  . With this notation, the map Z  → T ,

(x, ) → (y, )

is an isomorphism, so that e(Z  ) = e(T  ) = e(T ) − e(Z) (1.1.12.5)

= e(Pd−1 ) e(X) + Ld e(Xsing ) − e(P1 ) e(FX ).

§ 1. KAPRANOV’S MOTIVIC ZETA FUNCTION

373

(2)

Let U = X∗ be the open subset of X (2) consisting of unordered pairs {x, y} of points in X such that x = y. Then U is an open subset of X (2) . The complement of U is the image of the diagonal morphism X → X (2) . This diagonal morphism is a finite universal injection, and it is separable because the characteristic of k is different from 2. Thus it is a closed immersion, and (1.1.12.6)

e(X (2) ) = e(U ) + e(X).

Consider the morphism ϕ : U → G(2, V ) which maps an unordered pair {x, y} of distinct points of X to the line (xy). Let Y = ϕ−1 (FX ) be the closed subset of U consisting of pairs such that the line (xy) is contained in X. Let ∈ FX . (2) The fiber ϕ−1 ( ) is the space (P1κ() )∗ of unordered pairs of distinct points in  P1κ() . Consequently, the morphism ϕ induces a piecewise trivial (2)

fibration from Y to FX with fiber (P1 )∗ . This implies (2)

e(Y ) = e((P1 )∗ ) e(FX ) = (e((P1 )(2) ) − e(P1 )) e(FX ) (1.1.12.7)

= L2 e(FX ).

Let U  = U Y , and let C be the reduced closed subscheme of U  × X whose points correspond to tuples ({x, y}, z) such that x, y, and z are collinear. Let ψ : C → W  be the morphism that maps ({x, y}, z) to (z, ), where is the line through x and y. Then the morphism ψ and the projection morphism C → U  are both bijective on points with coordinates in a field extension of k. Thus, they are piecewise isomorphisms, and e(U  ) = e(W  ). Putting everything together, we find: e(X (2) ) = e(X) + e(U ) = e(X) + e(Y ) + e(U  ) = e(X) + L2 e(FX ) + e(W  ) = e(X) + L2 e(FX ) + e(W ) − e(Z) − e(Z  ) = e(X) + L2 e(FX ) + e(Pd ) e(X) − e(P1 ) e(FX ) − e(Pd−1 ) e(X) − Ld e(Xsing ) + e(P1 ) e(FX ) = (1 + Ld ) e(X) + L2 e(FX ) − Ld e(Xsing ), as claimed. To get a grasp on the interest of relation (1.1.12.1), let us now apply the motivic measure given by the Euler–Poincaré polynomial. For simplicity, we assume that d = 2 and that X is a smooth cubic surface in P3 . The cohomology of hypersurfaces in Pd+1 is known; in particular, one has EP(X) = 1 + 7t2 + t4 . Similarly, the cohomology of the symmetric product X (2) is computed via the Künneth formula (see §1.1.5.2, as well as example 1.2.8 below). In the

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present case, this gives 1 EP(X (2) ) = 1 + 7t2 + ( 8 · 7 + 1)t4 + 7t6 + t8 = 1 + 7t2 + 29t4 + 7t6 + t8 . 2 Since EP(L) = t2 , we get t4 EP(FX ) = (1 + 7t2 + 29t4 + 7t6 + t8 ) − (1 + t4 )(1 + 7t2 + t4 ) = 27t4 , whence the relation (1.1.12.8)

EP(FX ) = 27.

If k is algebraically closed, this implies that (FX )red is the union of 27 points; there are thus 27 lines on a smooth cubic surface in P3 . Of course, Formula (1.1.12.1) furnishes more precise cohomological information about FX . We refer to the original paper of Galkin and Shinder (2014) for further examples. For an application to the existence of lines on cubic hypersurfaces defined over finite fields, see also Debarre et al. (2017). 1.2. Definition of Kapranov’s Motivic Zeta Function Definition 1.2.1 (Kapranov 2000). — Let X be a quasi-projective kvariety. Its motivic zeta function is the formal power series with coefficients in K0 (Vark ) given by ∞  e(X (n) )tn . (1.2.1.1) ZX (t) = n=0

For every motivic measure μ, one also defines ∞  μ(X (n) )tn . (1.2.1.2) ZX,μ (t) = n=0

In particular, we write ZX,uh (t) for the image of ZX (t) in the ring K0uh (Vark )[[t]]. Proposition 1.2.2. — Let X be a quasi-projective k-variety, let Y be a closed subscheme of X, and let U = X Y be its complement. Then, ZX (t) = ZU (t)ZY (t). Proof. — This follows from proposition 1.1.7. Corollary 1.2.3. — There exists a unique additive invariant on Vark with values in the multiplicative group 1 + tK0 (Vark )[[t]] that maps a quasiprojective k-variety X to its motivic zeta function ZX (t). In other words, there exists a unique group homomorphism K0 (Vark ) → 1 + tK0 (Vark )[[t]] that maps the class of a quasi-projective k-variety X to ZX (t). We can use this homomorphism to define the motivic zeta function for arbitrary kvarieties.

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375

Remark 1.2.4. — Let X be a quasi-projective k-variety. By proposition 1.1.11, one has euh ((X × A1 )(n) = euh (X (n) ) × Ln . Consequently, ZX×A1 ,uh (t) = ZX,uh (Lt). By proposition 1.2.2, it follows that the same formula holds for every kvariety; more generally for every class ξ ∈ K0 (Vark ), one has ZLξ,uh (t) = Zξ,uh (Lt). Example 1.2.5. — We have seen in example 1.1.3 that (A1 )(n)  An , so that ∞  1 . ZA1 (t) = Ln tn = 1 − Lt n=0 More generally, it follows from example 1.1.11 that for every k-variety X, one has ZX×Am ,uh (t) = ZX,uh (Lm t). In particular, ZAm ,uh (t) =

∞ 

Lmn tn =

n=0

1 . 1 − Lm t

For any positive integer m, the projective space Pm has an open subset isomorphic to Am , with a complement projective space Pm−1 of lower dimension. By induction, one deduces from proposition 1.2.2 that 1 . ZPm ,uh (t) = (1 − t)(1 − Lt) . . . (1 − Lm t) Example 1.2.6. — Let X be an elliptic curve. By example 1.1.9, one has e(X (n) ) = e(Pn−1 ) e(X) for every positive integer n. Consequently,  (1 + L + · · · + Ln−1 ) e(X)tn ZX (t) = 1 + n1

= 1 + e(X)

∞  m=0

= 1 + e(X)

Lm

∞ 

tn = 1 + e(X)

n=m+1

∞  m=0

Lm

tm+1 1−t

t 1 + (e(X) − 1 − L)t + Lt2 = . (1 − t)(1 − Lt) (1 − t)(1 − Lt)

Example 1.2.7. — Assume that k is a finite field; then we can apply the motivic measure given by the cardinality of the set of rational points. The points of X (n) (k) are the effective 0-cycles of degree n on X; any such divisor is a sum of closed points of X of degrees n1 , . . . , nr , where n1 + · · · + nr = n. Therefore, one has ZX,Card (t) =

∞  n=0

Card(X (n) (k))tn =

 x∈|X|0

1 = ζX (t), 1 − tdeg(x)

the Hasse–Weil zeta function of X. (We have written |X|0 for the set of closed points of X and deg(x) for the degree of a closed point.) By a fundamental theorem of Dwork (1960), one knows that ζX (t) is a rational function.

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Example 1.2.8 (Macdonald 1962; Burillo 1990) Let k be the field of complex numbers, and choose for the motivic measure μ the Hodge realization χHdg with values in K0 (pHS), the Grothendieck ring of polarizable Hodge structures. For every pure polarizable Hodge structure M , let us define power series λM and σM with coefficients in the ring K0 (pHS) by:   [Altp (M )](−t)p and σM (t) = [Symp (M )]tp . λM (t) = p0

p0 

Their constant term is 1. If M and M are pure polarizable Hodge structures, one has λM ⊕M  (t) = λM (t)λM  (t)

and σM ⊕M  (t) = σM (t)σM  (t).

Consequently, the maps M → λM and M → σM extend to group morphisms from K0 (pHS) to the multiplicative group 1 + tK0 (pHS)[[t]]. Since Altp (M ) = 0 for p large enough, λM is a polynomial. Moreover, one has  [Symp (M ) ⊗ Altq (M )](−1)p tp+q = 1, λM (t)σM (t) = p0 q0

because of the decomposition of the tensor product Symp (M ) ⊗ Altq (M )  S(p+1,1q−1 ) (M ) ⊕ S(p,1q ) (M ) using Schur functors; we refer to section 4 of Heinloth (2007) for more details. This implies that σM (t) = 1/λM (t) is a rational function. Let now X be a projective smooth complex variety. We have seen that the total cohomology ring  i Hsing,c (X (n) (C), Q) i0

is isomorphic to the Sn -invariant part in ⎛ ⎞⊗n   i i Hsing,c (X n (C), Q)  ⎝ Hsing,c (X(C), Q)⎠ . i0

i0

To shorten notation, we will write H(X) for Hsing,c (X(C), Q). Taking the direct sum over n, this gives an isomorphism of graded algebras: ⎛ ⎞Sn    ⊗n ⎝ ⎠ H i (X (n) )  H i (X) n0 i0

n0

 Sym

i0

 ∗ i0 i even

 i H i (X) ⊗ Alt∗ H (X) , i0 i odd

§ 1. KAPRANOV’S MOTIVIC ZETA FUNCTION

377

because under the Künneth isomorphism, transpositions act by antihomomorphisms on the odd part of cohomology. Consequently,  (−1)i [H i (X (n) )]tn ZX,μ (t) = n0 i0

=



i0 i even

(1.2.8.1)

=





σ[H i (X)] (t)

λ[H i (X)] (t)

i0 i odd

(−1)i+1 λ[H i (X)] (t) .

i0

We thus conclude that ZX,μ (t) is a rational function. Since K0 (VarC ) is generated by classes of projective smooth complex varieties, it follows that the same holds for every complex variety. In particular, the image of ZX (t) under the Hodge–Deligne characteristic, or the Euler–Poincaré polynomial, or the Euler characteristic, is a rational function. Let us make the last case explicit. If Eu denotes the Euler characteristic, observe that for every pure Hodge structure M of rank d, one has  d tn = (1 − t)d , Eu(λM (t)) = n n0

and Eu(σM (t)) = 1/(1 − t)d . Consequently, (1.2.8.2) ZX,Eu (t) = Eu(ZX,μ (t)) =



(1 − t)(−1)

i+1

bi (X)

=

i0

1 , (1 − t)Eu(X)

a formula due to Macdonald (1962). (1.2.9). — These examples suggest that (for any field k) the motivic zeta function ZX of a k-variety X could be a rational function. As theorems 1.3.1 and 1.4.3 below will show, this holds for curves but not in general. 1.3. Motivic Zeta Functions of Curves Theorem 1.3.1 (Kapranov). — Let X be a smooth, projective, geometrically connected curve of genus g over a field k. Assume that X admits a divisor of degree 1. Then there exists a unique polynomial P ∈ K0 (Vark )[t] of degree 2g such that, in K0 (Vark )[[t]], P (t) . ZX (t) = (1 − t)(1 − Lt)  Moreover, writing P (t) = an tn , one has the following relations: a0 = 1 a2g−n = Lg−n an a2g = Lg .

for 0  n  g

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Remark 1.3.2. — After inverting L, one can rewrite the relations satisfied by the coefficients of P as the equality P (t) = Lg t2g P (1/Lt) in Mk [t]. Consequently, the rational function ZX (t) satisfies the following functional equation: Lg t2g P (1/Lt) (1 − 1/Lt)(1 − 1/t) P (t) = ZX (t). = (1 − t)(1 − Lt)

Lg−1 t2g−2 ZX (1/Lt) = L−1 t−2

Proof. — The uniqueness of such a polynomial P follows from the fact that the polynomial (1 − t)(1 − Lt) is invertible in K0 (Vark )[[t]], since its constant coefficient is invertible. The proof of this theorem is similar to Artin’s proof of the rationality and functional equation of zeta functions of algebraic curves over finite fields. In particular, it relies on the theorem of Riemann–Roch and Serre’s duality theorem. As we have seen in example 1.1.10, these theorems furnish an explicit formula for the symmetric products X (n) , at least when n is sufficiently large. Let J be the Jacobian variety of X. By equations (1.1.10.1) and (1.1.10.2), one has the following relations: if n > 2g − 2,

e(X (n) ) = e(Pn−g ) e(J)

(1.3.2.1) and (1.3.2.2)

e(X (2g−2) ) = e(Pg−2 ) e(J) + Lg−1 .

Therefore, setting e(Pm ) = 0 when m < 0, we have  ZX (t) = e(X (n) )tn =

2g−2 

(e(X (n) ) − e(Pn−g ) e(J))tn +

n=0

Let us set P1 (t) = ZX,1 (t), where

2g−2 n=0

∞ 

e(Pn−g ) e(J)tn .

n=0

(e(X

ZX,1 (t) =

(n)

∞ 

) − e(P

n−g

) e(J))tn . Then, ZX (t) = P1 (t) +

(1 + L + · · · + Ln−g ) e(J)tn

n=g

=

∞ n−g  

Lm e(J)tn

n=g m=0 ∞ 

= e(J)

∞ 

Lm tn

m=0 n=m+g ∞  m m+g

= tg e(J)

L t

m=0

1 1−t

§ 1. KAPRANOV’S MOTIVIC ZETA FUNCTION

= tg e(J)

379

1 . (1 − t)(1 − Lt)

Writing P (t) = (1 − t)(1 − Lt)P1 (t) + tg e(J), we thus have ZX (t) =

(1.3.2.3)

P (t) , (1 − t)(1 − Lt)

so that ZX (t) is a rational function, as claimed. By construction, deg(P1 )  2g − 2, so that deg(P )  2g. Write P1 (t) = b0 + b1 t + · · · + b2g−2 t2g−2 and P (t) = a0 + a1 t + · · · + a2g t2g , where a0 , . . . , a2g , b0 , . . . , b2g−2 are elements of K0 (Vark ). By definition, one has  e(X (n) ) if n  g − 1, bn = e(X (n) ) − e(Pn−g ) if g  n  2g − 2. If g = 0, we conclude that P (t) = 1, which proves the required relations in this case. Let us now assume that g  1. By equation (1.1.10.3), one has, for every integer n  g − 1, b2g−2−n = e(X (2g−2−n) ) − e(Pg−2−n ) e(J) = Lg−1−n e(X (n) ) = Lg−1−n bn . In particular, b2g−2 = Lg−1 b0 = Lg−1 . By the definition of the polynomial P (t), we have aj = bj − bj−1 (L + 1) + bj−2 L / {0, . . . , 2g − 2}. This yields for every j =  g, where we set bj = 0 for j ∈ a0 = 1, a2g = Lg , and a2g−n = b2g−n − (L + 1)b2g−n−1 + b2g−n−2 L = Lg−n (Lbn−2 − (L + 1)bn−1 + bn ) = Lg−n an for all 0  n < g. Remark 1.3.3. — It follows from the proof of theorem 1.3.1 that P (1) = e(J). Moreover, with the notation there, if k ⊂ C then χHdg (an ) is pure of weight n, for every n ∈ {0, . . . , 2g}. More precisely, example 1.2.8 shows that 2g n (−1)n [Hsing (X(C), Q)]tn ZX,Hdg (t) = n=0 , (1 − t)(1 − [Q(1)]t) so that n χHdg (an ) = (−1)n [Hsing (X(C), Q)].

A similar formula holds for χét (an ).

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Remark 1.3.4. — Let X be a projective irreducible smooth curve of genus 0 over a field k. Let d be the index of X, the smallest positive integer such that X admits a divisor of degree d. For every n  0, X (n) is a Severi–Brauer variety, that is, it becomes isomorphic to Pn after base change to an algebraic closure of k. Consequently, X (n) is isomorphic to Pn if and only if it has a k-rational point, which is equivalent to the property that d divides n. Litt (2015) gives a formula that computes explicitly the class e(X (n) ). In particular, this formula implies that the power series (1 − td )(1 − Ld td )ZX (t) is a polynomial. On the other hand, when d > 1 and k has characteristic zero, it is not true that (1 − t)(1 − Lt)ZX (t) is a polynomial. Indeed, one has  (1 − t)(1 − Lt)ZX (t) = (e(X (n) ) − (1 + L) e(X (n−1) ) + L e(X (n−2) ))tn n0

and the coefficient of t is congruent to e(X (n) ) − e(X (n−1) ) modulo L. If d divides n, then the classes of X (n) and X (n−1) are different modulo L by corollary 2/6.1.10, since X (n) has a k-rational point, but X (n−1) does not. n

1.4. Motivic Zeta Functions of Surfaces Proposition 1.4.1. — Let k be an algebraically closed field. Let X and Y be proper smooth connected surfaces over k. If X and Y are birational, then there is an integer m ∈ Z such that e(X) − e(Y ) = mL in K0 (Vark ). In particular, ZX (t) is rational if and only if ZY (t) is rational. Proof. — If Y is the blow-up of X at a closed point, then e(Y ) = e(X) + L. Consequently, the first result follows from the fact that every birational isomorphism of proper smooth connected surfaces over k is a composition blowing ups and blowing downs of proper smooth connected surfaces at closed points. Let thus m ∈ Z such that e(Y ) = e(X) + mL. By example 1.2.5, we have 1 ZY (t) = ZX (t)ZA1 (t)m = ZX (t) . (1 − LT )m In particular, ZY (t) is rational if and only if ZX (t) is rational, as claimed. Corollary 1.4.2. — If X is a rational proper smooth surface over an algebraically closed field k, then ZX,uh is rational. Proof. — Since ZP2 ,uh (t) is rational (example 1.2.5), the corollary follows immediately from proposition 1.4.1. Theorem 1.4.3 (Larsen and Lunts 2004). — Let k be an algebraically closed field of characteristic zero. Let X be a projective smooth connected surface over k. Then ZX (t) is rational if and only if the Kodaira dimension of X equals −∞.

§ 1. KAPRANOV’S MOTIVIC ZETA FUNCTION

381

On the positive side, this theorem asserts that if X is either a rational surface or a ruled surface, over a field of characteristic zero, then ZX (t) is rational. Recall that K0uh (Vark ) = K0 (Vark ) in this case. Consequently, if X is rational, then the assertion follows from corollary 1.4.2. On the other hand, if X is a ruled surface, it is birational to a product C × P1 , where C is a projective smooth connected curve over k. We may in fact assume that X = C × P1 ; in this case, remark 1.2.4 asserts that ZX (t) = ZC (Lt), so that the assertion follows from theorem 1.3.1. On the negative side, this theorem asserts that ZX (t) is never rational in any other case. We will only prove here the following weaker statement from Larsen and Lunts (2003). Proposition 1.4.4. — If h0 (X, ωX )  2, then there exists a motivic measure μ with values in a field such that ZX,μ (t) is not rational. We first define the motivic measure that shall be used in the proof of proposition 1.4.4. Lemma 1.4.5. — Let X and Y be birational smooth proper k-varieties. For every integer m, one has 0 m h0 (X, Ωm X ) = h (Y, ΩY ).

Proof. — There exists an open subset U of X such that codim(X U )  2 and a birational morphism u : U → Y . For every integer m, this induces a mor0 m U)  2 phism u∗ : H 0 (Y, Ωm Y ) → H (U, ΩX ); it is injective. Since codim(X m and X is smooth, ΩX is locally free, and every section of Ωm X on U extends uniquely to X. We thus obtain an injective morphism α from H 0 (Y, Ωm Y ) 0 m 0 m to H 0 (X, Ωm X ). Consequently, h (X, ΩX )  h (Y, ΩY ), and the other inequality follows by symmetry. (1.4.6). — For every smooth and projective k-variety X, we let 

dim(X)

(1.4.6.1)

ψ(X) =

m h0 (X, Ωm X )u = HD(X)(u, 0).

m=0

By lemma 1.4.5, this is a birational invariant of projective smooth k-varieties. Moreover, if X and Y are projective and smooth, one has ψ(X × Y ) = ψ(X)ψ(Y ). Let C be the submonoid of (Z[T ], ×) consisting of polynomials with positive leading coefficient; one has ψ(X) ∈ C for any k-variety X. The invariant ψ extends uniquely to a motivic measure, still denoted ψ, with values in the monoid algebra Z[C]. As an example, one has ψ(L) = ψ(P1 ) − ψ(1) = (1) − (1) = 0, which reflects the fact that ψ is a birational invariant. Let P be the set of irreducible polynomials in Z[T ] with positive leading coefficient. The ring Z[T ] is a unique factorization domain, and the polynomial 1 is its only unit with positive leading coefficient. Consequently, associating to a polynomial in C its unique decomposition in irreducible factors such that each factor has positive leading coefficient, we construct an

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CHAPTER 7. APPLICATIONS

isomorphism from the monoid C to the monoid (N(P ) , +). This implies that the monoid algebra Z[C] is isomorphic to the ring of polynomials in an infinite, countable, set of variables. In particular, it is an integral domain; we write H for its field of fractions and μ for the associated H -valued motivic measure. (1.4.7). — Let X be an irreducible k-variety of dimension d, and let Y be a proper smooth k-variety that is birational to X. By lemma 1.4.5, we may define pg (X) = h0 (Y, ωY ), because this value is independent of Y . Lemma 1.4.8. — Let X1 , . . . , Xm , Y be irreducible k-varieties of dimension m  d; assume that there exist integers n1 , . . . , nm such that ψ(Y ) = i=1 ni ψ(Xi ) in H . If dim(Y ) = d and pg (Y ) = 0, then there exists an index i such that dim(Xi ) = d and pg (Y ) = pg (Xi ). Proof. — Since k has characteristic zero, we can write the class of Y projective variety Y  , birational to Y , in K0 (Vark ) as the class of a smooth  plus a linear combination aj Yj of classes of smooth projective varieties of strictly smaller dimension. The analogous statement holds for each of the Xi :  , and integers ai,j , such that there are smooth projective varieties Xi and Yi,j   ) < dim(Xi ), and e(Xi ) = e(Xi ) + ai,j Yi,j . dim(Xi ) = dim(Xi ), dim(Yi,j We thus obtain a relation m     ni ψ(Xi ) + ni ai,j ψ(Yi,j )− aj ψ(Yj ). ψ(Y  ) = i=1

i,j

j 

By definition of the monoid algebra, the term ψ(Y ) appears on the righthand side, so that there exists either an integer i ∈ {1, . . . , m} such that  ), or an integer j such ψ(Y  ) = ψ(Xi ), or a pair (i, j) such that ψ(Y  ) = ψ(Yi,j    that ψ(Y ) = ψ(Yj ). By hypothesis, ψ(Y ) has degree d = dim(Y  ) and its  ) < d for every pair (i, j), leading coefficient is pg (Y  ) = pg (Y ). Since dim(Yi,j   ), resp. and dim(Yj ) < d for every integer j, their associated polynomial ψ(Yi,j   ψ(Yj ), has degree < d. We thus must be in the first case: ψ(Y ) = ψ(Xi ) for some i ∈ {1, . . . , m}. This implies that dim(Xi ) = dim(Xi ) = d and pg (Xi ) = pg (Xi ) = pg (Y  ) = ph (Y ), as was to be shown. Proposition 1.4.9 (Göttsche). — Let X be a connected smooth projective surface. Then, for any integer n  1,  pg (X) + n − 1 pg (X (n) ) = . n Proof. — We only sketch the proof, referring to (Larsen and Lunts 2004, proposition 7.2) for more details. The variety X (n) is singular, but one can construct a resolution of singularities in the following way. Denote by X [n] , the Hilbert scheme parameterizing closed subschemes of length n on X. Fogarty (1968) proved that it is smooth and connected and that the Hilbert–Chow morphism X [n] → X (n) , sending a length n subscheme to its associated 0cycle, is birational; it is thus a resolution of singularities. It is also known,

§ 1. KAPRANOV’S MOTIVIC ZETA FUNCTION

383

see for example (Beauville 1983, proposition 5), that this resolution of singularities is crepant. Consequently, we have isomorphisms H 0 (X [n] , ωX [n] )  H 0 (X n , ωX n )Sn  (H 0 (X, ωX )⊗n )Sn . This yields the indicated value for pg (X (n) ). (1.4.10). — We now conclude the proof of proposition 1.4.4. Assume that pg (X)  2 and that the image ZX,μ (t) of ZX (t) in H [[t]] is rational. Let P be a monic polynomial in H [[t]] such that P (t)ZX,μ (t) is a polynomial, and let m = deg(P ). Then, for n large enough, the columns of the matrix ⎛ ⎞ ψ(X (n) ) ψ(X (n+1) ) ... ψ(X (n+m) ) ⎜ ψ(X (n+1) ) ψ(X (n+2) ) . . . ψ(X (n+m+1) )⎟ ⎜ ⎟ ⎜ ⎟ .. .. .. .. ⎝ ⎠ . . . . ψ(X (n+m) )

ψ(X (n+m+1) )

ψ(X (n+2m) )

...

are linearly dependent, so that its determinant vanishes. This determinant is computed by the formula  m m     (n+i+σ(i)) (n+i+σ(i)) , ε(σ) ψ(X )= ε(σ)ψ X i=0

σ∈Sm+1

i=0

σ∈Sm+1

where ε(σ) ∈ {±1} is the signature of the permutation σ ∈ Sm+1 acting on {0, . . . , m}. For every σ ∈ Sm+1 , one has  m m   (n+i+σ(i)) = X 2(n + i + σ(i)) = 2(m + 1)n + 2m(m + 1) dim i=0

i=0

= 2(m + 1)(m + n), and pg (

m 

X (n+i+σ(i)) ) =

i=0

m   p − 1 + n + i + σ(i) p−1

i=0

,

where p = pg (X). Moreover, the variety X (n) ×X (n+2) ×. . . X (n+2m) appears only once, namely, for σ = id. By lemma 1.4.8, it follows that there exists a nontrivial permutation σ ∈ Sm+1 such that pg (

m 

X (n+i+σ(i)) ) = pg (

i=0

that is:

m   p − 1 + n + i + σ(i) i=0

p−1

m 

X (n+2i) ),

i=0

m   p − 1 + n + 2i = . p−1 i=0

Using that p  2, we rewrite this equality as m p−2   i=0 j=0

(p − 1 + n + i + σ(i) − j) =

m p−2   i=0 j=0

(p − 1 + n + 2i − j).

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CHAPTER 7. APPLICATIONS

Both sides of this equality are polynomials in n, and at least one permutation σ must achieve this equality for infinitely many values of n. This implies that these polynomials are equal, and we get σ = id, which is absurd. Consequently, ZX,μ (t) is not rational, as was to be shown. Remark 1.4.11. — Observing that the motivic measure μ of proposition 1.4.4 satisfies μ(L) = 0, Denef and Loeser (2004, conjecture 7.5.1) were led to ask the following question: Is the image of ZX (t) in Mk [[T ]] rational for every k-variety X—that is, after inverting L? To the best of our knowledge, this question is still open.

1.5. Rationality of Kapranov’s Zeta Function of Finite Dimensional Motives (1.5.1). — Let M ∈ Motrat,Q be a Chow motive. One associates with the motive M its motivic zeta function by setting  (1.5.1.1) ZM (t) = [Symn (M )]tn ∈ K0 (Motrat,Q )[[t]], n0 n

where Sym (M ) is the nth symmetric product of M ; see §2/5.3.7. The ! relations Sym (M ⊕ N ) = r+s= Symr (M ) ⊗ Syms (N ) translate into the relation ZM ⊕N (t) = ZM (t)ZN (t). Consequently, there exists a unique group morphism K0 (Motrat,Q ) → (1 + K0 (Motrat,Q )[[t]], ×),

ξ → Zξ (t)

such that Z[M ] (t) = ZM (t) for every M ∈ Motrat,Q . (1.5.2). — Recall from proposition 2/5.3.3 that there exists a motivic measure χrat,Q : K0 (Vark ) → K0 (Motrat,Q ), with values in the Grothendieck group of Chow motives with Q-coefficients, which associates with the class of a projective smooth k-variety X the class of its motive Mrat,Q (X) in K0 (Motrat,Q ). By del Baño Rollin and Navarro Aznar (1998, corollary 2.4), one has moreover χrat,Q (e(X (n) )) = [Symn (Mrat,Q (X))], for every n ∈ N∗ and every smooth projective k-variety X. Thus, we find: χrat,Q (ZX (t)) = ZMrat,Q (X) (t) in the ring K0 (Motrat,Q )[[t]]. Proposition 1.5.3 (André 2004, proposition 13.3.3.1) Let M be a Chow motive in Motrat,Q . If M is finite dimensional

§ 1. KAPRANOV’S MOTIVIC ZETA FUNCTION

385

(see §2/5.3.12), then there exist polynomials P, Q ∈ K0 (Motrat,Q )[t] with constant coefficient equal to 1 such that ZM (t) = P (t)/Q(t), in the ring K0 (Motrat,Q )[[t]]. Proof. — By assumption, there exists an evenly finite dimensional motive M + and an oddly finite dimensional motive M − such that M ∼ = M − ⊕ M + . Then we have ZM (t) = ZM + (t)ZM − (t), so that it suffices to prove the assertion for M + and M − . By the definition of a finite dimensional motive, there exists a positive integer m such that Symm (M − ) = Altm (M + ) = 0. It immediately follows that ZM − (t) is a polynomial with constant term 1 and degree at most m. The rationality of the first factor ZM + (t) relies on the following formula: ⎞−1 ⎛  ZM + (t) = ⎝ Altn (M + )(−t)n ⎠ , n0

which shows that ZM + (t) is the inverse of a polynomial with constant term 1 and degree at most m. The argument is sketched in André (2004, §13.3.3), and a detailed proof has been given by Ivorra (2014, proposition 11.17). A different argument, already alluded to in example 1.2.8, has been given by Heinloth (2007). Corollary 1.5.4. — Assume that the finite dimensionality conjecture of Kimura and O’Sullivan is valid (2/5.3.12). Then the power series ZM (t) ∈ Motrat,Q [[t]] is rational, for every Chow motive M . Example 1.5.5. — By example 2/5.3.10, we deduce from proposition 1.5.3 that ZMrat,Q (X) (t) is rational when X is a smooth and projective curve, a smooth and projective ruled surface, or an abelian variety (of arbitrary dimension). We have seen that, for curves and ruled surfaces, rationality already holds at the level of the Grothendieck ring of varieties but that it fails for abelian surfaces (theorems 1.3.1 and 1.4.3). Remark 1.5.6. — Kahn (2009) has shown that the motivic zeta function of a finite dimensional motive satisfies a functional equation. See also Heinloth (2007) for the case of motives associated with abelian varieties.

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§ 2. VALUATIONS AND THE SPACE OF ARCS 2.1. Divisorial Valuations and Discrepancies (2.1.1). — Let X be an integral separated scheme. Let ν be a valuation on the field k(X); let Rν be its valuation ring, that is, the set of elements a ∈ k(X) such that ν(a)  0. The ring Rν is a local ring, and its residue field is called the residue field of the valuation ν. One says that ν is centered on X, or that ν is a valuation on X, if there exists a point x ∈ X such that OX,x ⊂ Rν . If this holds, the set of points x ∈ X such that ν(a) > 0 for every a ∈ mx is an irreducible Zariski-closed subset of X; its generic point is called the center of ν and denoted by cX (ν). (1) It follows from the definitions that for any open subscheme U of X, the valuation ν is centered on U if and only if cX (ν) ∈ U , and then cU (ν) = cX (ν). Assume, moreover, that U is affine, say U = Spec(A); then cU (ν) is the prime ideal of all a ∈ A such that ν(a) > 0. (2.1.2). — Let f : Y → X be a birational morphism of integral separated schemes. It induces an isomorphism k(X)  k(Y ). Let ν be a valuation on the field k(Y ). If ν is centered on Y , then it is centered on X, and one has cX (ν) = f (cY (ν)). Assume moreover that f is proper; if ν is centered on X, it follows from the valuative criterion of properness that ν is centered on Y as well. In particular, if k is a field, Y is a proper k-scheme, and ν is a valuation of the field k(Y ) that is trivial on k, then ν is centered on Y . (2.1.3) Definition of Divisorial Valuations. — Let k be a field. Let X be an integral separated k-variety, and let k(X) be its field of functions. Let E be a prime divisor in X. Assume that X is normal at the generic point of E. Then the local ring OX,E is a discrete valuation ring, and ordE is the extension of its normalized valuation to its field of fractions k(X). This valuation is centered on X, and its center cX (ordE ) is the generic point of E. More generally, let X  be a normal integral k-variety, and let p : X  → X be a proper birational morphism, inducing an isomorphism of fields k(X)  k(X  ). Let E be a prime divisor in X  . We say that E is a divisor over X. The valuation ordE on k(X  ) induces a valuation, naturally denoted ordE , on the field k(X); this valuation is centered on X, and its center is equal to the generic point of p(E). By definition, a divisorial valuation on X is a valuation ν on k(X) with values in Z which is of the form q ordE , for such a triple (X  , p, E) and a positive integer q  1. The integer q is called the index of ν and denoted by qν .

(1) Some authors define the center to be the irreducible closed subset of which c (ν) is X the generic point.

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(2.1.4). — Let p1 : X1 → X and p2 : X2 → X be two proper birational morphisms, where X1 and X2 are normal, let E1 be a prime divisor in X1 , and let E2 be a prime divisor in X2 . Assume that ordE1 = ordE2 . Assume that there exists a birational morphism q : X2 → X1 such that p1 ◦ q = p2 . Let U1 be the largest open subset of X1 over which q is an isomorphism. Since X1 is normal, codim(X1 U1 , X1 )  2; in particular, U1 ∩ E1 = ∅. The prime divisors E2 ∩ q −1 (U1 ) and q −1 (E1 ∩ U1 ) on q −1 (U1 ) both define the valuation ordE1 on k(X); hence, they are equal. In general, let U be a dense open subset of X above which p1 and p2 are isomorphisms, and let Y ⊂ X1 × X2 be the Zariski closure of the image of U −1  by the map (p−1 1 , p2 ) : U → X1 × X2 ; let then X be the normalization of Y in its field of fractions, and let q1 and q2 the compositions of the normalization morphism with the natural projections from Y to X1 and X2 . Let U1 be the largest open subset of X1 over which q1 is an isomorphism; define U2 accordingly. Then E = q1−1 (E1 ∩ U1 ) is a prime divisor of Y and the valuations ordE and ordE1 on k(X) coincide. This implies that E = q2−1 (E2 ∩ U2 ). This shows that we can identify the set of divisorial valuations on X of index 1 with the projective limit of the set of triples (X  , p, E) where X  is an integral normal k-variety, p : X  → X is a proper birational morphism, and E is a prime divisor in X  , ordered by the dominance relation. (2.1.5). — Let I be a sheaf of fractional ideals on X. For every valuation ν on X, with center x, one defines ν(I ) = inf f ∈Ix ν(f ). If I is the ideal sheaf of a closed subscheme Y , one also writes ν(Y ). Similarly, if I = OX (−D) for some (possibly not effective) divisor D, one denotes it by ν(D). (2.1.6) Discrepancy of a Valuation. — Assume that X is Q-Gorenstein. Let ν be a divisorial valuation on X. Let X  be normal integral k-variety, let p : X  → X be a proper birational morphism, let E be a prime divisor in X  , and let q be an integer such that ν = q ordE . Let kE be the coefficient of E in the relative canonical divisor KX  /X . The integer kν = qkE is called the discrepancy of the valuation ν. It is clear that this definition only depends on ν, and not on the triple (X  , p, E). (2.1.7). — Finally, we mention the following theorem of Zariski (1939); we refer to Artin (1986, theorem 5.2, p. 230) or (Kollár 1996, chapter VI, theorem 1.4, p. 287) for a proof.

Theorem (Zariski). — Let X be an integral separated k-variety, and let ν be a discrete valuation on the field k(X) which is trivial on k and centered on X. a) The valuation ν is divisorial if and only if the residue field of the valuation ring Rν has transcendence degree dim(X) − 1 over k. b) Assume that ν is divisorial. Let (Xn ) be the sequence of k-varieties such that X0 = X, and for n  0, Xn+1 is the normalization of the blow-up

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of Xn along the closure of the center of ν. Then for sufficiently large n  0, the center of ν on Xn is the generic point of a prime divisor. 2.2. Valuations Defined by Algebraically Fat Arcs (2.2.1). — Let k be a perfect field and let X be an integral separated kvariety. Let γ be a point of the arc space of X, and let K = k(γ) be its residue field. Then we can view γ as a k-morphism γ : Spec(K[[t]]) → X. Recall (remark 4/4.3.7) that γ is called algebraically thin if its image is contained in a strict closed subscheme of X; otherwise, the morphism γ is dominant, and one says that γ is algebraically fat. (2) If the arc γ is algebraically fat, then it gives rise to a morphism of fields from k(X) to K((t)); if we compose it with the t-adic valuation, we obtain a discrete valuation: ordγ : k(X)× → Z,

f → ordt (γ ∗ (f )).

Observe that under the morphism γ, the generic point of Spec(K[[t]]) is mapped to the generic point of X, while the closed point is mapped to the center of the valuation ordγ . If C is the closure of {γ} in L∞ (X), then we will also write ordC instead of ordγ . Observe that, for every f in k(X)× , we have ordC (f ) = inf ordγ  (f ) γ

where the infimum is taken over all the algebraically fat arcs γ  in C. Example 2.2.2. — Let X be a smooth separated k-variety, and let E be an −1 irreducible divisor on X. The constructible closed subset θX,0 (E) of L∞ (X) is irreducible; we denote by γ its generic point. Then ordγ coincides with the divisorial valuation ordE on k(X). Definition 2.2.3 (Ishii 2008, Def. 2.8). — Let k be a perfect field and let X be an integral separated k-variety. Let ν be a discrete valuation on k(X). One defines a closed subset WX (ν) of L∞ (X) as the closure of the set of algebraically fat arcs γ ∈ L∞ (X) such that ordγ = ν. Remark 2.2.4. — If ν is not centered on X, then WX (ν) = ∅. More generally, let CX (ν) = {cX (ν)}, and let us show that θ0 (WX (ν)) ⊂ CX (ν).  (ν) be the set of algebraically fat arcs γ such that ordγ = ν, so Let WX  (ν). Let γ ∈ W  (ν). Let U be an open subscheme of X, that WX (ν) = WX X and let f ∈ OX (U ) such that ν(f ) > 0. Then ordt (γ ∗ (f )) = ν(f ) > 0; hence,  (ν)) ⊂ C(ν); since f (γ(0)) = 0, that is, θ0 (γ) ∈ C(ν). This shows that θ0 (WX θ0 is continuous and C(ν) is closed, one thus has  (ν)) ⊂ θ (W  (ν)) ⊂ C(ν), θ0 (WX (ν)) = θ0 (WX 0 X (2) The paper (Ein et al. 2004) calls them simply thin or fat, we explained in remark 4/4.3.7 the change of terminology.

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as claimed. When ν is a divisorial valuation, we will show below that θ0 (WX (ν)) = C(ν). Example 2.2.5. — Let X be a smooth separated k-variety, let E be a smooth prime divisor on X, and let ν = qordE for some positive integer E. Then WX (ν) coincides with the contact locus Contq (X, E) defined in  (ν) is contained in Contq (X, E) ⊂ section 5/3.4.4. To see this, note that WX q Cont (X, E) by definition. The contact locus Contq (X, E) is closed, and  (ν) contains the it is irreducible because X and E are smooth. Moreover, WX q  (ν) is equal generic point of Cont (X, E). Thus the closure WX (ν) of WX q to Cont (X, E). Lemma 2.2.6 (cf. Ishii 2008, prop. 2.9). — Let X and X  be integral separated k-varieties, let p : X  → X be a birational morphism, and let ν be a divisorial valuation on X which is centered on X  . Then WX (ν) = p∗ (WX  (ν)).  (ν) be the set of algebraically fat arcs γ on X such that Proof. — Let WX  (ν). Define W  (ν) accordingly. Let γ ∈ ordγ = ν, so that WX (ν) = WX X  WX  (ν); then p∗ (γ) = p ◦ γ is a fat arc on X and ordp∗ (γ) = ν, so that  p∗ (γ) ∈ WX (ν). Since p∗ : L∞ (X  ) → L∞ (X) is continuous,  (ν)) ⊂ p (W  (ν)) = W (ν). p∗ (WX  (ν)) = p∗ (WX  ∗ X X

Since WX (ν) is closed, it then follows that p∗ (WX  (ν)) ⊂ WX (ν). On the other hand, let γ ∈ L∞ (X) be an algebraically fat arc on X such that ordγ = ν; let K be its residue field and view γ as a morphism γ : Spec(K[[t]]) → X. Let ξ : Spec(K((t))) → X be the morphism deduced by restriction to the generic point; since γ is fat, the image of ξ is the generic point of X. The morphism p : X  → X being birational, there exists a unique morphism ξ  : Spec(K((t))) → X  such that p ◦ ξ  = ξ. Let Rν ⊂ k(X) be the valuation ring of ν; since by assumption, ν is centered on X  , the generic point Spec(k(X)) → X  extends uniquely to a morphism ϕ : Spec(Rν ) → X  . For every x ∈ X  and every f ∈ OX  ,x , one has ordt (ξ ∗ (f )) = ordγ (f ) = ν(ϕ∗ (f ))  0. This shows that ξ  extends uniquely to a morphism γ  from Spec(K[[t]]) to X  . Since p ◦ ξ  = ξ and p is separated, we have p∗ (γ  ) = γ. Moreover, ordγ  = ordγ = ν by construction. Thus   (ν)), (ν) ⊂ p∗ (WX WX 

so that WX (ν) ⊂ p∗ (WX  (ν)).

Theorem 2.2.7. — Let k be a perfect field, let X be a connected smooth separated k-variety, and let ν be a divisorial valuation on X. Then WX (ν)

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is an irreducible homogeneous closed constructible subset of L∞ (X) (in the sense of (5/3.4.1)) such that codim(WX (ν), L∞ (X)) = qν + kν . Moreover, θ0 (WX (ν)) = {cX (ν)}. Proof. — For every u ∈ Gm and every fat arc γ ∈ L∞ (X), the valuations defined by γ(t) and by the reparameterized arc γ(ut) are equal, so that WX (ν) is stable under the Gm -action. Since it is closed, it is homogeneous. This already implies that θ0,X (WX (ν)) is closed in X, by lemma 5/3.4.2. First assume that ν = q ordE , for some integer q  1 and some smooth prime divisor E in X. Then WX (ν) = Contq (X, E) by example 2.2.5, so that the result follows from proposition 5/3.4.6. In this case, one has kν = 0 and qν = q, and θ0 (WX (ν)) = E = cX (ν). For the general case, let h1 : X1 → X be a proper birational morphism of k-varieties such that X1 is normal and the center of ν on X1 is a prime divisor E of X1 . Then E meets the regular locus of X1 . Since k is perfect, there exists a dense smooth open subscheme Y of X1 such that E ∩ Y is a smooth prime divisor. One has ν = qν ordE , so that WY (ν) coincides with Contqν (E). By the smooth case, it is irreducible, closed, constructible, and of codimension qν ; moreover, θY,0 (WY (ν)) = E. By definition, one also has kν = ordE (Jach ) = ordWY (ν) (Jh ). Since WX (ν) = h∗ (WY (ν)), it follows from proposition 5/3.3.6 that WX (ν) is a closed constructible subset of codimension qν +kν of L∞ (X). It is also irreducible, because WY (ν) is irreducible. Finally, θ0 (WX (ν)) contains h(θ0 (WY (ν))) = h(E); hence, θ0 (WX (ν)) equals {cX (ν)} since it is closed and irreducible. This concludes the proof. Theorem 2.2.8. — Let k be a perfect field, let X be separated k-variety, and let C be an irreducible closed of L∞ (X) which does not dominate X. Let γ be the Then ordγ is a divisorial valuation on X whose center of θ0 (C).

a connected smooth constructible subset generic point of C. is the generic point

Proof. — Let K be the residue field of L∞ (X) at γ; then we can view γ as a point in X(K[[t]]). The arc γ is algebraically fat because C is constructible. Let Z = γ(0); one also has Z = θ0 (C), so that the assumption that C does not dominate X is equivalent to the fact that Z is a strict subset of X. We set ν = ordγ ; this is a valuation on k(X) whose center is the generic point of Z. Assume that Z is not a divisor in X. Since k is perfect, there exists a dense open subscheme U of X such that Z ∩ U is smooth and nonempty. Let then h1 : X1 → U be the blowing-up of U along Z ∩ U ; observe that X1 is a smooth k-variety and ν has a center on X1 . Let C1 be the inverse image of C in L∞ (X1 ). This is a closed constructible subset of L∞ (X1 ), and h∗ (C1 ) is

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dense in C. Since Z is not a divisor, the Jacobian ideal of h1 is nontrivial, and ordC1 (Jh1 )  1. Consequently, codim(C1 )  codim(C) − 1. We repeat this process as long as the center of ν is not a divisor. We thus construct a sequence of morphisms hn : Xn → X of smooth k-varieties and closed constructible subsets Cn ⊂ L∞ (Xn ) such that h∗ (Cn ) is dense in C, and codim(Cn )  codim(C) − n. Necessarily, this process has to stop. This shows that there exists a smooth separated k-variety Y and a birational morphism h : Y → X such that the center of ν on Y is the generic point of a smooth prime divisor E of Y . Then ν = qν ordE , so that ν is a divisorial valuation.

2.3. Minimal Log Discrepancies and the Log Canonical Threshold Let k be a perfect field. Let X be an integral separated Q-Gorenstein k-variety, and let Y be a closed subscheme of X. In this section, we review some classical numerical invariants associated with the pair (X, Y ). (2.3.1). — Let ν be a divisorial valuation on k(X) which is centered on X. For every real number c, one sets (2.3.1.1)

aν (X, cY ) = kν + qν − cν(Y ).

It is called the log discrepancy of ν with respect to the pair (X, cY ). If ν = ordE , for some proper birational morphism p : X  → X with X  normal and some prime divisor E ⊂ X  , one also writes aE (X, cY ) = aν (X, cY ). If X = Y , one sets lct(X, X) = mld(X, X) = 0. From now on, we assume that Y = X. The minimal log discrepancy of the pair (X, cY ) is then defined by the formula (2.3.1.2)

mld(X, cY ) = inf aν (X, cY ) = inf (kν + qν − cν(Y )) ν

ν

where the infimum is taken over the set of all divisorial valuations on k(X) which are centered on X. One says that the pair (X, cY ) is log canonical (lc) if mld(X, cY )  0. If (X, ∅) is log canonical and Y is nonempty, then one also defines the log canonical threshold lct(X, Y ) of the pair (X, Y ) by the formula (2.3.1.3)

lct(X, Y ) = inf ν

kν + qν , ν(Y )

where the infimum is taken over the set of all divisorial valuations on k(X) which are centered on X and whose center lies in Y . Thus lct(X, Y ) is the supremum of all nonnegative real numbers c such that (X, cY ) is log canonical, hence the terminology “log canonical threshold.” It follows from the definitions that for every open covering (Ui ) of X, one has lct(X, Y ) = inf lct(Ui , Y ∩ Ui ) i

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and mld(X, cY ) = inf mld(Ui , c(Y ∩ Ui )). i

(2.3.2). — If Z is a closed subscheme of X, the preceding notions can be localized around Z as follows. The minimal log discrepancy along Z of the pair (X, cY ) is defined by the formula (2.3.2.1)

mldZ (X, cY ) =

inf

cX (ν)∈Z

aν (X, cY ),

where the infimum is taken over the set of all divisorial valuations on k(X) which are centered on X and whose center is contained in Z. Similarly, if (X, ∅) is log canonical on an open neighborhood of Z and Y ∩Z is nonempty, then one defines the log canonical threshold of the pair (X, Y ) along Z by the formula (2.3.2.2)

lctZ (X, Y ) =

kν + qν , cX (ν)∩Z=∅ ν(Y ) inf

where the infimum is taken over the set of all divisorial valuations on k(X) centered on X whose center lies in Y and such that the closure of the center meets Z. It follows from the definitions that mldZ (X, cY ) = mldZ (U, c(Y ∩ U )) and lctZ (X, Y ) = lct(U, Y ∩U ) for every open neighborhood U of Z in X. Remark 2.3.3. — The classical definitions of the log canonical threshold and of the minimal log discrepancy only involve divisorial valuations of index 1; however, they coincide with the present one. In the case of the log canonical threshold, this follows from homogeneity (the expression (kν + qν )/ν(Y ) is invariant under rescaling the valuation ν). For the minimal log discrepancy, this is a consequence of Kollár and Mori (1998, corollary 2.31), which asserts that the classical mld is either nonnegative or −∞. Proposition 2.3.4. — Assume that the ground field k has characteristic 0 and that mldZ (X, Y ) = −∞. Let p : X  → X be a log resolution of the in the union of the pair (X, Y ), and let Ei , i ∈ I, be the prime components  exceptional locus of p and the strict transform of Y . Let mi Ei be the total  transform of Y ; thus, IY OX  = O(− mi Ei ). Let ki be thediscrepancy of Ei with respect to X for every i ∈ I; equivalently, KX  /X = ki Ei .  a) Let D be the inverse image of Z on X , and assume that either Z = X or D + Ei is a divisor with strict normal crossings. Then (2.3.4.1)

mldZ (X, Y ) =

min (ki + 1 − mi )  0.

p(Ei )⊂Z

b) The pair (X, ∅) is log canonical on an open neighborhood of Z, and if Y ∩ Z = ∅, then (2.3.4.2)

lctZ (X, Y ) =

min

p(Ei )∩Z=∅

ki + 1  0. mi

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Sketch of proof. — As in the proof of Kollár and Mori (1998, corollary 2.31), one can compute the discrepancies given by exceptional divisors on blow-ups of X  . It turns out that they are smaller than those of the Ei , and that if one of them is strictly negative, then they can be arbitrarily negative. This implies the proposition. Corollary 2.3.5. — Assume that k has characteristic zero and that (X, ∅) is log canonical. Then lctZ (X, Y ) is a rational number, and it is the maximum of the set of all nonnegative real numbers c such that (X, cY ) is log canonical in an open neighborhood of Z. Remark 2.3.6. — Still assuming that k has characteristic zero, another consequence of proposition 2.3.4 is that the function x → lctx (X, Y ) is lower semicontinuous on the set of closed points of X. The analogous property for the minimal log discrepancy, conjectured by Ambro (1999), has been proved by Ein et al. (2003), when X is smooth, and by Ein and Mustaţˇ a (2004) when X is a normal local complete intersection. 2.4. Arc Spaces and the Log Canonical Threshold Theorem 2.4.1 (Zhu 2013). — Let k be a perfect field; let X be a smooth integral separated k-variety; let Y and Z be closed subschemes of X. Let CZ be the set of all irreducible closed constructible subsets C of L∞ (X) which do not dominate X and such that θ0,X (C) is contained in Y and meets Z. Then one has the following equalities: (2.4.1.1) (2.4.1.2) (2.4.1.3)

codim(C) ordC (Y ) dimZ (Lm (Y )) = dim(X) − sup m+1 m0

lctZ (X, Y ) = inf

C∈CZ

= dim(X) − lim sup m0

dimZ (Lm (Y )) m+1

In particular, one has lctZ (X, Y )  codimZ (Y, X). In this statement, for any integer m and any subscheme A of Lm (X), one denotes by dimZ (A) the maximal dimension of an irreducible component of A which meets sm,X (Z); moreover, codimZ (Y, X) is the minimal codimension of an irreducible component of Y which meets Z. Proof. — We first prove the equality: lctZ (X, Y ) = inf

C∈CZ

codim(C) . ordC (Y )

By definition, lctZ (X, Y ) = inf ν

qν + kν , ν(Y )

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where ν ranges over all divisorial valuations on X such that {cX (ν)} meets Z. For every divisorial valuation ν on X, theorem 2.2.7 asserts that WX (ν) is an irreducible closed constructible subset of L∞ (X) which does not dominate X and such that codim(WX (ν), L∞ (X)) = qν + kν , and ordWX = ν; moreover, {cX (ν)} meets Z if and only if θ0,X (WX (ν)) meets Z. Consequently, inf

C∈CZ

codim(WX (ν)) qν + kν codim(C)  = , ordC (Y ) ordWX (ν) (Y ) ν(Y )

hence lctZ (X, Y )  inf C∈CZ (codim(C)/ ordC (Y )). On the other hand, let C be an irreducible closed constructible subset of L∞ (X) which does not dominate X and such that θ0,X (C) meets Z; let ν = ordC . One has C ⊂ WX (ν); hence, codim(C) codim(WX (ν)) qν + kν  =  lctZ (X, Y ). ordC (Y ) ordWX (ν) (Y ) ν(Y ) This implies the reverse inequality, lctZ (X, Y )  inf C (codim(C)/ ordC (Y )). Let now m be an integer. Since X is smooth and integral, the jet space Lm (X) is smooth, integral, and of dimension (m + 1) dim(X), so that one has codimZ (Lm (Y ), Lm (X)) dimZ (Lm (Y )) = . dim(X) − m+1 m+1 Let Am be an irreducible component of Lm (Y ) which meets sm,X (Z) and such that dimZ (Lm (Y )) = dim(Am ). Let us view Am as a closed integral −1 (Am ). Then A is a closed irreducible subscheme of Lm (X), and let A = θX,m constructible subset of L∞ (X), and codim(A) = codim(Am , Lm (X)) = codimZ (Lm (Y ), Lm (X)). m m One has θ0,X (A) = θ0,X (θm,X (A)) = θ0,X (Am ) ⊂ Y , so that θ0,X (A) ⊂ Y as well; in particular, it is distinct from X. Moreover, one has ordγ (Y )  m + 1 −1 for every γ ∈ θm,X (Lm (Y )), so that ordA (Y )  m + 1. Consequently,

lctZ (X, Y ) 

codimZ (Lm (Y ), Lm (X)) codim(A) = . ordA (Y ) m+1

We thus have shown the inequality codimZ (Lm (Y ), Lm (X)) . m0 m+1

lctZ (X, Y )  inf

Let then C be an irreducible closed constructible subset of L∞ (X) which does not dominate X and such that θ0,X (C) is contained in Y and meets Z. We will prove that codimZ (Lm (Y ), Lm (X)) codim(C)  inf . m0 ordC (Y ) m+1 Let m = ordC (Y ) − 1; by assumption, one has m  0. Then C ⊂ −1 Cont(m+1) (Y ) = θm,X (Lm (Y )). Since C is irreducible, there exists an −1 (S). The closed irreducible component S of Lm (Y ) such that C ⊂ θm,X

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−1 m set θm,X (S) is clearly homogeneous and thus contains s∞,X (θ0,X (S)) by lemma 5/3.4.2. This implies that S contains sm,X (θ0,X (C)); in particular, S meets sm,X (Z). Consequently, −1 (S)) codim(C)  codim(θm,X

= codim(S, Lm (X))  codimZ (Lm (Y ), Lm (X)). It remains to prove the equality: dimZ (Lm (Y )) dimZ (Lm (Y )) = lim sup . m + 1 m+1 m→∞ m0 sup

It holds obviously if the supremum on the left is not achieved. On the other hand, let us assume that there is an integer p such that dimZ (Lp (Y )) dimZ (Lm (Y )) = . m + 1 p+1 m0 sup

Let Ap be an irreducible component of Lp (Y ) which meets sp,X (Z) and such −1 that dimZ (Lp (Y )) = dim(Ap ); let also A = θX,p (Ap ). The above arguments show that A is an irreducible closed constructible subset of L∞ (X) and that lctZ (X, Y ) = codim(A)/ ordA (Y ). Let q  1 and let C = WX (q ordA ). By homogeneity of the expression (qν + kν )/ν(Y ), C is an irreducible closed constructible subset of L∞ (X) such that lctZ (X, Y ) = codim(C)/ ordC (Y ). Let m = ordC (Y ) − 1 = q ordA (Y ) − 1; the above arguments show that codim(C)  codimZ (Lm (Y ), Lm (Y )); hence, dimZ (Lm (Y )) dimZ (Lp (Y )) = . m+1 p+1 Since m  q − 1, this concludes the proof. Corollary 2.4.2. — Let k be a perfect field. Let X be a smooth integral separated k-variety, and let Y be a closed subscheme of X. Then one has lct(X, Y ) = inf

C∈C

codim(C) ordC (Y ) dim(Lm (Y )) m+1 m0

= dim(X) − sup

= dim(X) − lim sup m→∞

dim(Lm (Y )) , m+1

where C is the set of all irreducible closed constructible subsets of L∞ (X) which do not dominate X. In particular, one has lct(X, Y )  codim(Y, X). Proof. — This follows from theorem 2.4.1, applied with Z = Y . Corollary 2.4.3. — Let k be a perfect field.

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a) Let X, X  be smooth integral separated k-varieties, let Y be a closed subscheme of X, and let Y  be a closed subscheme of X  . If there exists a closed immersion of Y into Y  , then one has dim(X  )−lct(X  , Y )  dim(X)− lct(X, Y ). b) If Y and Y  are isomorphic, then dim(X  ) − lct(X  , Y  ) = dim(X) − lct(X, Y ). c) Let X be a smooth integral separated k-variety, let Y  be a closed subscheme of X  , and let Y be a closed subscheme of Y  . Then for every closed subset Z of X, one has lctZ (X, Y )  lctZ (X, Y  ). Proof. — Let j : Y → Y  be a closed immersion. For every integer m, it induces a closed immersion of Lm (Y ) into Lm (Y  ), hence dim(Lm (Y ))  dim(Lm (Y  )). This implies the first assertion, and the second follows by symmetry. The third assertion follows also from the first one, by restricting to arbitrary open neighborhoods of Z in X. Corollary 2.4.4. — Let k be a perfect field and let K be a perfect extension of k. Let X be a smooth geometrically integral separated k-variety, and let Y be a closed subscheme of X; one has lct(XK , YK ) = lct(X, Y ). More generally, for every closed subscheme Z of X, one has lctZK (XK , YK ) = lctZ (X, Y ). Proof. — The formation of jet schemes commutes with base change, so that one has Lm (XK ) = Lm (X)K and Lm (YK ) = Lm (Y )K for every integer m  0. Consequently, dim(Lm (Y )) = dim(Lm (Y )K ) = dim(Lm (YK )), and the first assertion follows from corollary 2.4.2. The proof of the second assertion is analogous, using theorem 2.4.1. Remark 2.4.5. — Let p be a prime number, let k be a field of characteristic p which is not perfect, and let a ∈ k k p . Let X = A1k = Spec(k[T ]) and let Y = V (T p − a). Then Y is a prime divisor of X, and the (T p − a)adic valuation ν of k(T ) satisfies qν = 1, kν = 0, and ν(Y ) = 1. One has lct(X, Y ) = 1. On the other hand, let K be a perfect extension of k, and let b ∈ K be such that bp = a. Then XK = A1K and YK = V ((T − b)p ). The (T − b)-adic valuation μ of K(T ) satisfies qμ = 1, kμ = 0, and μ(Y ) = 1. Consequently, lct(XK , YK ) = 1/p. This shows that in corollary 2.4.4, the assumption that the base field k be perfect cannot be omitted. The same example shows that it cannot be omitted from theorem 2.4.1 and corollary 2.4.2 neither. Corollary 2.4.6 (Semicontinuity of the Log Canonical Threshold) Let S be a reduced scheme, let X be a smooth separated S-variety with geometrically integral fibers, and let Y be a closed subscheme of X. For every section τ : S → Y , the map lctX/S,τ : S → R,

s → lctτ (s) (Xκ(s) , Yκ(s) )

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on S decreases by specialization. Here κ(s) is an algebraic closure of the residue field at s. If S is of finite type over a field of characteristic zero (or, more generally, S is a quasi-excellent Q-scheme), then lctX/S,τ is lower-semicontinuous. Proof. — For simplicity of notation, we write Xs = Xκ(s) and similarly for Y . For every point s ∈ S, theorem 2.4.1 implies the relation dimτ (s) (Lm (Ys )) dim(Xs ) − lctτ (s) (Xs , Ys ) = sup m+1 m0 Let us denote by p the canonical morphism from X to S. By Chevalley’s theorem, the function y → dimy (Lm (Yp(y) )) is upper semicontinuous on Lm (Y /S) (ÉGA IV3 , théorème 13.1.3). This implies that the function s → dimτ (s) (Lm (Ys )) is upper semicontinuous on S. In particular, it increases by specialization. Consequently, the function s → dim(Xs ) − lctτ (s) (Xs , Ys ) increases by specialization as well. Since X is flat over S with nonempty fibers, s → dim(Xs ) is locally constant on S, so that the function lctX/S,τ decreases under specialization. Now assume that S is a quasi-excellent Q-scheme. We will show that lctX/S,τ is lower-semicontinuous. Since the assertion is local on S, we may assume that S is Noetherian. It suffices to show that it is constructible (that is, its fibers are constructible subsets of S), because constructible subsets that are closed under specialization are closed. Thus it is enough to find a nonempty open subscheme U of S such that lctX/S,τ is constant on U . Let h : X  → X be a log resolution for (X, Y ). Such a log resolution exists by our assumption that S is a quasi-excellent scheme of characteristic zero Temkin (2008). Let Ei , i ∈ I be the prime components in the union of the exceptional locus of h and the strict transform of Y . Replacing S by a nonempty open →S subset and restricting h accordingly, we can arrange that X  → S and Ei are smooth and surjective, for every i ∈ I, and that the restriction of Ei to each fiber of X  over S is a strict normal crossings divisor. Then the base change of h to each geometric point s of S is a log resolution of (Xs , Ys ). Using proposition 2.3.4 to compute the log canonical threshold on these log resolutions, we see that the function lctX/S,τ is constant on S. The semicontinuity of the log canonical threshold in characteristic zero was first proven by Varchenko (1982) in the case of a hypersurface; Demailly and Kollár (2001) proved a generalization in the framework of analytic multiplier ideals. The proof we have given here is due to Mustaţă (2002). Corollary 2.4.7 (Inversion of Adjunction). — Let k be a perfect field, let X be a smooth separated k-variety, and let Y be a closed subscheme of X. Let V be a smooth closed subscheme of X, and let Z be a closed subscheme of V such that Y ∩ Z = ∅. The following inequality holds: lctZ (X, Y )  lctZ (V, Y ∩ V ).

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Proof. — Working locally on X, we may assume that X is affine and that V is a global complete intersection. By induction, this allows to assume that V is a hypersurface in X, defined by one element of OX (X). Let m be an integer, let T be an irreducible component of Lm (Y ∩V ) that meets sm,V (Z), and let S be an irreducible component of Lm (V ) that contains T . The construction of the jet schemes shows that Lm (Y ∩ V ) is a closed subscheme of Lm (Y ) whose ideal is generated by m + 1 elements. It thus follows from Krull’s Hauptidealsatz that dim(T )  dim(S) − (m + 1). Consequently, dimZ (Lm (Y ∩ V ))  dimZ (Lm (Y )) − (m + 1). Applying theorem 2.4.1, we then have dimZ (Lm (Y ∩ V )) m+1 m0  dimZ (Lm (Y )) −1  (dim(X) − 1) − sup m+1 m0

lctZ (V, Y ∩ V ) = dim(V ) − sup

dimZ (Lm (Y )) m+1 m0

 dim(X) − sup  lctZ (X, Y ).

This concludes the proof of the corollary. Corollary 2.4.8. — Let k be a perfect field, let X be a connected smooth separated k-variety, and let Y  , Y  be closed subschemes of X. Let Z be a closed subscheme of X. One has lctZ (X, Y  ∩ Y  )  lctZ (X, Y  ) + lctZ (X, Y  ). Proof. — Since the formation of jet schemes is compatible with products, it follows from theorem 2.4.1 that lctZ (X, Y  ) + lctZ (X, Y  ) = lctZ×Z (X × X, Y  × Y  ). Let then Δ be the diagonal of X ×k X. By corollary 2.4.7, one has lctZ×Z (X × X, Y  × Y  )  lctZ×Z (Δ, Δ ∩ (Y  × Y  )) = lctZ (X, Y  ∩ Y  ). This concludes the proof. Corollary 2.4.9. — Let k be a perfect field, let X be a connected smooth separated k-variety, and let Y be a closed subscheme of X such that Y = X. For every closed point y ∈ Y , one has the inequality: lcty (X, Y )  1/ ordy (Y ). In particular, lctZ (X, Y ) > 0 for every closed subscheme of X such that Y ∩ Z = ∅. Recall that ordy (Y ) is the least upper bound of all integers n such that IY,y ⊂ mnX,y .

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Proof. — We argue by induction on dim(X). If dim(X) = 1, then there exists an open neighborhood U of y in X such that Y ∩ U = {y}. Then, lcty (X, Y ) = lcty (U, Y ∩ U ) = 1/ ordy (Y ), by the very definition of the log canonical threshold. Let us now assume that dim(X) > 1. There exists an irreducible hypersurface V of X such that y is a smooth point of V and such that ordy (Y ∩ V ) = ordy (Y ). Let U be an open neighborhood of y such that V ∩ U is smooth. By corollary 2.4.7, one has lcty (X, Y ) = lcty (U, Y ∩ U )  lcty (V, Y ∩ V ). By induction, this proves that lcty (X, Y )  1/ ordy (Y ∩ V ) = 1/ ordy (Y ), as was to be shown. Since lctZ (X, Y ) = inf y∈Y ∩Z lcty (X, Y ) and ordy (Y ) is bounded from above when y ∈ Y , this implies the final statement of the corollary. Remark 2.4.10. — There are similar results for the minimal discrepancies. The proofs are analogous albeit slightly more technical, and we refer the interested reader to the original papers, notably (Ein and Mustaţˇ a 2004) in characteristic zero and Ishii and Reguera (2013) in arbitrary characteristic. Theorem 2.4.11 (Ein and Mustaţˇ a 2004). — Let k be a field of characteristic zero. Let Y be a normal integral separated k-variety, and assume that Y is a locally complete intersection. Then the following equivalences hold: a) The variety Y has log canonical singularities if and only if the jet scheme Lm (Y ) has pure dimension for every m  0. b) The variety Y has canonical singularities if and only if the jet scheme Lm (Y ) is irreducible for every m  0. c) The variety Y has terminal singularities if and only if the jet scheme Lm (Y ) is normal for every m  0. Proof. — This is theorem 1.3 in Ein and Mustaţˇ a (2004); the second equivalence had previously been proven by Mustaţă (2001). Here, we will only sketch a proof of the special case where Y is a divisor in a smooth separated k-variety X; this case was treated in Ein et al. (2003). Denote by d the dimension of Y . Then the jet scheme Lm (Y ) has an irreducible component of dimension d(m+1) for every m  0, namely, the closure of the jet scheme Lm (Ysm ) of the smooth locus of Y . On the other hand, each irreducible component of Lm (Y ) has dimension at least d(m + 1), because Lm (Y ) can locally be defined by m + 1 equations in the jet scheme Lm (X), which has pure dimension (d + 1)(m + 1). Thus, Lm (Y ) has pure dimension if and only if it has dimension at most d(m + 1). By corollary 2.4.2, this is equivalent to saying that the pair (X, Y ) has log canonical singularities. It is proven in Kollár (1992, §17.6–7) that (X, Y ) has log canonical singularities if and only if Y has log canonical singularities; this follows from a stronger form of inversion of adjunction than the one stated in corollary 2.4.7. This settles the first assertion.

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The proofs of the second and third assertions rely on a variant of theorem 2.4.1 for minimal log discrepancies. By means of this variant, one shows that Y has canonical, resp. terminal singularities, if and only if (θm,Y )−1 (Ysing ) has codimension at least one, resp. two, in Lm (Y ). Since Lm (Y ) has pure dimension, (θm,Y )−1 (Ysing ) has codimension one if and only if it does not contain an irreducible component of Lm (Y ); but this is equivalent to saying that Lm (Y ) is irreducible, because Lm (Y ) (θm,Y )−1 (Ysing ) = Lm (Ysm ) is irreducible. Since Lm (Y ) is locally defined by a regular sequence in the smooth Y scheme Lm (X) ×X Y , a point y in Lm (Y ) is smooth if and only if Y is smooth at θm,Y (y). Thus (θm,Y )−1 (Ysing ) coincides with the singular locus of Lm (Y ). As Lm (Y ) is locally a complete intersection, it is normal if and only if it is regular in codimension one; thus, Lm (Y ) is normal if and only if (θm,Y )−1 (Ysing ) has codimension at least two in Lm (Y ). This concludes the sketch of the proof. 2.5. The Nash Problem (2.5.1). — Let k be a field, let X be an integral k-variety, and let Xsing be −1 its singular locus. We define the Nash space of X as N (X) = θ0,X (Xsing ). It is a constructible closed subset of L∞ (X). Definition 2.5.2. — Let k be a field of characteristic zero, and let X be an integral separated k-variety. One says that a divisorial valuation ν on X is essential if, for every resolution of singularities h : Y → X, the center of ν on Y is the generic point of an irreducible component of h−1 (Xsing ). Equivalently, one says that a divisor over X is essential if the associated divisorial valuation is essential. It follows from proposition 4 of Abhyankar (1956), see also Kollár and Mori (1998, prop. 1.3), that a divisor over X which is not ruled is essential. Proposition 2.5.3 (Nash 1995 (4) ). — Let k be a field of characteristic zero, let X be an integral separated k-variety, and let Xsing be its singular locus. a) Every irreducible component of the Nash space N (X) is fat, and the associated valuation is an essential divisorial valuation of index 1 on X. b) The valuations associated with two distinct components are distinct. Proof. — Let W be an irreducible component of N (X). By lemma 3/4.3.2, the generic point of W has a generization ϕ which is not contained in L∞ (Xsing ) and whose base point is contained in Xsing . By definition, one has ϕ ∈ N (X) and W ⊂ {ϕ}; by the definition of an irreducible component, the arc ϕ is the generic point of W , hence W ⊂ L∞ (Xsing ). (4) This

paper had been written in 1968.

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Let then h : Y → X be a resolution of singularities. Let Z = h−1 (Xsing ), and let (Zi )i∈I be the family of its irreducible components. The map L∞ (h) is continuous and, by proposition 3/4.4.2, induces a continuous bijection from ∞ −1 ) (Z) L∞ (Z) to N (X) L∞ (Xsing ). Since Y is smooth, the sets (θ0,Y ∞ −1 ∞ −1 ) (Zi ), for i ∈ I, are the irreducible components of (θ0,Y ) (Z). (θ0,Y It follows that there exists a unique irreducible component Zi of Z such that W

∞ −1 (W ∩ L∞ (Xsing )) = L∞ (h)((θ0,Y ) (Zi ))

L∞ (Xsing ).

∞ −1 Then γ is the image by L∞ (h) of the generic point γi of (θ0,Y ) (Zi ). Observe that γi is a fat point, because Y is smooth. Since h is surjective, the point ϕ is fat as well; hence, W is fat. Moreover, if we denote by Y  → Y the blow-up of Y at Zi and by E the unique divisor on Y  that dominates Zi , then the valuation ordγ on k(X) is equal to the divisorial valuation ordE , because γ is the image of the generic point of θY−1 ,0 (E). This description also shows that to two distinct components, W corresponds to distinct irreducible components Zi of Z, whence the proposition.

Remark 2.5.4. — If the characteristic of k is a prime number p, the Nash space of X may have thin irreducible components. As observed in example 2.13 of Ishii and Kollár (2003), this happens, for instance, when X is the hypersurface of A3k defined by the polynomial xp − y p z ∈ k[x, y, z]. In this case (Xsing )red = V (x, y) is isomorphic to A1k . Let us show that any arc on X that specializes to the arc (0, 0, t) is contained in L∞ (Xsing ). Let us consider a local k-algebra A, with maximal ideal m, and three power series p p x, y, z ∈ A[[t]] such that  x n−y z = 0 and (x, y,n z) ≡ (0, 0, t) (mod m). Writing n yn t , and z = zn t , we thus have xn , yn ∈ m for x = xn t , y = every n, zn ∈ m for every n = 1, and z1 − 1 ∈ m. Let us prove that x = y = 0. Let m  0 be an integer such that xn = yn = 0 for every n < m. Since R is a k-algebra, one has p = 0 in R; hence, the equation xp = y p z implies p p z0 and ym z1 = 0. Since z1 ∈ 1 + m, it is invertible; consequently, xpm = ym ym = 0, and then xm = 0. By induction, this proves that x = y = 0; hence, (x, y, z) ∈ L (Xsing ). In other words, every irreducible component of N (X) that contains the arc (0, 0, t) is contained in L∞ (Xsing ). Since (Xsing )red  A1k , its arc space L∞ (Xsing ) is irreducible. This shows that L∞ (Xsing ) is an irreducible component of N (X); by definition, this component is thin. In fact, we have proven a stronger statement: L∞ (Xsing ) is an irreducible component of L∞ (X), and it is the unique component that contains the arc (0, 0, t). (2.5.5). — The mapping associating with an irreducible component W of the Nash space N (X) the essential divisorial valuation νW on X is called the Nash map. By the previous proposition, it is injective. In particular, the set of irreducible components of N (X) is finite, which is a particular case of proposition 3/4.5.3.

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Nash (1995) raised the question whether this map is surjective. In other words, does every essential divisorial valuation ν of index 1 on X correspond to some irreducible component of the Nash space? In fact, this paper marked the introduction of the notions of arc/jet schemes for the study of singularities of algebraic and analytic varieties, and Nash’s question motivated a lot of research in singularity theory. Moreover, Nash expected that the answer would be positive in dimension two while suggesting that it could be more complicated in higher dimensions. The following theorems, established almost 50 years after Nash wrote his paper, show that his insight was indeed correct! Theorem 2.5.6 (Fernández de Bobadilla and Pe Pereira 2012) Let k be an algebraically closed field of characteristic zero. Let X be an integral surface defined over k. Then the Nash map is bijective. The proof of this theorem builds on (Reguera 2006) and (Fernández de Bobadilla 2012), among others, and goes beyond the scope of this book. Let us also mention that many specific cases of surface singularities (quasiordinary, quotient, rational double points,. . . ) had been treated before; see the references quoted in the paper (Fernández de Bobadilla and Pe Pereira 2012). (2.5.7). — However, the Nash map fails to be bijective in general: Ishii and Kollár (2003) gave counterexamples in dimension  4, and de Fernex (2013) completed the picture in dimension  3. Let us only mention the following result from Johnson and Kollár (2013, example 2), refering the interested reader to that paper for more general results on cA-type singularities. Let m be an integer  2, and let Xm be the hypersurface in A4C defined by the polynomial x21 + x22 + x23 + xm 4 . The following properties hold: a) The Nash space N (Xm ) is irreducible. b) If m is odd and m  5, then there are two essential divisors over Xm . c) If m is even and m  2 or m = 3, then there is only one essential divisor over Xm . Remarkably, this example was studied already in Nash (1995), but at the time Nash was unable to determine the exact number of irreducible components of N (X) and essential divisors in the case where m is odd. It remains an interesting question in singularity theory to understand the image of the Nash map. Here are three positive results. Theorem 2.5.8 (Ishii and Kollár 2003, Theorem 3.16) Let k be an algebraically closed field. Let X be an affine toric k-variety. Then the Nash map is bijective. Theorem 2.5.9 (Lejeune-Jalabert and Reguera 2012) Let k be a field which is algebraically closed and uncountable, and let X be an integral separated k-variety. Every divisorial valuation of index 1 on X that is associated with a non-uniruled divisor belongs to the image of the Nash map.

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Theorem 2.5.10 (de Fernex and Docampo 2016) Let k be an uncountable algebraically closed field of characteristic zero. Let X be an integral separated k-variety. Every divisorial valuation of index 1 on X which is associated with a divisor on a minimal model over X belongs to the image of the Nash map. Recall that a minimal model over X is a projective birational morphism f : Y → X, where Y is a normal variety with terminal singularities such that the canonical class KY is relatively nef over X. For this reason, de Fernex and Docampo (2016) call terminal the divisorial valuations of index 1 which are associated with a divisor on a minimal model. When minimal desingularizations exist, terminal valuations coincide with essential valuations; hence, the Nash map is surjective; this is in particular the case for surfaces and gives another proof of theorem 2.5.6.

§ 3. MOTIVIC VOLUME AND BIRATIONAL INVARIANTS 3.1. Motivic Power Series k [[T ]] of power series with coLet k be a field. We consider the ring M  k . Let F = k [[T ]]. Let r be a real number; efficients in M Fn T n ∈ M one says that F converges on the closed disk of valuative radius r if one has dim(Fn ) − nr → −∞ when n → +∞. The infimum ρ(F ) of all these real numbers r is the valuative radius of convergence of F ; it is given by the formula  dim(Fn ) ρ(F ) = lim sup . n n→∞ The set of power series F such that F converges on the closed disk of k {T Lr }. As k [[T ]] which we denote by M valuative radius r is a subring of M a consequence, the set of power series F such that ρ(F ) < r is a subring of k {T Lr }, and we denote it by M k {T Lr }† . M k {T Lr }. For every inte(3.1.1). — Let r be a real number and let F ∈ M  −ns k , and its limit ger s such that s  r, the series Fn L converges in M is denoted by F (L−s ). Moreover, the map F → F (L−s ) is a ring morphism k {T Lr } to M k . from M More generally, let s be a rational number such that s  r; let us write s = m/d, where m and d are coprime integers. The formula ∞  n=0

Fn L−ns =

∞ d−1   p=0 q=0

Fqd+p L−(qd+p)m/d =

d−1  p=0

L−pm/d

∞ 

Fqd+p L−qm

q=0

k [L1/d ] = M k [T ]/(T d − L). Again, the defines F (L−s ) as an element of M −s map F → F (L ) is a ring morphism.

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k generated by all fractional powers One may introduce the extension of M k -module, it is free with basis (Lr )r∈Q∩[0,1[ . of L; as an M k [T ] is invertible (3.1.2). — Let (a, b) ∈ N>0 ×Z. The element 1−Lb T a of M k [[T ]]; its inverse is given by the power series: in M ∞ 

Lbn T an ,

n=0

and its valuative radius of convergence is equal to b/a. In particular, 1−Lb T a k [T ]. is not a zero divisor in M k [[T ]] generated by the imLet us consider the subring Mk [[T ]]rat of M k , and the power series age Mk of Mk in M  Lb T a (L − 1) = (L − 1) Lbn T na , b a 1−L T n1

for all (a, b) ∈ N>0 × Z. For every real number r, we denote by Mk {T Lr }†rat the subring of Mk [[T ]] obtained by restricting oneself to the pairs (a, b) such that b/a < r. This is a subring of Mk {T Lr }† . k [[T ]]. Let (a, b) ∈ N>0 × Z be such that Lemma 3.1.3. — Let F ∈ M b/a > ρ(F ); let m ∈ N>0 and let G = F/(1 − Lb T a )m . Then one has ρ(G)  b/a. More precisely, if F (L−b/a ) = 0, then ρ(G) = b/a; if m = 1 and F (L−b/a ) = 0, then ρ(G) = ρ(F ). Proof. — Since the valuative radius of convergence of (1 − Lb T a )−1 is equal to b/a, one has ρ(G) = ρ((1 − Lb T a )−m F )  sup(b/a, ρ(F )) = b/a. If ρ(G) < b/a, then one can evaluate the expression F = (1 − Lb T a )m G at T = L−b/a ; this implies F (L−b/a ) = 0. Conversely, let us assume that F (L−b/a ) = 0 and m = 1. Without loss of generality, we assume that a and b a−1 k [[T ]]; are coprime and write F = j=0 T j Fj (T a ), where F0 , . . . , Fa−1 ∈ M a−1 −jb/a −b/a −b −b ) = j=0 L Fj (L ), we see that Fj (L ) = 0 for since one has F (L every j. This reduces the problem to the case where a = 1. Let us then write ∞ F = n=0 Fn T n . Since F (L−b ) = 0, one has ∞  ∞    G(T ) = Fn T n Lnb T n =

n=0 ∞ 

T

 n

n=0 ∞ 

=−

n=0

n=0 n 



(n−m)b

Fm L

m=0



Tn



m>n

 Fm L(n−m)b

,

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405

  so that G = Gn T n , with Gn = m>n Fm L(n−m)b . Let r be a real number such that b > r > ρ(F ), and let n0 ∈ N be such that dim(Fn ) − nr  0 for n  n0 . For every integer n such that n  n0 , one has dim(Gn ) − nr  sup (dim(Fm ) + (n − m)b) − nr mn+1

 sup (m − n)(r − b) mn+1

 0. This implies that ρ(G)  r, as was to be shown. Proposition 3.1.4 (Topological Realization). — There exists a unique ring morphism μtop : Mk [[T ]]rat → Q(s) satisfying the following properties: a) For every k-variety X, one has μtop (X) = Eu(X), the Euler characteristic of X. b) For every (a, b) ∈ N>0 × Z, one has  Lb T a 1 . μtop (L − 1) = 1 − Lb T a as − b It will follow from the construction below that the poles of a rational function in the image of μtop are rational numbers. Proof. — One has Eu(1) = Eu(L) = 1, so that these assumptions imply that μtop (1 − Lb T a ) = 0; hence, μtop (T ) = 1. It is thus obvious that there is at most one morphism of rings satisfying the given requirements. Let ψ : Mk [T ] → Z[u, u−1 , T ] be the unique ring morphism mapping T to T and extending the Euler–Poincaré polynomial motivic measure from Mk to Z[u, u−1 ]. One has ψ(L) = u2 ; hence, ψ(1 − Lb T a ) = 1 − u2b T a = 0 for every (a, b) ∈ N>0 × Z. Consequently, the morphism ψ extends uniquely to a ring morphism ψ  from Mk [[T ]]rat to the subring R of Q(u, T ) generated by Z[u, u−1 , T ] and the elements (1 − u−2 )u2b T a /(1 − u2b T a ). When we set T = u−2s and pass to the limit u → 1, one has u2b T a 1 . → 1 − u2b T a as − b Consequently, there exists a unique ring morphism λ : R → Q(s) such that λ(u) = λ(T ) = 1 and λ((1 − u−2 )u2b T a /(1 − u2b T a )) = 1/(as − b). The composition μtop = λ ◦ ψ is the desired ring morphism. (1 − u−2 )

3.2. The Jacobian Ideal (3.2.1). — Let k be a field, and let h : X  → X be a birational morphism of k-varieties of dimension d. Recall that the Jacobian ideal of h is defined as the Fitting ideal: Jach = Fitt0 Ω1X  /X . It measures where h fails to be an open immersion, and it played a crucial role in the change of variables formula for motivic integrals. If X  is smooth

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over k, then the proof of lemma 5/3.1.3 reveals that Jach is the unique ideal sheaf such that the image of h∗ ΩdX/k → ΩdX  /k equals Jach ·ΩdX  /k . We will now explain how the Jacobian ideal is related to the discrepancies defined in section 2.1.6, starting with the case where X and X  are smooth. Proposition 3.2.2. — Let h : X  → X be a birational morphism of smooth k-varieties. Let V be the largest open subscheme of X  where h is an open immersion, and denote by j the open immersion V → X  . Then Jach is an invertible ideal sheaf on X, and the associated Cartier divisor is precisely the relative canonical divisor KX  /X . Proof. — Let d be the dimension of X and X  . We can use the fundamental exact sequence h∗ Ω1X/k → Ω1X  /k → Ω1X  /X → 0 to compute the Fitting ideals of Ω1X  /X . By definition, Jach = Fitt0 Ω1X  /X is the determinant ideal of the morphism of locally free sheaves h∗ Ω1X/k → Ω1X  /k ; thus, it is invertible and characterized by the property that h∗ ΩdX/k = Jach ·ΩdX  /k . Corollary 3.2.3. — Let h : X  → X and g : X  → X  be birational morphisms of k-varieties. Assume that X  and X  are smooth. Then Jach◦g = Jacg ·(Jach OX  ). If g factors through the blow-up of Jach , then Jach◦g is invertible on X  . Proof. — Let d be the dimension of X, X  , and X  . The image of (h ◦ g)∗ ΩdX/k → ΩdX  /k equals (Jacg ·(Jach OX  ))ΩdX  /k , which implies the expression for Jach◦g . The ideal Jacg is invertible by proposition 3.2.2. If g factors through the blow-up of Jach , then Jach OX  and thus Jach◦g are invertible. (3.2.4). — More generally, assume that X is Q-Gorenstein. We denote by d the dimension of X. Let U be the smooth locus of X, and let i : U → X be the canonical immersion. Let KX be a canonical divisor on X, and let m  1 be an integer such that mKX is Cartier; then ωX,m = i∗ ((ΩdU/k )⊗m ) is a line bundle on X. Let π : (ΩdX )⊗m → ωX,m be the canonical morphism of OX -modules. Let α ∈ Ker(π); since π is an isomorphism over U , one has α|U = 0; hence, α is torsion. Conversely, since ωX,m is locally free, every torsion section of (ΩdX )⊗m is contained in Ker(π). Consequently, the kernel of π is the torsion submodule of (ΩdX )⊗m . Since ωX,m is locally free of rank 1, there exists a unique ideal sheaf JX,m on X such that the image of π is equal to JX,m ωX,m . Proposition 3.2.5. — Let h : X  → X be a birational morphism of integral separated k-varieties, and assume that X is Q-Gorenstein and X  is smooth. Let m be a positive integer such that mKX is Cartier. Then (Jach )m = JX,m OX  (−mKX  /X ) where we view OX  (−mKX  /X ) as a fractional ideal on X  .

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Proof. — Denote by d the dimension of X and X  . Set V = X  (∪i Ei ) and consider the open immersion j : V → X  . Then by the definition of the relative canonical divisor, we have the following equality of sub-OX  -modules of j∗ ΩdV /k : h∗ ωX,m = (ΩdX  /k (−KX  /X ))⊗m . Moreover, the image of h∗ ΩdX → ΩdX  /k equals Jach ΩdX  /k , and the image of h∗ (ΩdX )⊗m → h∗ ωX,m equals JX,m (h∗ ωX,m ). The result follows. Corollary 3.2.6. — Let h : X  → X be a birational morphism of integral separated k-varieties, and assume that X is Q-Gorenstein and X  is smooth. Let m be a positive integer such that mKX is Cartier. Let (Ei )i∈I be the family of exceptional prime divisors of h. For every i, we denote ai and μi are the multiplicities of h∗ JX,m and Jach along Ei . Then the relative canonical divisor of h can be expressed as  1 KX  /X = μi − ai Ei . m i∈I

Proof. — This follows immediately from proposition 3.2.5. 3.3. Motivic Igusa Zeta Functions (3.3.1). — Let k be a field and let X be an integral k-variety. Let I be a nonzero coherent sheaf of ideals on X, and let Y be the closed subscheme of X defined by I . In §4/4.4.3, we have defined a function: ordI : L∞ (X) → Z ∪ {+∞} on the arc scheme L∞ (X) of X: for every γ ∈ L∞ (X), ordI (γ) is the infimum of the orders of the elements γ ∗ f , where f ∈ Iγ(0) {0}. Since Y = X, the closed set L∞ (Y ) of L∞ (X) has measure zero. Moreover, the function ordI (γ) is finite and constructible on L∞ (X) L∞ (Y ). We thus can define the motivic Igusa zeta function of I by the formula  ∞  n T ordI dμX = μX (ord−1 (3.3.1.1) Z(X, I ; T ) = I (n))T L∞ (X)

n=0

X [[T ]]. For every subscheme j : W → X, we define the motivic Igusa zeta in M function of I along W as the image of Z(X, I ; T ) under the base-change X [[T ]]→ M * morphism j ∗ : M W [[T ]], and we denote it by ZW (X, I ; T ). Example 3.3.2. — Let X be a k-variety of dimension d. Let Y be a closed subscheme of X, and denote by I the defining ideal sheaf of Y in X. Let W be a subscheme of X, and assume that X is smooth along W . In this case, the motivic Igusa zeta function ZW (X, I ; T ) can be computed in terms of the jet spaces of Y .

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n For every integer n, let us define Ln (Y )W = (θY,0 )−1 (W ∩ Y ), and let An ∞ be the set of arcs γ ∈ L∞ (X) such that ordI (γ) = n and θX,0 (γ) ∈ W . One has  ∞ −1 Am = (θX,n ) (Ln (Y )W ), mn

and since X is smooth along W , this implies the relation  e(W )L−d − μL∞ (X) (Am ) = e(Ln (Y )W )L−(n+1)d . mn

A straightforward computation then furnishes the equality:  (1 − T ) e(Ln (Y )W )L−(n+1)d T n = e(W )L−d − ZW (X, I ; T ). n0

The motivic Igusa zeta function can be computed explicitly on a log resolution of (X, I ). In order to establish this formula, we first prove an elementary lemma. Lemma 3.3.3. — Let X be a smooth k-variety of pure dimension d, and let E = E1 + . . . + Er be a reduced divisor with strict normal crossings on X. We denote by IEi the defining ideal sheaf of Ei on X, and we set D = ∩ri=1 Ei . Let n be an element of Nr>0 , and denote by Bn the set of points γ in L∞ (X) such that ordIEi (γ) = ni for every i in {1, . . . , r}. Then Bn is a constructible r subset in L∞ (X) and μX (Bn ) = e(D)(L − 1)r L−d− i=1 ni in MX . Proof. — Around every point of EI , we can find an open neighborhood U and an étale morphism h : U → Adk such that Ei ∩ U is defined by the equation xi ◦ h = 0, for every i in {1, . . . , r}. By the additivity of the motivic measure, it suffices to prove the result after replacing X by U . The morphism h induces isomorphisms Lm (U ) → Lm (Adk ) ×Ad U k

that commute with the truncation morphisms. Therefore, it is sufficient to consider the case where X = Adk and Ei is the hyperplane defined by xi = 0, for every i in {1, . . . , r}. If we set m = max{n1 , . . . , nr }, then it is clear that Bn is a constructible set of L∞ (Adk ) of level  m. More precisely, it is the inverse image under θAd ,m of the locally closed subset of k

Lm (Adk ) ∼ = Spec k[xi,j | 1  i  n, 0  j  m] defined by the equations xi,ni = 0 and xi,j = 0 for i = 1, . . . , r and j < ni . It follows that r μX (Bn ) = e(D)(L − 1)r L−d− i=1 ni in MX . Theorem 3.3.4. — Let k be a field, let X be an integral k-variety, and let I be a nonzero coherent sheaf of ideals on X. Denote by Y the closed subscheme of X defined by I . Assume that the pair (X, Y ) admits a log resolution h : X  → X such that the Jacobian ideal Jach is invertible on X  . Let (Ei )i∈I

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be the family of irreducible components of the union of the exceptional locus of h with the strict transform of Y . For every i ∈ I, let mi be the multiplicity of I OX  along Ei , and let μ i be the multiplicity of Jac h along Ei . For every subset J of I, we set EJ = j∈J Ej and EJ◦ = EJ j∈I J Ej . Then one has  (L − 1)L−1−μj T mj  Z(X, I ; T ) = L− dim(X) e(EJ◦ ) 1 − L−1−μj T mj J⊂I

j∈J

X [[T ]]. in M Proof. — We denote by E the exceptional locus of h, and by D its image in X. Then L∞ (E) has measure 0 in L∞ (X  ), and L∞ (D) has measure 0 in L∞ (X). Moreover, for every field extension k  of k, the morphism h induces a bijection L∞ (X  )(k  )

L∞ (E)(k  ) → L∞ (X)(k  )

L∞ (D)(k  )

because h is proper and induces an isomorphism X  E → X the change of variables formula (theorem 6/4.3.1) implies that  T ordI OX  L− ordjach dμX  Z(X, I ; T ) =

D. Thus,

L∞ (X  )

X [[T ]]. For every n ∈ NI , let Bn be the set of arcs γ ∈ L∞ (X  ) such in M that ordIEi (γ) = ni for every i in I, where IEi denotes the defining ideal  sheaf of Ei onX  . For every γ ∈ Bn , one has ordI OX  (γ) = mi ni and μi ni . Let J be the subset of I consisting of indices i such ordjach (γ) =  that ni > 0. Then by applying lemma 3.3.3 to the divisor j∈J Ei restricted to X  i∈J / Ei , we find that  −d− ni i∈I μX  (Bn ) = e(EJ◦ )(L − 1)Card(J) L in MX  . It follows that    Z(X, I ; T ) = μX  (Bn )T mi ni L− μi ni n∈NI

=

 

e(EJ◦ )(L − 1)Card(J) L−d−

J⊂I n∈NJ >0

=



e(EJ◦ )L−d (L − 1)Card(J)

J⊂I

=







L−

ni

L−



μi ni



(μi +1)ni

 T

 T

mi ni

mi ni

n∈NJ >0

e(EJ◦ )L−d (L − 1)Card(J)

J⊂I

 L−(muj +1) T mj 1 − L−(μj +1) T mj j∈J

X [[T ]], as was to be shown. in M Corollary 3.3.5. — Assume that (X, I ) admits a log resolution h : X  → X with invertible Jacobian ideal Jach . We keep the notations of theorem 3.3.4. Then for every subscheme W of X, the zeta function ZW (X, I ; T )

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lies in M W [[T ]]rat , and the valuative radius of convergence of its image in Mk [[T ]] satisfies the formula  μi + 1 ρ(ZW (X, I ; T )) = − inf i mi where the index i ranges over all the elements of I such that h−1 (W )∩Ei = ∅ and mi > 0. Proof. — By base change to W , theorem 3.3.4 implies that   L−(μj +1) T mj e(EJ◦ ∩ h−1 (W ))L−d (L − 1)Card(J) . ZW (X, I ; T ) = 1 − L−(μj +1) T mj J⊂I j∈J This immediately implies the formula for ρ(ZW (X, I ; T )). Corollary 3.3.6. — Let X be smooth and separated. Assume either that W = X or that k has characteristic zero. Assume moreover that (X, I ) admits a log resolution h : X  → X with invertible Jacobian ideal Jach (this condition is always satisfied when k has characteristic zero, by corollary 3.2.3). Then ρ(ZW (X, I ; T )) is equal to the opposite of the log canonical threshold of the pair (X, I ) along W . Proof. — If k has characteristic zero, then the assertion follows from proposition 2.3.4 and corollary 3.3.5; thus we may assume that X = W . Since X is smooth, we can rewrite ZW (X, I ; T ) in terms of the jet schemes of Y , the closed subscheme defined by I , as in example 3.3.2:  e(Ln (Y ))L−(n+1)d T n = e(X)L−d − Z(X, I ; T ). (1 − T ) n0

Comparing the valuative radii of convergence of both sides, we obtain the formula:  1 dim(Lm (Y )) − dim(X) = ρ(Z(X, I ; T )). lim m m By theorem 2.4.1, the left-hand side of this formula is precisely − lctW (X, I ). Remark 3.3.7. — Combining corollary 3.3.6 with the formula for the radius of convergence in corollary 3.3.5, we obtain a generalization of proposition 2.3.4 to positive characteristic in the case W = X. Corollaries 3.3.5 and 3.3.6 also yield an alternative proof of the formula for the log canonical threshold in terms of dimensions of jet schemes (theorem 2.4.1) in the case where k has characteristic zero and W = X. The log canonical threshold initially appeared in singularity theory under the name of complex singularity exponent as an abscissa of convergence. To simplify notation, let us assume that X = AdC and that W = {0}; let (f1 , . . . , fm ) be a finite generating family of the ideal I . Then lct0 (X, I ) is the supremum of all positive real numbers λ such that (|f1 |2 + · · · + |fm |2 )−λ is locally integrable around the origin.

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411

Motivic integration thus yields a purely algebraic characterization of the log canonical threshold which is close in spirit to the complex analytic point of view while allowing generalizations, for example, in positive characteristic. (3.3.8) A Refinement of the Motivic Igusa Zeta Function. — Assume that k has characteristic zero. When, moreover, X is smooth away from the support of I , it is possible to define the motivic Igusa zeta function Z(X, I ; T ) X [[T ]], such that the formula in as an element of MX [[T ]], rather than M theorem 3.3.4 holds already over MX [[T ]]. Let h : X  → X be a log resolution of (X, I ) such that h is an isomorphism over the complement of Y , the closed subscheme of X defined by I . Then the function ordjach takes only finitely many values on each of the fibers of ordI OX  . Consequently, the motivic integral  T ordI OX  L− ordjach dμX  L∞ (X  )

is defined as an element of MX [[T ]], and we can take this object as the def( inition of the refined zeta function Z(X, I ; T ). Since every pair of log resolutions can be dominated by a third one, the change of variables formula guarantees that this definition does not depend on the choice of the log resolution h. (3.3.9). — From the explicit formula given by theorem 3.3.4, we can also prove a functional equation for the motivic Igusa zeta function. Recall from corollary 2/5.1.8 that the ring Mk admits a unique involution D such that D(e(V )) = e(V )L− dim(V ) for every proper smooth integral k-variety V ; moreover, one has D(L) = L−1 . Analogously to §3.1.2, we define Mk [[T ]]rat as the subring of Mk [[T ]] generated by Mk and the power series (L − 1)

 Lb T a = (L − 1) Lbn T na , b a 1−L T n1

for all (a, b) ∈ N>0 × Z. Note that the substitution of T by 1/T defines a map: Mk [[T ]]rat → Mk [[T ]],

(L − 1)

Lb T a L−1 − 1 → . b a 1−L T 1 − L−b T a

Proposition 3.3.10 (Functional Equation). — Let k be a field of characteristic zero. Let X be a proper k-variety of pure dimension d, let I be a coherent ideal sheaf on X, and let Y = V (I ) be the closed subscheme defined by I . Assume that X is smooth away from Y . One has DZ(X, I ; T ) = Ld Z(X, I ; 1/T ) in Mk [[T ]]rat .

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Proof. — We adopt the notations from theorem 3.3.4, and we assume that h is an isomorphism over X Y . To compute DZ(X, I ; T ), we need to rewrite the formula   1 − L−1 Z(X, I ; T ) = L−d e(EJ◦ ) L−kj T aj 1 − L−kj −1 T aj J⊂I

j∈J

in terms of proper smooth k-varieties only. For every subsetJ of I, the ◦ ◦ implies the equality e(EJ ) = K⊃J e(EK ) disjoint union EJ = K⊃J EK  ◦ Card(J)−Card(K) and, by inversion, e(EJ ) = K⊃J (−1) e(EK ). Consequently, one has Z(X, I ; T )    (−1)Card(J)−Card(K) e(EK ) = L−d J⊂I K⊃J



j∈J



1 − L−1 L−kj T aj 1 − L−kj −1 T aj 

1 − L−1 L−kj T aj 1 − L−kj −1 T aj K⊂I J⊂K j∈J   1 − L−1 −kj aj = L−d (−1)Card(K) e(EK ) L T 1− 1 − L−kj −1 T aj = L−d

(−1)Card(K) e(EK )

K⊂I

= L−d

 J⊂I

= L−d



(−1)Card(J)

j∈K

 1 − L−kj −1 T aj − (1 − L−1 )L−kj T aj (−1)Card(J) e(EJ ) 1 − L−kj −1 T aj j∈J

(−1)Card(J) e(EJ )

J⊂I

 1 − L−kj T aj . 1 − L−kj −1 T aj

j∈J

By the definition of a divisor with strict normal crossings, EJ is smooth and purely of dimension d − Card(J) for every subset J of I. It is also proper, because it is closed in the proper k-variety X. Under the involution D of Mk , one thus has DZ(X, I ; 1/T )   1 − Lkj T −aj (−1)Card(J) e(EJ )LCard(J)−d = Ld 1 − Lkj +1 T −aj J⊂I

=



(−1)Card(J) e(EJ )

J⊂I

=



j∈J

 j∈J

(−1)Card(J) e(EJ )

J⊂I

kj

L

L T

−aj

(L−kj T aj − 1) − 1)

Lkj +1 T −aj (L−kj −1 T aj

 1 − L−kj T aj − 1 1 − L−kj −1 T aj

j∈J

= Ld Z(X, I ; T ). This concludes the proof of the corollary. (3.3.11). — In the prologue, we have introduced Igusa’s local zeta function (§1/3) and Denef–Loeser’s topological zeta function (§1/3.3) and established

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413

some of their properties. In the remainder of this section, we give a slight generalization of these definitions and show that they are specializations of the motivic Igusa zeta function. (3.3.12). — Let K be a local field; let R be its valuation ring, m its maximal ideal, and k its residue field; let q = Card(k). Let X be a separated R-scheme of finite type whose generic fiber X = X ⊗R K is smooth; let π : X (R) → X (k) be the reduction map. Let μX be the model measure on X (R), as defined in §1/1.6.9. For every subscheme W of X ⊗R k, we write ]W [ = π −1 (W ) in X (R). Let I be a coherent sheaf of ideals on X . The function ordI : X (R) → N ∪ {+∞} is defined as follows: view a point x ∈ X (R) as a morphism εx : Spec(R) → X ; then ε−1 x I · R is an ideal in R. If it is zero, we set ordI (x) = +∞; otherwise, it is of the form mn , for some n ∈ N, and we set ordI (x) = n. The geometric variant of Igusa’s local zeta function can be defined by  q −s ordI (x) dμX (x). ZW (X , I ; s) = ]W [

(3.3.13). — Let F be a number field with ring of integers OF . Let X be a separated OF -scheme of finite type such that X = X ⊗OF F is smooth. Let I be a coherent sheaf of ideals on X . Let W be a subscheme of X , and set W = W ⊗OF F . The motivic Igusa zeta function ZW (X, I ; T ) is then defined as an object of MW [[T ]]rat , and we can view it as an object in MW [[T ]]rat by forgetting the W -structure. Its étale realization χét (ZW (X, I ; T )) is an element of K0 (RepGF Q )[[T ]]rat . For every finite place v of F , let qv be the cardinality of the residue field of v, and let us choose a Frobenius element Frobv at v. One has Tr(Frobv |χét (1 − Lb T a )) = 1 − qvb T a . By substituting T = qv−s , where s is seen as a formal parameter, we get the nonzero element 1−qvb−as . This furnishes a ring morphism μv from MF [[T ]]rat to the ring Q(qv−s ). Proposition 3.3.14. — Let F be a number field with ring of integers OF . Let X be a separated OF -scheme of finite type such that X = X ⊗OF F is smooth. Let I be a sheaf of ideals on X and W a subscheme of X , and set W = W ⊗OF F . For every finite place v of F , we denote of residue field at v by Fv , and the cardinality of this field by qv . Then there exists an integer N such that for every finite place v of F such that the characteristic of Fv is larger than N , one has  qv−s ordI (x) dμX (x). μv (ZWF (XF , IF ; qv−s )) = ]W (Fv )[

Proof. — Let W = W ⊗OF F . We consider a resolution of the singularities h : Y → X of the pair (X, I ). Let N  1 be an integer and set A = OF [1/N ]. When N is sufficiently large, then h can be extended to a proper A-birational morphism Y → X ⊗OF A such that Y is smooth over Spec(A) and such that

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the zero locus of I OY is a divisor that has strict normal crossings with the exceptional locus of h, relatively over Spec(A). The result now follows from the comparison between formula 3.3.4 for the motivic integral and Denef’s formula for p-adic integrals explained in §1/3.2. (3.3.15). — Let us now explain the relation of the motivic Igusa zeta function with the topological zeta function. Let k be a field, let X be an integral k-variety, let I be a sheaf of ideals on X, and let W be a subscheme of X. Given a log resolution h : X  → X of (X, I ) as in the statement of theorem 3.3.4, the topological zeta function Ztop,W (X, I ; s) is the element of Q(s) defined by   1 Eu(h−1 (W ) ∩ EJ◦ ) . (3.3.15.1) Ztop,W (X, I ; s) = 1 + kj + saj J⊂I

j∈J

Proposition 3.3.16. — Under the homomorphism μtop : M [[T ]]rat → Q(s) of proposition 3.1.4, the motivic Igusa zeta function ZW (X, I ; T ) specializes to the topological zeta function. Proof. — By definition of μtop , one has the relations μtop (h−1 (W ) ∩ EJ◦ ) = Eu(h−1 (W ) ∩ EJ◦ ), μtop (L) = 1, and μtop ((1 − L−1 )L−ki −1 T ai (1 − L−ki −1 T ai )) = 1/(ai s + ki + 1), for every i. Thus under the morphism μ , the formula for the motivic zeta function in theorem 3.3.4 specializes to the expression (3.3.15.1) for the topological zeta function. 3.4. Stringy Invariants In this section, we assume that k has characteristic zero. (3.4.1). — Let X be an integral Q-Gorenstein separated k-variety (§A/2.4.2), and let m be a positive integer such that mKX is Cartier. Consider the coherent ideal sheaf JX,m on X defined in section 3.2.4. Proposition 3.4.2. — One has ρ(Z(X, JX,m ; T )) < −1/m if and only if X has log terminal singularities. Proof. — Consider a log resolution of the pair (X, JX,m ). Then X has log terminal singularities if and only if the discrepancies of the exceptional components of the resolution are strictly larger than −1. By corollaries 3.2.6 and 3.3.5, this is equivalent to saying that ρ(Z(X, JX,m ; T )) < −1/m. Thus if X has log terminal singularities, we can evaluate the motivic zeta function Z(X, JX,m ; T ) at T = L1/m . This leads to the following definition: Definition 3.4.3 (Gorenstein volume). — Let X be an integral separated k-variety with log terminal singularities. Let m  1 be an integer such

§ 3. MOTIVIC VOLUME AND BIRATIONAL INVARIANTS

415

that mKX is a Cartier divisor. The Gorenstein volume of X is the element k [L1/m ] given by of M μGor (X) = Z(X, JX,m ; L1/m ). More generally, one defines the Gorenstein measure on L∞ (X) by the converging integral  μGor (A) = 1A LordJX,m /m dμX , L∞ (X)

for every measurable subset A of L∞ (X). The Gorenstein volume μGor (X) of X is thus equal to μGor (L∞ (X)). n that the Gorenstein volume It follows from the relation JX,nm = JX,m does not depend on the choice of the integer m such that mKX is Cartier. Proposition 3.4.4. — Let h : X  → X be a log resolution of X, and write  ki Ei . KX  /X = i∈I

Then



μGor (X) = L− dim(X)

e(EJ◦ )

J⊂I

 j∈J

L−1 . Lkj +1 − 1

Proof. — Let m be a positive integer such that mKX is Cartier. Then it follows from theorem 3.3.4 that we can write  L−1  L− dim(X) e(EJ◦ ) Lkj +1 − 1 J⊂I



j∈J

is equal to Z(X , O(−mKX  /X ); L ). Now the change of variables formula for motivic integrals implies that Z(X  , O(−mKX  /X ); L1/m ) does not depend on the choice of the resolution X  . Thus we may assume that h : X  → X is a log resolution for (X, JX,m ). Let Fi , i ∈ I  , be the prime divisors in the union of the exceptional locus of h and the zero locus of JX,m OX  . For every i in I  , let μi and ai be the multiplicities of Jach and JX,m OX  along Fi . Then theorem 3.3.4 shows that   (L − 1)L−μj Laj /m e(FJ◦ ) μGor (X) = L− dim(X) 1 − L−μj Laj /m J⊂I  j∈J 1/m

= L− dim(X)



e(FJ◦ )

J⊂I 

= L− dim(X)



J⊂I

e(EJ◦ )



j∈J



j∈J

L−1 Lμj −aj /m − 1 L−1 , Lkj +1 − 1

where the final step follows from corollary 3.2.6.

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Example 3.4.5. — If X is smooth, then one can take m = 1; then JX,1 = OX and μGor is the usual motivic measure on L∞ (X). In particular, one has μGor (X) = L− dim(X) e(X). Example 3.4.6. — Let X be an integral separated k-variety with log terminal singularities, and let p : Y → X be a crepant resolution. Then one has μGor (X) = L− dim(Y ) e(Y ). (3.4.7) Stringy Euler Characteristic and Stringy Hodge Numbers. — Let X be a separated integral k-variety with log terminal singularities. The motivic measure associated with the Hodge–Deligne polynomial extends to a ring morphism: E : Mk [L1/m ] → Z[[u−1/m , v −1/m ]][u, v]. The stringy Hodge–Deligne invariant of X is defined by (3.4.7.1)

HDstr (X) = HD(Ldim(X) μGor (X)).

If X is smooth, then HDstr (X) = HD(X), the usual Hodge–Deligne polynomial of X. In general, the stringy Hodge–Deligne invariant of X can be computed on a log resolution as above:   uv − 1 . HD(EJ◦ ) Estr (X) = (uv)νi +1 − 1 J

i∈J

This formula shows in particular that Estr (X) belongs to Q(u1/m , v 1/m ). The stringy Euler characteristic of X is then defined by the formula   1 . Eu(EJ◦ ) (3.4.7.2) Eustr (X) = νi + 1 J

i∈J

Assume that Estr (X) ∈ Z[u, v]. Then one defines the stringy Hodge numbers of X by the formula  p q Estr (X) = (−1)p+q hp,q str (X)u v . p,q

When X is smooth, one has μ (X) = L− dim(X) e(X). Thus when X is smooth and proper, the stringy Hodge numbers of X are defined and coincide with the usual Hodge numbers of X. Gor

Remark 3.4.8. — Assume that X is proper and of pure dimension d. Rewriting the formula for Estr (X) in terms of the smooth and proper kvarieties EJ as in the proof of proposition 3.3.10, one sees that the stringy Hodge–Deligne invariant of X satisfies the symmetry formula Estr (X; 1/u, 1/v) = (uv)−d Estr (X; u, v). If the stringy Hodge numbers of X are defined, this implies in particular that for all (p, q), one has d−p,d−q (X). hp,q str (X) = hstr

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417

(3.4.9). — Suppose that X is proper. Assuming that the stingy Hodge numbers of X are defined, it is an open question (Batyrev 1999a, conjecture 3.10) whether they are nonnegative and vanish for max(p, q) > dim(X), like in the smooth case. An even more fundamental problem is to interpret these numbers as the dimensions of some “orbifold cohomology groups,” such as those of Chen and Ruan (2004). Particular cases have been studied: crepant resolutions (example 3.4.6), toroidal Gorenstein singularities (Batyrev and Dais 1996), ADE singularities (Schepers 2006), quotient singularities (Yasuda 2004), etc. For related work, see also Veys (2004), Ito (2004), Schepers and Veys (2007), and Schepers and Veys (2009).

3.5. The Theorem of Batyrev–Kontsevich (3.5.1). — The history of motivic integration began in a 1995 seminar lecture at Orsay given by Maxim Kontsevich, where he introduced the definition of the motivic integral and stated the change of variables formula. Its goal was to generalize the theorem of Batyrev (1999a) about the equality of Betti numbers of two complex birational Calabi–Yau varieties so as to prove the equality of Hodge numbers. The original proof of Batyrev’s theorem uses p-adic integration and has been presented in §1/2. We have also explained there how Kontsevich’s theorem follows. In this section, we present the motivic integration approach. We also explain the subsequent developments concerning K-equivalence of singular varieties. (3.5.2). — Let k be a field; let X and Y be proper integral Q-Gorenstein kvarieties. Recall (definition A/2.5.1) that a K-equivalence between X and Y is a diagram Z f

X

g

Y

where Z is a smooth k-variety and f and g are proper birational morphisms such that KZ/X = KZ/Y ; if such a diagram exists, then one says that X and Y are K-equivalent. This condition obviously implies that X and Y are birational. (3.5.3). — Lemma A/2.5.3 furnishes geometric hypotheses on X and Y where birational equivalence implies K-equivalence. Modulo existence of resolution of singularities for k-varieties of dimension dim(X) is, for example, the case when X and Y are Calabi–Yau varieties, meaning that their canonical classes KX and KY are trivial (example A/2.5.4).

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Theorem 3.5.4 (Yasuda 2004). — Let X and Y be integral separated kvarieties with log terminal singularities. If X and Y are K-equivalent, then μGor (X) = μGor (Y ). Proof. — This follows immediately from proposition 3.4.4. Corollary 3.5.5 (Kontsevich 1995). — Let X and Y be smooth integral separated k-varieties. If X and Y are K-equivalent, then e(X) = e(Y ) in Mk . Let us moreover assume that X and Y are proper and smooth. If k = C, the corollary implies in particular that X and Y have isomorphic Hodge structures (corollary 2/4.3.8). In general (the field k still having characteristic zero), this implies that X and Y have the same Hodge numbers. If X and Y are birational proper Calabi–Yau varieties, in the sense that their canonical class is trivial modulo numerical equivalence, then they are K-equivalent (see example A/2.5.4). Thus we recover from corollary 3.5.5 the celebrated theorem of Kontsevich! Remark 3.5.6. — We have discussed in chapter 2 the geometric content of an equality of the form e(X) = e(Y ) in Mk . For example, theorem 2/6.3.2 shows that it does not imply the existence of a piecewise isomorphism in general. However, Ivorra and Sebag (2012) asked whether, in the context of Kontsevich’s theorem, this equality could be lifted to Mk , K0 (Vark ), or even to K0+ (Vark ), that is, to the existence of a piecewise isomorphism between X and Y . A first result in that direction can be found in Sebag (2010b), where this assertion is indeed proved in dimension  4. In another direction, one can ask whether this equality can be lifted to an isomorphism of Chow motives (see §2/5.3). (3.5.7) McKay Correspondence. — Let G be a finite subgroup of SL(n, C) acting on the affine space AnC , and let X = AnC /G. In general, the variety X is singular, but it is always Gorenstein and has at most canonical singularities. Assume that X admits a crepant resolution p : Y → X. The McKay correspondence predicts that the cohomology of Y is related to the set of conjugacy classes of G; in particular, it predicts that the Euler characteristic of Y is equal to the number of conjugacy classes in G. It has been proved by Batyrev (1999b). Let L∞ (X)0 ⊂ L∞ (X) be the set of arcs based at the origin in X. Denef and Loeser (2002c) have given a more precise formula, which computes, in , the Gorenstein volume μGor (L∞ (X)0 ) as a an appropriate quotient of M −w(g) sum of terms of the form L , where g runs along a set of representatives of conjugacy classes in G, and w(g), “the age of g,” is an integer associated with the eigenvalues of g ∈ SL(n, C).

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§ 4. DENEF–LOESER’S ZETA FUNCTION AND THE MONODROMY CONJECTURE 4.1. Motivic Zeta Functions Associated with Hypersurfaces (4.1.1) A Naive Definition. — Let k be a field. Let X be a smooth connected k-variety. Let f : X → A1k be a flat morphism, and denote by X0 the closed subscheme of X defined by f = 0. We consider the coherent sheaf I := f OX of ideals of OX . For every integer n  1, ord−1 I (n) is a con−1 structible subset of L (X) contained in θX,0 (X0 ). Since X is assumed to be smooth, it has a measure in MX0 . Then, following definition 3.3.1, we set  n (4.1.1.1) Zfnaive (T ) = ZX0 (X, I ; T ) = μX (ord−1 I (n))T n1

in MX0 [[T ]]. For every subscheme j : W → X0 , we also define the power naive (T ) by j ∗ Zfnaive (T ); this is an element of MW [[T ]]. As a direct series Zf,W application of theorem 3.3.4, we obtain the following statement: Theorem 4.1.2. — Let W be a subscheme of X0 . Let h : X  → X be a log resolution of the pair (X, f OX ) that is an isomorphism over X X0 . Let (Ei )i∈I be the family of irreducible components of h−1 (X0 ) ∪ Exc(h). For every i ∈ I, let ai be the multiplicity of f ◦ h along Ei , and let ki be thediscrepancy of Ei with respect to X. For every subset J of I, let EJ = j∈J Ej and EJ◦ = EJ j∈I J Ej . Then one has naive Zf,W (T ) =

 J⊂I

e(h−1 (W ) ∩ EJ◦ )

 (L − 1)L−kj −1 T aj 1 − L−kj −1 T aj

j∈J

in MW [[T ]]. naive In particular, Zf,W (T ) belongs to MW [[T ]]rat if k has characteristic zero.

(4.1.3) A Refined Definition. — For various purposes developed in the following paragraphs, it is useful to adapt definition 4.1.1 in order to obtain a richer object. For every positive integer n, we set: An = {x ∈ L∞ (X); ordI (x) = n and ac(f (x)) = 1} = {x ∈ L∞ (X); f (x) = tn (1 + tϕ(t)) with ϕ(t) ∈ κ(x)[[t]]} Recall that ac(f (x)) is the angular component of the arc f (x); see §6/5.1.1. The sets An are constructible closed subsets of L∞ (X). Indeed, if we set 4 5 (4.1.3.1) Zn = x ∈ Ln (X); f (x) ≡ tn (mod (tn+1 )) , −1 then we have An = θn,X (Zn ). Moreover, since n is positive, the image of the ∞ is contained in X0 . constructible set An under the truncation morphism θX,0 Thus it is meaningful to consider the elements e(Zn ) ∈ K0 (VarX0 ) and the

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power series (4.1.3.2)

Zf (T ) =

 n1

μX (An )T n =



e(Zn )L−(n+1) dim(X) T n

n1

in MX0 [[T ]]. We also define, for every subscheme j : W → X0 , the power series Zf,W (T ) as j ∗ Zf (T ). We call this object the motivic zeta function of f with support on W , or simply the motivic zeta function of f if W = X0 . In particular, if x is a closed point on X0 with residue field κ(x), then Zf,x (T ) is an element of Mκ(x) [[T ]] which is called the local motivic zeta function of f at x. Remark 4.1.4. — Assume that k has characteristic zero. Let R be a ring, and let ∼ be an adequate equivalence relation on algebraic cycles. By proposition 2/5.3.3, one knows that χ∼,R : Mk → K0 (Mot∼,R )(k) is a motivic measure. Then, by specializing the motivic zeta function with respect to this motivic measure, we obtain a zeta function χ∼,R (Zf (T )) ∈ K0 (Mot∼,R )(k). This is the object that was originally defined by Denef and Loeser (1998). The passage to motives was necessary to define an isotypical decomposition of the zeta function that reflects the character twists of Igusa’s p-adic zeta function. Example 4.1.5. — Assume that k has characteristic different from 2. Let X = A2k = Spec(k[T1 , T2 ]), and let f : X → A1k be the morphism defined by f = (T1 )2 T2 . Then the support of X0 is the union of the coordinate axes E1 and E2 in A2k . Let n be a positive integer, let K be a field extension of k, and let ϕ = (ϕ1 , ϕ2 ) be an element of 2

L∞ (X)(K) = K[[t]] . ordt (ϕ1 ) + ordt (ϕ2 ) = n and Then ϕ belongs to An if and only if 2 j ac(ϕ1 )2 ac(ϕ2 ) = 1. Expanding ϕi (t) as j0 ci,j t for i = 1, 2, we see that An is the disjoint union of closed subsets An, for ∈ {0, . . . , !n/2"} where An, is defined by the conditions that c1,j = 0 for j < , c2,j = 0 for j < n − 2 , and (c1, )2 c2,n−2 = 1. The last equation defines a closed subvariety of the two-dimensional torus G2m,k which is isomorphic to G1m,k . The projection from An, to X0 maps ϕ to the couple (c1,0 , c2,0 ). Hence, we find that

n/2  ◦ −n−2 ( μX (An ) = e(E1 )L + (L − 1)L−n−2 =1

( ◦ is the degree two étale cover of E ◦ ∼ where E 1 1 = Gm,k obtained by taking a square root of the coordinate T1 . Bringing all of these terms together, it follows that Zf (T ) is equal to   −1 −2 3 ( ◦ )L−1 T 2 T T (L − 1)L L e( E 1 . + + L−2 1 − L−1 T 2 1 − L−1 T (1 − L−1 T 2 )(1 − L−1 T )

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421

(4.1.6). — Assume that (X, X0 ) admits a log resolution h : X  → X that is an isomorphism over X X0 . Then Denef and Loeser (2002a) have given an explicit formula for the motivic zeta function Zf (T ) similar to the one in theorem 3.3.4. Let (Ei )i∈I be the family of irreducible componentsof h−1 (X0 ). Let J ⊂ I be a nonempty set. Recall that we set EJ = j∈J Ej and ( ◦ → E ◦ as follows. For every i E ◦ = EJ \ Ei . We define the étale cover E J

i∈J

J

J

in I, we denote by ai the multiplicity of f ◦ h along Ei . Set aJ = gcdj∈J (aj ). ◦ Then around every point of E find an open subscheme U of X  on J , we acan j which we can write f ◦h as u j∈J xj where xj = 0 is an equation for Ej ∩U and u is an invertible function on U . We consider the étale cover U  → U defined by U  = Spec(O(U )[T ]/(uT aJ − 1)). By base change, it induces an étale cover of EJ◦ ∩ U . These local étale covers do not depend on any choices (◦ → E ◦ . and glue together to an étale cover E J J

Theorem 4.1.7. — Let X be a smooth connected k-variety of dimension d, let f : X → A1k be a flat morphism, and denote by X0 the closed subscheme of X defined by f = 0. Let h : X  → X be a log resolution of the pair (X, X0 ) that is an isomorphism over X X0 . Let (Ei )i∈I be the family of irreducible components of X0 = h−1 (X0 ). For every i ∈ I, let ai be the multiplicity of Ei in X0 , and let ki be the discrepancy of Ei with respect to X. Then one has   L−kj −1 T aj (J◦ ) Zf (T ) = L−d (L − 1)Card(J)−1 e(E 1 − L−kj −1 T aj ∅=J⊂I

j∈J

in MX0 [[T ]]. Proof. — The proof is similar to the one in theorem 3.3.4 but requires a few refinements. We denote by IEi the defining ideal sheaf of Ei in X  . For every n in NI , we denote by An the set of arcs γ in L∞ (X  ) such that ordIEi (γ) = ni for every i in I and ac((f ◦ h)(γ)) = 1. Then for every arc γ   in An , we have that ordt (h ◦ f )(γ) = i∈I ai ni and ordjach (γ) = i∈I ki ni . Hence, the change of variables formula yields    − ki ni a n i∈I μX  (An )L T i∈I i i Zf (T ) = n∈NI {0}

in MX0 [[T ]]. We fix a nonzero element n in NI , and we denote by J the set of indices i ∈ I such that ni = 0. Then it suffices to show that  ( ◦ )L−d− j∈J nj (4.1.7.1) μX  (An ) = (L − 1)Card(J)−1 e(E J in MX0 . We choose a bijection of J with {1, . . . , r}. We denote by m the maximum of the numbers ni , i ∈ I. Then the set An is a constructible subset of L∞ (X  ) of level  m. By the scissor relations in the Grothendieck ring, we may assume that X  is affine and endowed with an étale morphism ϕ : X  → Adk = Spec(k[z1 , . . . , zd ])

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such that Ej is defined  by the equation zj ◦ ϕ = 0 for every j ∈ J. Then we can write h ◦ f = u j∈J (zj ◦ ϕ)aj for some invertible function u on X  . The morphism ϕ induces an isomorphism Lm (X  ) → X  ×Ad Lm (Adk ). k

This isomorphism allows us, for every field extension K of k, to identify the n-jets in Lm (X  )(K) with the tuples (γ, x) ∈ (K[t]/(tm+1 ))d × X  (K) ∞  such that ϕ(x) = γ(0). Such a tuple (γ, x) lies in θX  ,m (An ) if and only if  aj ordt (γj ) = nj for every j ∈ J and u(x) j∈J ac(γj ) = 1. Let Y be the closed subscheme of EJ◦ ×k Spec(k[vj±1 | 1  j  r])  a defined by the equation u j∈J vj j = 1. We consider the morphism Y ×k Spec(k[wi, | 1  i  d, ni + 1   m]) → Lm (Adk ) ∼ = Spec(k[zi, | 0  i  d, 0   m]) defined by zj,nj → vj for 1  j  r, zi, → 0 whenever < ni , zi,0 → zi ◦ ϕ for r +1  i  d and zi, → wi, for all the remaining couples (i, ). It induces a morphism of X  -schemes:  md− nj j∈J g : Y ×k Ak → X  ×Ad Lm (Adk ). k

∞  The morphism g is an immersion whose image is precisely θX It  ,m (An ). follows that  md− nj ∞  j∈J e(θX  ,m (An )) = e(Y )L

in MX0 . ( ◦ ×k Gr−1 . Finally, we show that the EJ◦ -scheme Y is isomorphic to E J m,k Choosing an automorphism of Zr that maps (a1 , . . . , ar ) to (aJ , 0, .  . . , 0), we a can construct a k-automorphism of k[vj±1 | 1  j  r] that sends j∈J vj j to v1aJ . After this change of coordinates, the equation of Y takes the form uv1aJ = 1. Then by projection onto the first coordinate and the last r − 1 ( ◦ ×k coordinates of Grm,k , we find an isomorphism of EJ◦ -schemes Y → E J Gr−1 m,k . This yields  nj ∞  r−1 md− (◦ j∈J e(θX L  ,m (An )) = e(EJ )(L − 1) which implies (4.1.7.1) by the definition of the motivic volume. Corollary 4.1.8. — For every closed point x of X0 , we have Zf,x (T ) = L−d

 ∅=J⊂I

in Mx [[T ]].

( ◦ ∩ h−1 (x)) (L − 1)|J|−1 e(E J

 j∈J

L−kj T aj 1 − L−kj T aj

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423

4.2. The Motivic Nearby Fiber (4.2.1). — Assume that k has characteristic zero. We keep the notation of theorem 4.1.7. Denef and Loeser (2002b) have defined the motivic nearby fiber ψf of the morphism f by taking a formal limit of the motivic zeta function Zf (T ) for T → +∞. More precisely, this limit is defined by means of the following proposition. Proposition 4.2.2. — We denote by MX0 [[T ]]ζ the subring of MX0 [[T ]] generated by MX0 and the geometric series Lb T a /(1−Lb T a ), for all integers a, b with a > 0. Then there exists a unique morphism of MX0 -algebras lim : MX0 [[T ]]ζ → MX0 [[T ]]

T →+∞

that maps each of the geometric series Lb T a /(1 − Lb T a ) to −1. Proof. — The only thing to check is that this morphism is well-defined. Every element in MX0 [[T ]]ζ can be written as a quotient P (T )/Q(T ) of polynomials of the same degree with coefficients in MX0 such that the leading coefficients of P (T ) and Q(T ) are invertible. The quotient of these leading coefficients does not depend on the choice of P (T ) and Q(T ), and this is precisely P (T ) . lim T →+∞ Q(T ) Theorem 4.1.7 guarantees that Zf (T ) lies in MX0 [[T ]]ζ , so that we can make the following definition: Definition 4.2.3. — Let k be a field of characteristic zero. Let X be a smooth connected k-variety, and let f : X → A1k be a flat morphism. The motivic nearby fiber of f , denoted by ψf , is the element of MX0 defined by ψf = −Ldim(X) lim Zf (T ). T →+∞



Proposition 4.2.4. — Let h : X → X be a log resolution of the pair (X, X0 ) that is an isomorphism over X X0 . Let (Ei )i∈I be the family of irreducible components of h−1 (X0 ). Then one has  (J◦ )(1 − L)Card(J)−1 ψf = e(E ∅=J⊂I

in MX0 . Proof. — This follows immediately from the formula for the motivic zeta function in theorem 4.1.7. Example 4.2.5. — If the morphism f : X → A1k is smooth, then ψf = 1 in MX0 .

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(4.2.6). — For every subscheme p : W → X0 , we define the motivic nearby fiber of f with support on W by the formula ψf,W = p∗ ψf

∈ MW .

If x is a closed point of X0 , then the object ψf,x is also called the motivic Milnor fiber of f at x. In terms of the log resolution h in theorem 4.1.7, we can write it explicitly as  (J◦ ∩ h−1 (x))(1 − L)Card(J)−1 e(E ψf,x = ∅=J⊂I

in Mx . (4.2.7). — One should think of the object ψf as the class of a fiber of f that lies infinitesimally close to X0 . Equivalently, one can view it as the motivic incarnation of the complex of nearby cycles of the morphism f . Likewise, if k = C, one can think of ψf,x as the motivic incarnation of the topological Milnor fiber Fx of f at the point x. This viewpoint is justified by the fact that ψf,x has the same cohomological invariants as the Milnor fiber. In particular, it was proven in (Denef and Loeser 1998, §4.2) that the Hodge realization of ψf,x (see §2/3.2) coincides with the class of the limit mixed Hodge structure on the cohomology of Fx in the Grothendieck group of Hodge structures. Ivorra and Sebag (2013) have shown that the image of the realization of ψf in a suitable Grothendieck ring of triangulated motives (see §2/3.7) coincides with the class of the nearby motivic sheaf introduced by Ayoub (2007a,b). The analogous result holds for ψf,x . In particular, ψf and ψf,x can be interpreted as Euler characteristics of complexes of Chow motives. Ayoub et al. (2017) complete this picture by showing that the motivic rigid framework introduced by Ayoub (2015) also provides a comparison between the analytic Milnor fiber introduced in Nicaise and Sebag (2007b) (see §6.6.3) and the motivic Milnor fiber ψf,x . Remark 4.2.8. — The motivic zeta function and motivic nearby fiber defined by Denef and Loeser (2001) are endowed with an additional structure, namely, an action of the profinite group scheme of roots of unity over k. This structure is important because it encodes the monodromy action on the nearby cycles complex of f . In order to define this group action on ψf , one needs to refine the change of variables formula to an equivariant setting. For motivic integrals on formal schemes, this was worked out by Hartmann (2015). We have not presented the equivariant version here because it adds another layer of technical complexity in the proofs. 4.3. Lefschetz Numbers of the Monodromy (4.3.1). — Let X be a connected and smooth complex variety, endowed with a flat morphism f : X → A1C . Let Fx be the Milnor fiber associated with f an at x ∈ X0 (C) (see (1/3.4.3)). We denote by Mx the monodromy

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425

∗ operator which acts on Hsing (Fx , C). Then the monodromy zeta function ζf,x is an alternating product of characteristic polynomials of Mx :  m+1 m (4.3.1.1) ζf,x (T ) = det(Id − T Mx ; Hsing (Fx , C))(−1) . m0

A’Campo (1975) gave an explicit formula for ζf,x (T ) in terms of a log resolution of the pair (X, X0 ): Theorem 4.3.2 (A’Campo 1975). — Let h : X  → X be a log resolution of the pair (X, X0 ) that is an isomorphism over X X0 . Let (Ei )i∈I be the family of irreducible components of h−1 (X0 ). For every i ∈ I, let ai be the multiplicity of Ei in h−1 (X0 ). Then one has  ◦ −1 ζf,x (T ) = (1 − T ai )− Eu(Ei ∩h (x)) . i∈I

(4.3.3). — Let us recall that Euler characteristic for singular cohomology with compact support induces a motivic measure Eu : MC → Z. We extend it into a ring morphism:   Eu : MC [[T ]]→ Z[[T ]], an T n → Eu(an )T n . n0

n0

Then we have the following comparison theorem: Theorem 4.3.4. — Let X be a smooth connected complex variety endowed with a flat morphism f : X → A1C . Then Eu(Zf,x (T )) = T

d log(ζf,x (T )) dT

for every x ∈ X0 (C). Proof. — By theorem 4.3.2, we have d log(ζf,x (T ))  T ai T = ai Eu(Ei◦ ∩ h−1 (x)) dT 1 − T ai i∈I ai  (i◦ ∩ h−1 (x)) T = Eu(E . 1 − T ai i∈I

By corollary (4.1.8), this is precisely Eu(Zf,x (T )). (4.3.5). — Theorem 4.3.4 yields a cohomological interpretation of the motivic zeta function that was first obtained by Denef and Loeser (2002a). For every n  0, the n-th Lefschetz number of the monodromy transformation Mx at x is the integer  Λf,x (n) = (−1)m Tr(Mxn ; H m (Fx , C)). m0

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Theorem 4.3.6 (Denef and Loeser 2002a). — Let X be a smooth connected complex variety, endowed with a flat morphism f : X → A1C . Let x be a closed point on X0 . We have the following equality  Eu(Zf,x (T )) = Λf,x (n)T n n>0

in Z[[T ]]. Going back to the definition of Zf,x , this equality can be rewritten as follows. For every integer n > 0, let Zn,x be the closed subset of Ln (X) consisting of the n-jets ϕ such that θ0n (ϕ) = x and f (ϕ(t)) ≡ tn mod tn+1 . Since the n-th coefficient of Zf,x (T ) is given by e(Zn,x )L− dim(X)(n+1) , the theorem is equivalent to the equality Eu(Zn,x ) = Λf,x (n), for every integer n > 0. Proof. — In view of theorem 4.3.4, it suffices to show that  d log(ζf,x (T )) . Λf,x (n)T n = T dT n>0 This follows from the following classical result in linear algebra. Lemma 4.3.7. — Let k be a field of characteristic zero, let V be a finite dimensional k-vector space, and let M be an endomorphism of V . Then  d log(det(1 − M T )) . Tr(M n )T n = −T dT n>0 Proof. — To prove the formula, we may extend the scalars to an algebraic closure of k. We may then assume that V admits a basis in which the matrix of M is upper triangular, with diagonal (a1 , . . . , ad ). Then Tr(M n ) = d d n i=1 ai and det(1−M T ) = i=1 (1−ai T ). Consequently, the given formula boils down to the relation  d log(1 − aT ) , an T n = −T dT n>0 itself a consequence of the power series expansion 1 an T n , log(1 − aT ) = − n n>0 for every a ∈ k. Remark 4.3.8. — Ultimately, the proof of theorem 4.3.6 relies on an explicit computation of both sides of the equality using a log resolution of (X, X0 ), and the verification that the obtained formulas coincide. Hrushovski and Loeser (2015) have given an alternative proof that does not use resolution of singularities.

§ 5. MOTIVIC INVARIANTS OF NON-ARCHIMEDEAN ANALYTIC SPACES

427

4.4. The Motivic Monodromy Conjecture Denef and Loeser (1998) gave a motivic generalization of Igusa’s monodromy conjecture presented in 1/3.4.8. Conjecture 4.4.1. — Let k be a subfield of the field C of complex numbers. Let X be a smooth k-variety endowed with a flat morphism f : X → A1k of k-schemes. Then there exists a finite subset S ⊂ N∗ × N∗ such that Zf (T ) and Zfnaive (T ) belong to 6  7 1 Mk T, 1 − L−b T a (a,b)∈S and, for every (a, b) ∈ S, there is a point x ∈ X0 (C) such that exp(2iπb/a) is an eigenvalue of the monodromy transformation Mx acting on the singular cohomology of the Milnor fiber Fx of f at x. Conjecture 4.4.1 implies the original conjecture of Igusa, at least for sufficiently large primes. Proposition 4.4.2. — Let F be a number field and let f be a polynomial in F [X1 , . . . , Xd ]. If conjecture 4.4.1 holds for the naive motivic zeta function Zfnaive (T ), then conjecture 1/3.4.8 holds for the Igusa zeta function Zϕ (f, s) at every finite place v of OF of sufficiently large residue characteristic. Proof. — This follows from the fact that Zfnaive (T ) specializes to Zϕ (f, s) for sufficiently large primes; see proposition 3.3.14. (4.4.3). — This conjecture has essentially been proven in the same cases as those of Igusa’s monodromy conjecture (see 1/3.4.14). It holds in particular when X is a surface, as well as for some special classes of singularities. The general case remains wide open. In §(6.7) we will discuss an analogous problem for degenerations of Calabi– Yau varieties.

§ 5. MOTIVIC INVARIANTS OF NON-ARCHIMEDEAN ANALYTIC SPACES In this section and the next, we will explain how one can use motivic integration on formal schemes to attach motivic invariants to non-Archimedean analytic spaces. The invariants we discuss here are the motivic integral of a volume form and the motivic Serre invariant. Both of these were first introduced in Loeser and Sebag (2003). These invariants should be viewed as geometric upgrades of the integral of a volume form on a p-adic manifold and the p-adic Serre invariant (see section 1/1). We will explain how they specialize to their p-adic counterparts in section 5.4. We will use the theory of Néron

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smoothenings to define motivic integrals in the localized Grothendieck ring . This provides M (or its modified variant), rather than the completion M finer information, since we do not know whether the completion morphism  is injective. The refinement is especially pertinent when we conM →M struct the motivic Serre invariant, which is well-defined only modulo the ideal  but not in M . (L − 1), which is the unit ideal in M Throughout this section, we adopt the following notation. We denote by R a complete discrete valuation ring with maximal ideal m, by K its quotient field and by k its residue field. All K-analytic spaces are assumed to be strictly K-analytic and Hausdorff. An analytic domain immersion of K-analytic spaces is a morphism f : Y → X where Y is an analytic domain in X and f is the embedding of Y in X. We make a similar notational convention as in chapter 6: let S be a kscheme. If R has equal characteristic, then MSR denotes the usual localized Grothendieck ring K0 (VarS )[L−1 S ] of varieties over the scheme S; if R has mixed characteristic, then MSR will denote the modified Grothendieck ring MSuh obtained by identifying the classes of universally homeomorphic Svarieties. If X is an S-variety, we will write e(X) instead of e(X/S) or euh (X/S) when it is clearly indicated that this class is considered in MSR . Likewise, we will write L instead of LS . 5.1. Néron Smoothening for Formal R-schemes Formally of Finite Type (5.1.1). — We first extend the construction of Néron smoothenings in section 4/3.4 to formal R-schemes that are formally of finite type, using the standard technique to produce formal R-schemes of finite type from formal R-schemes formally of finite type, called dilatation. Let Y be a flat formal R-scheme formally of finite type, and denote by I its largest ideal of definition. Let h : Y → Y be the formal blow-up of Y at I , and denote by X the largest open formal subscheme of Y where the ideal I OY is generated by m. The formal R-scheme X is called the dilatation of Y. Since I OY is an ideal of definition on Y , the formal scheme X is adic over R, and thus of finite type. Moreover, it is flat over R because Y is flat over R, and the morphism of formal k-schemes X0 → Y0 factors through a morphism of k-schemes X0 → Yred . The dilatation morphism X → Y is characterized by the following universal property: if X is a flat formal R-scheme of finite type and X → Y is a morphism of formal R-schemes such that X0 → Y0 factors through X0 → Yred , then there exists a unique morphism of formal R-schemes X → X whose composition with X → Y is the given morphism X → Y. This follows easily from the universal property of the formal blow-up; see Nicaise (2009, 2.22).

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Let us point out an interesting special case of the universal property: if X is a flat formal R-scheme of finite type such that X0 is reduced, then every morphism of formal R-schemes X → Y factors uniquely through the dilatation X → Y. In particular, if R is a finite unramified extension of R, then the map X(R ) → Y(R ) is a bijection. If Xη is quasi-smooth over K, this implies at once that Xη is néronian, because for every finite unramified extension K  of K, every K  -point on Xη is contained in the compact analytic domain Xη . Example 5.1.2. — If Y = Spf(R[[S1 , . . . , Sm ]]{T1 , . . . , Tn }) and π is a uniformizer of R, then the dilatation X of Y is given by X = Spf(R{

Sm S1 ,..., , T1 , . . . , Tn }). π π

(5.1.3). — If Y is a formal R-scheme formally of finite type, then a Néron smoothening of Y is a morphism of formal R-schemes h : X → Y such that X is smooth and of finite type over R, hη : Xη → Yη is an analytic domain immersion, and the pair (X, hη ) is a weak Néron model for Yη . The latter condition is equivalent to saying that the map X(R ) → Y(R ) is a bijection for every finite unramified extension R of R. Note that, since X is of finite type over R and X0 is reduced, every Néron smoothening factors uniquely through the dilatation of Y. If X → Y and X → Y are Néron smoothenings, then a morphism of Néron smoothenings X → X is a morphism of formal Y-schemes X → X. Such a morphism is unique if it exists, and in that case we say that X → Y dominates X → Y. Note that X → X is still a Néron smoothening. Proposition 5.1.4. — A formal R-scheme formally of finite type Y has a Néron smoothening X → Y if and only if Yη is néronian. If X → Y and X → Y are Néron smoothenings, then there exists a Néron smoothening of Y that dominates both X and X . Proof. — The Néronian condition is obviously necessary. If Yη is Néronian, then the same holds for the generic fiber of the dilation Y of Y, so that Y has a Néron smoothening by theorem 4/3.4.5. Composing this smoothening with the dilation X → Y , we obtain a Néron smoothening for Y. If X and X are Néron smoothenings of Y, then we can construct a smoothening that dominates both of them by taking a Néron smoothening of X ×Y X .

5.2. Motivic Integration of Volume Forms on Rigid Varieties (5.2.1). — Let X be a formal R-scheme of finite type of pure relative dimension d with quasi-smooth generic fiber Xη , and let ω be a differential form of degree d on Xη . Let R be an extension of R with maximal ideal m and fraction field K  , and let ψ be a point in X(R ). We denote by ψη : M (K  ) → Xη

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the morphism of analytic spaces deduced from ψ by passing to the generic fiber. Then Lψ = ψ ∗ ΩdX/R /(torsion) is a rank one lattice over R , and ω gives rise to an element ψη∗ ω of the K  -vector space: ψη∗ ΩdXη /K = Lψ ⊗R K  . We denote by ordω (ψ) the valuation of ψη∗ ω with respect to the lattice Lψ . In other words, ordω (ψ) equals ∞ if ψη∗ ω = 0, and otherwise it is equal to the unique integer such that (π  )− · ψη∗ ω is a generator of Lψ , for any uniformizer π  of R . (5.2.2). — Let X be a smooth formal R-scheme of finite type, and let h : X → X be a morphism of formal R-schemes such that hη : Xη → Xη is quasi-étale. We denote by Jach the Jacobian ideal of h, that is, the 0-th Fitting ideal of the sheaf of relative differentials ΩX /X . Since hη is quasiétale, Jach contains a power of the maximal ideal m of R. Let R be an extension of R with maximal ideal m , and let ϕ be an element of X (R ). Then the pullback ϕ−1 (Jach ) · R of the ideal sheaf Jach is equal to m , for some unique nonnegative integer , which we denote by ordJach (ϕ). Since ΩX /R is locally free, the natural morphism h(ϕ)∗ ΩX/R → ϕ∗ ΩX /R factors through a morphism of free R -modules of rank d h(ϕ)∗ ΩX/R /(torsion) → ϕ∗ ΩX /R that fits into an exact sequence of R -modules h(ϕ)∗ ΩX/R /(torsion) → ϕ∗ ΩX /R → ϕ∗ ΩX /X → 0. Since the formation of Fitting ideals commutes with base change, we see by taking determinants that the image of Lh(ϕ) → Lϕ is equal to (m )ordJach (ϕ) . Thus we obtain the following chain rule: (5.2.2.1)

ordω (h(ϕ)) = ordω (ϕ) − ordJach (ϕ).

(5.2.3). — If R has mixed characteristic, we assume that k is perfect. If R has equal characteristic, we choose a section for the residue morphism R → k. In this way, we can speak of the Greenberg scheme Gr(X) of a formal R-scheme of finite type X. Recall that with each point x of Gr(X), we associated in (4/3.3.7) a point ψx on X with coordinates in the extension R(κx ) of R of ramification index one, where κx denotes the residue field of Gr(X) at x if R has equal characteristic and the perfect closure of this residue field if R has mixed characteristic. Thus we obtain a function ordω : Gr(X) → Z ∪ {∞},

x → ordω (ψx ).

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(5.2.4). — Now assume that X is smooth over R and that ω is a volume form on Xη . We denote by π0 (X0 ) the set of connected components of the special fiber of X. For each connected component C of X0 , we define the order ordC (ω) of ω along C in the following way. Since X is smooth over R, the completed local ring RC := OX,ξ of X at the generic point ξ of C is an extension of R of ramification index one. We denote the corresponding point of X(RC ) by ψC , and we set ordC (ω) = ordω (ψC ). If π is a uniformizer of R, then ordC (ω) is equal to the unique integer such that π − ω extends to a generator of the line bundle ΩdX/R locally around ξ. Proposition 5.2.5. — Let X be a smooth formal R-scheme of finite type of pure relative dimension d, and let ω be a volume form on Xη . Then for every connected component U of X, the function ordω is constant on Gr(U) ⊂ Gr(X), with value ordU0 (ω). Proof. — Let π be a uniformizer in R. Multiplying ω with π − ordU0 ω shifts the function ordω by − ordU0 ω, so that we can reduce to the case where ordU0 ω = 0. This means that ω extends to a generator of ΩdX/R at the generic point of U0 . Then ω must generate the line bundle ΩdX/R at every point x of U0 . Indeed, if ωx is a local generator at x, then we can write ω = f ωx with f ∈ OX,x , and f is invertible in OX,x ⊗R K (since ω is a volume form on Xη ) and f is not identically zero in OX0 ,x = OX,x /(π) (since ordU0 (ω) = 0). Hence, f is invertible in OX,x . It follows that the function ordω is identically zero on Gr(U). Corollary 5.2.6. — Let X be a smooth formal R-scheme of finite type of pure relative dimension d, and let ω be a volume form on Xη . Then ordω is integrable on Gr(X) and, we have   L− ordω dμX = L−d e(C)L− ordC (ω) Gr(X)

in M

R X0

C∈π0 (X0 )

.

Proof. — This follows at once from proposition 5.2.5 and the definition of the motivic integral. Theorem 5.2.7. — We assume that the residue field k of R is perfect. a) Let Y be a formal R-scheme formally of finite type of pure relative dimension d such that Yη is Néronian. Let ω be a volume form on the quasismooth locus of Yη over K. If h : X → Y is a Néron smoothening, then the motivic integral  − ordh∗ ω η dμ ) ∈ M R L (hred )! ( X Yred Gr(X)

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only depends on Y and ω, and not on the chosen Néron smoothening. b) Let X be a Néronian K-analytic space of pure dimension d. Let ω be a volume form on the quasi-smooth locus of X over K. If X is a weak Néron model of X, then the motivic integral  L− ordω dμX ∈ MkR Gr(X)

only depends on X and ω, and not on the chosen weak Néron model X. Proof. — We start with the first assertion. By proposition 5.1.4 it is enough to prove the following claim: if h : X → Y is a Néron smoothening that dominates X → Y, then   − ord(h )∗ ω − ordh∗ ω R η dμ η L L dμX in MY . X = red Gr(X )

Gr(X)

This follows at once from the chain rule (5.2.2.1), proposition 4/3.5.1, and the change of variables formula for motivic integrals (theorem 6/1.2.5). The second assertion can be deduced from the first: by proposition 4/3.4.7, it is enough to compare the integrals computed on weak Néron models X and X of X such that X dominates X. Then the morphism X → X is a Néron smoothening, so that the first part of the theorem implies that the two integrals coincide in MXR0 and thus, a fortiori, in MkR . Definition 5.2.8. — With the notation and assumptions of theorem 5.2.7, we write   − ordh∗ ω η dμ ) ∈ M R |ω| := (hred )! ( L X Yred Gr(X)

Y

and



 |ω| := X

L− ordω dμX ∈ MkR

Gr(X)

and we call these expressions the motivic integrals of ω on Y and X, respectively. (5.2.9). — In practice, these motivic integrals can be computed using corollary 5.2.6. Note that the formula proven there shows, in particular, that the motivic integrals in definition 5.2.8 do not depend on the choice of the section k → R in the equal characteristic case (even though this choice is technically necessary in the construction in order to speak about the Greenberg schemes). (5.2.10). — The condition that k is perfect cannot be removed in theorem 5.2.7. For instance, set X = M (K{T }) and ω = dT . Then X = Spf(R{T }) is a weak Néron model of X, and  L− ordω dμX = 1. Gr(X)

However, if we remove from X a closed point of its special fiber whose residue field is a nontrivial purely inseparable extension k  of k, then the resulting

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formal scheme X is still a weak Néron model of X, and  L− ordω dμX = 1 − e(Spec(k  ))L−1 = 1. Gr(X )

Proposition 5.2.11. — Assume that k is perfect. Let Y be a formal Rscheme formally of finite type of pure relative dimension d such that Yη is Néronian. Let ω be a volume form on the quasi-smooth locus of Yη over K. Let h : Z → Y be a morphism of formal R-schemes formally of finite type such that hη is an open immersion and such that the induced map Zη (K  ) →  Yη (K  ) is bijective  ∗ K of K. Then the  for every finite unramified extension motivic integral Y |ω| is equal to the image of Z |hη ω| under the forgetful R morphism (hred )! : MZRred → MY . red Proof. — This is a trivial consequence of the definition of the motivic integral, since every Néron smoothening of Z gives rise to a Néron smoothening of Y by composition with h. In the situation of proposition 5.2.11, we will usually write that   |ω| = |ω| Y

Z

, leaving the pullback of ω to Zη and the application of the forgetful in M R implicit. morphism MZRred → MY red R Yred

Proposition 5.2.12. — Assume that k is perfect. Let Y be a formal Rscheme formally of finite type of pure relative dimension d such that Yη is Néronian. Let ω be a volume form on the quasi-smooth locus of Yη over K. Let Z be a locally closed subset of Yred , and denote by Z the formal completion of Y along Z. We still write ω for the restriction of ω to the quasi-smooth locus of the analytic domain Zη in  Y. Then Zη is Néronian, and the motivic integral Z |ω| is the image of Y |ω| under the base-change morphism: R → MZRred . MY red Proof. — We can refine {Z, Yred Z} into a finite partition {U1 , . . . , Ur } of Yred into locally closed subsets. Denote by Ui the formal completion of Y along Ui , and let Xi → Ui be a Néron smoothening, for every i. Let X be the disjoint union of the formal R-schemes Xi . Then the morphism X → Y is a Néron smoothening. Moreover, since X ×Y Z is a union of connected components of U, the morphism X ×Y Z → Z is a Néron smoothening, as well. Now it follows immediately from the definition of the motivic integral  that Z |ω| is the image of Y |ω| under the base-change morphism: R MY → MZRred . red

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5.3. The Motivic Serre Invariant Theorem 5.3.1. — We assume that the residue field k of R is perfect. a) Let Y be a formal R-scheme formally of finite type such that Yη is néronian, and let X → Y be a Néron smoothening. Then the class e(X0 ) of R /(L − 1) only depends on Y, and not on the choice of the Néron X0 in MY red smoothening X → Y. b) Let X be a Néronian K-analytic space, and let X be a weak Néron model of X. Then the class e(X0 ) of X0 in MkR /(L − 1) only depends on X, and not on the choice of the weak Néron model X. Proof. — We only need to prove the first assertion; the second part of the theorem follows from the first in the same way as in the proof of theorem 5.2.7. By proposition 5.1.4, it is enough to prove the following claim: if X → Y is a Néron smoothening that dominates X → Y, then e(X0 ) = e(X0 ) in MXR0 /(L − 1). Since k is perfect and the morphism h : X → X is still a Néron smoothening, it follows from proposition 4/3.5.1 that the map Gr(X )(k  ) → Gr(X)(k  ) is bijective for every field extension k  of k if R has equal characteristic (resp. for every perfect field extension k  of k if R has mixed characteristic). For every connected component U of X , the function ordJach is constant on Gr(U). We denote its value by cU . By theorem 5/3.2.2, we know that the image of Gr(U) in Gr(X) is a constructible set of motivic measure e(U0 )L−cU . As U runs through the set π0 (X ) of connected components of X , the images of the sets Gr(U) form a partition of Gr(X). Thus, by the scissor relations and the additivity of the motivic measure, we find  e(U0 ) = μX (Gr(X)) = e(X0 ) e(X0 ) = U∈π0 (X )

in MXR0 /(L − 1). Definition 5.3.2. — Assume that k is perfect. a) Let Y be a formal R-scheme formally of finite type such that Yη is Néronian. Then we define the motivic Serre invariant S(Y) of Y as S(Y) = e(X0 )

R ∈ MY /(L − 1) red

where X → Y is any Néron smoothening. b) Let X be a Néronian K-analytic space. Then we define the motivic Serre invariant S(X) of X as S(X) = e(X0 )

∈ MkR /(L − 1)

where X is any weak Néron model of X. (5.3.3). — By theorem 5.3.1, these definitions are independent of the choice of X. If Y is a formal R-scheme formally of finite type such that Yη is Néronian, then it follows from the definitions that S(Yη ) is the image of S(Y) R /(L − 1) → MkR /(L − 1) induced under the forgetful morphism f! : MY red

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by the structural morphism f : Yred → Spec(k). Observe that the group homomorphism f! is well-defined because the element f! (e(Z)(e(A1Yred ) − e(Yred ))) = f! (e(A1Z ) − e(Z))) = e(Z)(L − 1) in K0 (Vark ) for every Yred -scheme Z of finite type. Proposition 5.3.4. — Assume that k is perfect. a) Let Y be a formal R-scheme formally of finite type and of pure relative dimension such that Yη is Néronian. If ω is a volume form  on the quasismooth locus of Yη over K, then S(Yη ) is the image of Y |ω| under the R R projection morphism MY → MY /(L − 1). red red b) Let X be a Néronian K-analytic space of pure dimension. If ω is a volume form on the quasi-smooth locus of X over K, then S(X) is the image  of X |ω| under the projection morphism MkR → MkR /(L − 1).  Proof. — If X → Y is a Néron smoothening, then the image of Y |ω| in R /(L − 1) is equal to MY red  e(C) = e(X0 ) C∈π0 (X0 )

by corollary 5.2.6 and the scissor relations in the Grothendieck ring. The same argument applies to the second assertion. Proposition 5.3.5. — Assume that k is perfect. Let Y be a formal Rscheme formally of finite type. Assume that Y is regular, and denote by Ysm its R-smooth locus. Then Yη is Néronian. Moreover, S(Y) = e((Ysm )red ) in R /(L − 1), and S(Yη ) = e((Ysm )red ) in MkR /(L − 1). MY red Proof. — The same proof as for proposition 4/3.4.8 shows that, for every finite unramified extension R of R, every R -point on Y factors through Ysm . Thus Yη is Néronian, and every Néron smoothening of Ysm is also a Néron smoothening of Y. Therefore, we may assume that Y is smooth over R. Let {U1 , . . . , Ur } be a partition of Yred into k-smooth subvarieties, and denote by Ui the formal completion of Y along Ui , for every i. Let Xi be the dilatation of Ui . If we denote by X the disjoint union of the formal schemes Xi , then the morphism X → Y is a Néron smoothening. Moreover, (Xi )red is a locally trivial fibration over Ui = (Ui )red whose fibers are affine spaces, so that S(Y) = e(Xred ) = e(U1 ) + . . . + e(Ur ) = e(Yred ) R /(L − 1), and similarly S(Yη ) = e(Yred ) in MkR /(L − 1). in MY red 5.4. Comparison with p-adic Integration (5.4.1). — Suppose that k is finite of cardinality q, and let X be a Néronian K-analytic space of pure dimension d. Then the set X(K) of K-valued points on X has the structure of a compact K-analytic manifold; this structure is defined in a similar way as for smooth K-varieties (see section 1/1.6.3). Let

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X be a weak Néron model of X. The specialization morphism spX : Xη → X0 restricts to a map: sp ( X : X(K) = Xη (K) → X0 (k). We claim that each fiber of this map is an open submanifold of X(K) isomorphic to the open unit ball md in K d (which is also isomorphic to the closed unit ball Rd via division by a uniformizer). This is a generalization of lemma lemm/formfiber-smooth to formal schemes and can be proven as follows. Since X is smooth over R, the topological R-algebra OX,x is isomorphic to R[[T1 , . . . , Td ]]. Thus the open subspace ∼  sp−1 X (x) = Spf(OX,x )η of Xη is isomorphic to the open unit polydisk E d . It follows that sp ( −1 X (x) is d an open submanifold of X(K) isomorphic to m . Thus the choice of a weak Néron model X of X gives rise to a partition of X(K) into open submanifolds isomorphic to md , parameterized by the finite set X0 (k). Using this observation, we will now prove that motivic integrals on X specialize to p-adic integrals on X under the point counting realization. Proposition 5.4.2. — a) The Serre invariant S(X(K)) of X(K) is the image of the motivic Serre invariant of X under the point counting realization: MkR /(L − 1) → Z/(q − 1), e(Y ) → Card(Y (k)). b) Let ω be a volume form on the K-smooth locus of X, and let us denote by the same symbol the induced volume form on the K-analytic manifold   X(K). Then X(K) |ω| is the image of the motivic integral X |ω| under the point counting realization: MkR → Z[q −1 ],

e(Y ) → Card(Y (k)).

Proof. — The first statement follows immediately from (5.4.1), since S(X) = e(X0 ) and S(X(K)) = |X0 (k)|. For the proof of the second assertion, we make an analogous computation as in the proof of theorem 1/1.6.11. It is enough to prove the following claim: if x is a point of X0 (k) and we denote by c the order of ω along the connected component of X0 containing x, then −(c+d) . the volume of sp ( −1 X (x) with respect to the measure |ω| is equal to q Then the result follows from the formula for the motivic integral X |ω| in corollary 5.2.6, together with the additivity of the volume of |ω| on X(K) with respect to the partition {sp ( −1 X (x), | x ∈ X0 (k)}. So let us prove our claim. Multiplying ω with an element in K of valuation d −c, we can reduce to the case c = 0. Identifying sp ( −1 X (x) with m , we can write ω = f · dT1 ∧ . . . ∧ dTd

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on md , with f a K-analytic function of constant modulus one. By definition −d , the Haar measure of the measure |ω|, the volume of sp ( −1 X (x) is equal to q d of m .

5.5. The Trace Formula (5.5.1). — We will now explain a cohomological interpretation of the motivic Serre invariant by means of a trace formula in terms of the Galois action on the étale cohomology of K-analytic spaces. For the theory of étale cohomology for K-analytic spaces, we refer to Berkovich (1993). We make a few reminders in the setup that we will need. Assume that k is algebraically closed and of characteristic zero. Let K a be an algebraic closure of K, and a its completion. The Galois group Gal(K a /K) is isomorphic to denote by K the procyclic group μ (k) = lim μn (k) ← − n of roots of unity in k, via the isomorphism



Gal(K a /K) → μ (k), σ → a

σ(tn ) tn

n

n

where tn is any element in K such that (tn ) is a uniformizer in K. Any topological generator of Gal(K a /K) is called a monodromy operator. If k = C then there is a canonical monodromy operator, corresponding to the element (k). (exp(2πi/n))n of μ If X is a separated formal R-scheme formally of finite type and is a prime number, then to the generic fiber of X we can attach -adic cohomology spaces i a , Q ) Hét (Xη ⊗K K

for i  0. These are finite dimensional Q -vector spaces equipped with a continuous action of the Galois group Gal(K a /K), and they vanish for i > 2 dim(Xη ). These spaces can be computed by means of Berkovich’s nearby cycles complex RψX (Q ) of the formal scheme X (Berkovich 1996b), which is a bounded constructible complex of Q -vector spaces on the étale site of Xred , endowed with a continuous action of Gal(K a /K). There exists a Gal(K a /K)-equivariant isomorphism (5.5.1.1)

i a , Q )  Hi (Xred , RψX (Q )) Hét (Xη ⊗K K ét

for every i  0. The following theorem was proven in Nicaise (2009, 6.4); it refines (Nicaise and Sebag 2007b, 5.4). Theorem 5.5.2 (Trace Formula). — Assume that R is complete and that k is algebraically closed and of characteristic zero. Let X be a separated formal

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R-scheme formally of finite type such that Xη is quasi-smooth over K. Then 

2 dim(Xη )

Eu(S(Xη )) =

i a , Q )) (−1)i Tr(σ | Hét (Xη ⊗K K

i=0

for every monodromy operator σ. Proof. — We will only sketch the main lines of the argument and refer to theorem 6.4 of Nicaise (2009) for the details of the proof. Because of the isomorphism (5.5.1.1), we have 

2 dim(Xη )

i a , Q )) = (−1)i Tr(σ | Hét (Xη ⊗K K

i=0



q (−1)p+q Tr(σ | Hét (Xred , Rp ψX (Q ))).

p,q0

A small generalization of a result of Laumon (1981) shows that the righthand side of this expression does not change if we take étale cohomology with compact supports. Then it follows from a result by Deligne, published in Illusie (1981), that we can compute this expression by integrating the constructible function  (−1)p Tr(σ | Rp ψX (Q )x )) Xred (k) → Q : x → p0

with respect to the Euler characteristic. Using a suitable form of resolution of singularities, we can reduce to the case where X is regular and its special fiber is a divisor with strict normal crossings (see section 6.3). Then, locally in the étale topology, X is algebraizable: there exists a separated R-scheme of finite type Y and a subvariety Z of Y ⊗R k such that Y is regular, Y ⊗R k is a strict normal crossings divisor, and X is isomorphic to the formal completion of Y along Z. It follows from Berkovich’s comparison theorem for algebraic and formal nearby cycles that RψX (Q ) is canonically isomorphic to the restriction of the algebraic nearby cycles complex RψY (Q ) to Z. This means that Grothendieck’s local description of the -adic nearby cycles on a strict normal crossings divisor (SGA VII1 , exposé I, 3.3) is also valid for the formal scheme X. This description states that, when x is a point of Xred (k), and we denote by r the number of rig-irreducible components of X0 passing through x and by m the greatest common divisor of their multiplicities, then p , A Q Qr−1 ⊗ ( ) Rp ψX (Q )x ∼ =   where A is a set of cardinality m, and the monodromy operator σ acts on A by means of a transitive permutation, and trivially on the second factor in the tensor product. Thus the value  (−1)p Tr(σ | Rp ψX (Q )x ) p0

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is 0 unless r = m = 1, that is, X is smooth over R at x; in that case, this value equals 1. In this way, we find that 

2 dim(Xη )

i a , Q )) = Eu((Xsm )red ). (−1)i Tr(σ | Hét (Xη ⊗K K

i=0

By proposition 5.3.5, we have Eu((Xsm )red ) = Eu(S(Xη )).

§ 6. MOTIVIC ZETA FUNCTIONS OF FORMAL SCHEMES AND ANALYTIC SPACES 6.1. Definition of the Motivic Zeta Function (6.1.1). — Let X be a formal R-scheme formally of finite type with quasismooth generic fiber Xη , and let ω be a volume form on Xη . We will investigate how the motivic integral X |ω| behaves under finite totally ramified extensions of the discrete valuation ring R. This leads us to define the motivic zeta function of the pair (X, ω). Our main application is a non-Archimedean interpretation of Denef and Loeser’s motivic zeta function. (6.1.2). — For the remainder of this section, we fix the following conventions. We assume that k has characteristic zero, and we fix an isomorphism R ∼ = k[[π]]. For every positive integer n, we set R(n) = R[t]/(tn − π), and we denote by K(n) the quotient field of R(n). Then K(n) is a finite totally ramified extension of K of degree n. If k is algebraically closed, then K(n) is the unique degree n extension of K up to K-isomorphism, but this does not hold in general; this is the reason why we fixed the uniformizer π in R. The motivation for the assumption that k has characteristic zero is that we will need a suitable form of resolution of singularities to prove the main results in this section. For every formal R-scheme X, we set X(n) = X ⊗R R(n). Likewise, for every K-analytic space X, we set X(n) = X ⊗K K(n), and for every differential form ω on X, we denote by ω(n) the pullback of ω to X(n). Definition 6.1.3 (Motivic Zeta Functions). — Let X be a formal Rscheme formally of finite type. Assume that the generic fiber Xη is quasismooth over K, and let ω be a volume form on Xη . Then the motivic zeta function of the pair (X, ω) is the generating series:     Z(X,ω) (T ) = |ω(n)| T n ∈ MXred [[T ]]. n>0

X(n)

Likewise, the motivic zeta function of the pair (Xη , ω) is the generating series:     |ω(n)| T n ∈ Mk [[T ]]. Z(Xη ,ω) (T ) = n>0

Xη (n)

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Note that, by definition, we obtain Z(Xη ,ω) (T ) from Z(X,ω) (T ) by applying the forgetful morphism MXred → Mk to every coefficient of the generating series. The analytic version of the trace formula in theorem 5.5.2 immediately gives us a cohomological interpretation of the motivic zeta function. We denote by Eu(Z(Xη ,ω) (T )) the element of Z[[T ]] that we obtain by applying the Euler characteristic Eu to each coefficient of Z(Xη ,ω) (T ). Theorem 6.1.4 (Cohomological Interpretation of the Motivic Zeta Function) Let X be a formal R-scheme formally of finite type of pure relative dimension d, and assume that Xη is quasi-smooth over K. Let ω be a volume form on Xη . Then we have   2d   i n i s  , Q )) T n Eu(Z(X ,ω) (T )) = (−1) Tr(σ | H (Xη ⊗K K η

n0

i=0

for every monodromy operator σ. Proof. — For every n > 0, we have that   |ω(n)| = Eu(S(Xη (n))) Eu Xη (n)

by proposition 5.3.4. Now the result immediately follows from 5.5.2 and the fact that σ n is a topological generator for the inertia group of K(n). 6.2. Bounded Differential Forms (6.2.1). — In order to establish an explicit formula for the motivic zeta function, we need to introduce a special class of differential forms that we call bounded. Let X be a formal R-scheme formally of finite type. Let ω be a differential form of degree i on Xη , for some nonnegative integer i. We say that ω is X-bounded if it lies in the image of the natural map: (6.2.1.1)

ΩiX/R (X) ⊗R K → ΩiXη /K (Xη ).

Then it is clear that ω is also X -bounded for every formal R-model X of Xη that dominates X. If X is of finite type over R, then Xη is compact, the map (6.2.1.1) is bijective, and every differential form on Xη is X-bounded, but this fails in the general case. We say that ω is bounded if it is Y-bounded for some formal R-model Y of Xη . For i = 0, we recover the standard boundedness condition for an analytic function on Xη , by the following result. Proposition 6.2.2. — Let X be a flat formal R-scheme formally of finite type, and let f be an analytic function on Xη . Then the following are equivalent: a) The function f is X-bounded. b) The function f is bounded. c) The absolute value of f is bounded on Xη .

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Proof. — It follows immediately from the definitions that (a) implies (b). If f extends to a regular function on some formal R-model of Xη , then the absolute value of f is bounded by 1 on Xη , so that (b) implies (c). Thus, it suffices to deduce (a) from (c). Multiplying f with a suitable nonzero element of R, we may assume that |f (x)|  1 for every x ∈ Xη . We may also suppose that X is affine, say, X = Spf(A). The integral closure A of A in A ⊗R K is finite over A, by excellence of A. Thus, replacing X by Spf(A ), we can reduce to the case where A is integrally closed in A ⊗R K. In this case, it follows from de Jong (1995, 7.4.2) that the image of OX (X) → OXη (Xη ) consists precisely of the analytic functions on Xη whose absolute value is bounded by 1. Remark 6.2.3. — Our definition of X-boundedness is equivalent to the more cumbersome one in definition 2.11 of Nicaise (2009): the presheaf ΩiX ⊗R K on X is a sheaf, because X is quasi-compact and K is flat over R. 6.3. Resolution of Singularities for Formal Schemes (6.3.1). — Let Y be a regular formal R-scheme formally of finite type. We say that the special fiber Y0 has strict normal crossings if the following conditions are satisfied: a) For every point y of Y, there exist a regular system of local parameters (z1 , . . . , zr ) and a unit u in OY,y such that (6.3.1.1)

π=u

r 

ziNi

i=1

in OY,y , for some nonnegative integers N1 , . . . , Nr . b) Every rig-irreducible component of Y0 (as defined in section A/3.4.7) is regular. Condition (b) is equivalent to the property that every rig-irreducible component of Y0 is locally integral, by lemma 2.34 in Nicaise (2009). This follows from (a) if Y is of finite type over R (then Y0 is a scheme, and every integral scheme is locally integral). However, we do not know if (a) implies (b) in general, so we include the second condition explicitly in the definition. (6.3.2). — If X is a formal R-scheme formally of finite type with quasismooth generic fiber Xη , then a log resolution for X is a morphism h : Y → X of formal R-schemes formally of finite type such that hη : Yη → Xη is an isomorphism, Y is regular, and Y0 has strict normal crossings. If X is a K-analytic space, then an snc-model for X is a formal R-model X of X such that X is regular and X0 has strict normal crossings. Theorem 6.3.3 (Temkin 2009). — Every formal R-scheme X formally of finite type with quasi-smooth generic fiber Xη admits a log resolution. In particular, Xη has an snc-model. Proof. — This follows from theorem 1.1.13 in Temkin (2009).

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Remark 6.3.4. — For our purposes, it would be sufficient to know that theorem 6.3.3 holds when X is affine, because we can apply the additivity properties of motivic integrals. The affine case already follows from Temkin’s nonfunctorial embedded resolution for quasi-excellent Q-schemes (theorem 1.1 in Temkin (2008)) because an affine formal R-scheme X = Spf(A) formally of finite type is an snc-model if Spec(A) is regular and Spec(A ⊗R k) is a divisor with strict normal crossings on Spec(A). (6.3.5). — Now let Y be a regular formal R-scheme formally of finite type, and assume that Y0 has strict normal crossings. Since Y0 is itself a formal Rscheme formally of finite type, we can consider its rig-irreducible components as defined in (A/3.4.7). To each rig-irreducible component E of Y0 , we can attach a multiplicity in Z>0 , in the following way. Let y be a point on E and let r  N (6.3.5.1) π=u zj j j=1

be an expression of the form (6.3.1.1) in OY,y . Then, locally at y, the rigirreducible component E is defined by an equation zj = 0 with Nj > 0, because we are assuming that E is locally integral. The positive integer Nj does not depend on the choice of the expression (6.3.5.1), by the following intrinsic characterization: if we denote by P the stalk at y of the defining ideal sheaf of E in Y, then (OY,y )P is a discrete valuation ring with uniformizer zj , and Nj is equal to the length of the module (OE,y )P over (OY,y )P . Furthermore, Nj only depends on Y and E, and not on the choice of the point y, because this module remains invariant if we replace y by a generization. We call Nj the multiplicity of E in Y0 . We will write  Y0 = Ni E i i∈I

to indicate that Ei , i ∈ I are the rig-irreducible components of Y0 and that Ni is the multiplicity of Ei in Y0 , for every i ∈ I. (6.3.6). — In order to write down explicit formulas for the motivic zeta functions in definition 6.1.3, we need to introduce some additional notation. For every i ∈ I, we set Ei = (Ei )red . For every nonempty subset J of I, we denote by EJ the closed formal subscheme of Y defined by the sum of the ideal sheaves of the components Ej with j in J. By the definition of a strict normal crossings divisor, EJ is regular. We also write   Ej , EJ◦ = EJ Ei . EJ = (EJ )red = j∈J

i∈J /

These are locally closed subsets of the k-scheme Yred , and we endow them with their induced reduced structures. The subschemes EJ◦ form a partition of Yred . We denote by NJ the greatest common divisor of the multiplicities Nj with j ∈ J.

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443

Example 6.3.7. — Let Y = Spf(R[[x, y]]/(π − xN1 y N2 )). Then the rig-irreducible components of Y0 are the closed formal subschemes E1 = Spf(k[[x]]) and E2 = Spf(k[[y]]), and Y0 = N1 E1 + N2 E2 . Moreover,

E1◦

and

E2◦

◦ are empty, and E{1,2} = E{1,2} = Spec(k).

(6.3.8). — For every nonempty subset J of I, we will also need to consider ( ◦ over E ◦ , where μN denotes the finite étale group scheme of a μNJ -torsor E J J J NJ -th roots of unity over k. Denote by Z the normalization of Y ×R R(NJ ). We set (J◦ = Z ×Y EJ◦ . E ( ◦ via its Galois action on R(NJ ). The proof of Then μNJ acts on Z and E J (◦ . the following proposition gives a more explicit description of E J

( ◦ is a μN -torsor over E ◦ . If J = {i} Proposition 6.3.9. — The scheme E J J J (◦ . for some i in I, then Z is smooth over R(NJ ) along E i Proof. — It follows from equation 6.3.1.1 that, for every point y of EJ◦ , we can find an integral affine open neighborhood U of y in Y and regular functions u and f on U such that u is invertible and π = uf NJ on U. The statement we are trying to prove is local on Y, so that we may assume that Y = U. We ( over Y by define a μNJ -torsor Y ( = Spf(O(Y)[T ]/(T NJ − u)). Y ( as a formal scheme over R(NJ ) = R[t]/(tNJ − π) by sending We can view Y t to T f . Since Z is normal, the element t/f of the total ring of fractions of O(Z) lies in O(Z), because its NJ -th power is equal to u and, hence, lies in O(Z). ( by sending T to We define a morphism of Y ×R R(NJ )-schemes h : Z → Y t/f . Now we make the following observations. ( η is an isomorphism of K(NJ )-analytic – The morphism hη : Zη → Y spaces; in fact, Zη is canonically isomorphic to Yη ×K K(NJ ) because this latter space is normal. ( is étale over Y and, hence, regular. In particular, – The formal scheme Y it is normal. – The morphism h is finite because Z is finite over Y. These properties imply that h is an isomorphism (for instance, by using ( ◦ is isomorphic, as an E ◦ -scheme, to (de Jong 1995, 7.4.1)). In particular, E J J ( ×Y E ◦ . the μNJ -torsor Z J In the case where J = {i}, the completed local ring of Y at any point of Ei◦ is of the form k  [[π, z1 , . . . , zm ]]/(π − (z1 )Ni )

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for some field extension k  of k and some positive integer m. Computing the normalized base change to R(Ni ), we see at once that Z is smooth over R(d) ( ◦ (here we use that normalization commutes with completion for along E i excellent local rings).

6.4. Néron Smoothening After Ramification (6.4.1). — We return to our general setup: let X be a formal R-scheme formally of finite type such that the generic fiber Xη is quasi-smooth over K. As a first step toward an explicit formula for the motivic zeta functions in definition 6.1.3, we need to construct a Néron smoothening for the formal R(n)-scheme X ×R R(n), for every positive integer n. We will first construct a particular type of log resolution for X. Definition 6.4.2. — Let Y be a regular formal R-scheme formally of finite type such that  Ni E i Y0 = i∈I

has strict normal crossings. Let n be a positive integer. We say that Y is n-adapted if there do not exist a subset J of I of cardinality at least 2 and positive integers αj , j ∈ J such that EJ◦ is nonempty and  αj Nj . n= j∈J

The interest of this condition lies in the following property. Proposition 6.4.3. — Let Y be a regular formal R-scheme formally of finite type such that  Ni E i Y0 = i∈I

has strict normal crossings. Let n be a positive integer and denote by Z the normalization of Y ×R R(n). Let R be a finite unramified extension of R(n). If Y is n-adapted, then every point in Z(R ) lies in the R(n)-smooth locus Zsm of Z. Moreover, (Zsm )red is isomorphic as an Yred -scheme to the disjoint ( ◦ with i ∈ I such that Ni divides n. union of the torsors E i Proof. — By proposition 6.3.9, it is sufficient to show that, for every Rmorphism ψ : Spf(R → Y), the image of ψ is contained in Ei◦ for some i ∈ I with Ni |n. Denote the image of ψ by y, and let J be the unique nonempty subset of I such that the image of ψ lies in the stratum EJ◦ of Yred . Consider an equation  N zj j π=u j∈J

§ 6. MOTIVIC ZETA FUNCTIONS OF FORMAL SCHEMES

445

of the form (6.3.1.1) in OY,y . Evaluating both sides in ψ and taking valuations in R , we obtain the expression  vR (zj (ψ))Nj . n = vR (π) = j∈J

Note that vR (zj (ψ)) is positive for every j in J, because zj (ψ) lies in the maximal ideal of R . Our assumption that Y0 is n-adapted now implies that J = {i} for some i ∈ I. Proposition 6.4.4. — Let Y be a regular formal R-scheme formally of finite type such that Y0 has strict normal crossings, and write  Ni E i . Y0 = i∈I

Let n be a positive integer. Then we can construct a log resolution Y → Y such that Y0 is n-adapted, by repeated formal blow-ups with a center of the form EJ , where J is a subset of I of cardinality at least 2. Proof. — Successively blowing up Y at centers EJ with |J| > 1 preserves the regularity of Y and the fact that Y0 has strict normal crossings, because the formal schemes EJ are regular. Moreover, such a blow-up induces an isomorphism between the generic fibers. We will prove the following claim: by means of a finite number of successive blow-ups of Y at strata EJ with |J| > 1, one can arrange that, for every subset J of I such that |J| > 1 and EJ is nonempty, there exists an element j in J such that Nj > n. This property obviously implies that Y0 is n-adapted. In order to prove our claim, it is convenient to introduce a combinatorial invariant of Y: the dual intersection complex Δ(Y) of the special fiber Y0 . This is a finite Δ-complex whose vertex set is equal to I and such that, for every nonempty subset J of I, the set of faces spanned by the vertices vj , j ∈ J is the set of connected components of EJ . We label each vertex vi with the corresponding multiplicity Ni . Let J be a subset of I of cardinality at least 2, and let Y → Y be the blow-up at EJ . Then the dual intersection complex of Y is obtained from Δ(Y) by means of a stellar subdivision of all the faces spanned by the vertices vj , j ∈ J. Here the multiplicities of the vertices are preserved, and  the new vertices at the barycenters of the subdivided faces get multiplicity j∈J Nj . Now we can reformulate our claim in the following way: by means of a finite number of such stellar subdivisions, we can arrange that every face of positive dimension of the dual intersection complex has a vertex with multiplicity at least n + 1. Let m0 be the smallest positive integer such that Δ(Y) has a face of positive dimension whose vertices all have multiplicity at most m0 . Let A(m0 ) be the set of such faces. By the minimality of m0 , we know that for every face in A(m0 ), at least one of the vertices has multiplicity m0 . In particular, the sum of the multiplicities of the vertices is strictly larger than m0 . Thus by taking a stellar subdivision of the faces in any nonempty subset of A(m0 ),

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we always decrease the cardinality of A(m0 ). The result now follows from a double induction on m0 and the cardinality of A(m0 ). 6.5. A Formula for the Motivic Zeta Function (6.5.1). — Let X be a formal R-scheme formally of finite type of pure relative dimension d, and assume that Xη is quasi-smooth over K. Let ω be an X-bounded volume form on Xη . We will apply corollary 5.2.6 to compute the motivic integral  |ω(n)| X(n)

for every positive integer n. We will explain below how to use proposition 6.4.3 to construct Néron smoothenings of the formal R(n)-schemes X(n). The other ingredient we need is a method to compute the orders of ω(n) along the components of the special fibers of these Néron smoothenings. This requires a new definition: the order of ω along a rig-irreducible component of the special fiber of an snc-model of Xη . (6.5.2). — Let Y → X be a log resolution of X. We can view Y as a smooth formal k-scheme formally of finite type, by forgetting the R-structure, and consider the sheaf of differentials Ωd+1 Y/k . This is a line bundle on Y, and it   d+1 induces a line bundle ΩY/k on the K-analytic space Yη . It is proven in η

proposition 7.19 of Nicaise (2009) that the wedge product with dπ induces an isomorphism of OYη -modules: (6.5.2.1)

ΩdYη /K

dπ∧( · )



Ωd+1 Y/k

 η

.

Let E be a rig-irreducible component of Y0 of multiplicity N , and let y be a point of E. Let z = 0 be a local equation for E in Y at the point y. Since ω is X-bounded, π a ω extends to an element of ΩdY/R for some positive integer a. Then, locally around y, we can write dπ ∧ π a ω = z m vω0 where ω0 generates the stalk of Ωd+1 Y/k at y, the exponent m is a nonnegative integer, and v is an element of OY,y that is not divisible by z. We call (m + 1)−(a+1)N the order of ω along E. This number is clearly independent of the choice of a, and it is easy to see that it does not depend on y, either. If Y is of finite type over R and smooth along E (that is, N = 1), then this definition is equivalent to the one in section 5.2.4, because the isomorphism (6.5.2.1) extends to an isomorphism (6.5.2.2) locally at every point of E.

ΩdY/R

dπ∧( · )

Ωd+1 Y/k

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447

Remark 6.5.3. — The reason for the appearance of a+1 and m+1, rather than a and m, in the definition of the order of a volume form is that we are implicitly treating ω as a rational section of the logarithmic relative canonical line bundle of Y over R. Theorem 6.5.4 ((Nicaise and Sebag 2007b, corollary 7.7) and Nicaise 2009, corollary 7.13) Let X be a formal R-scheme formally of finite type of pure relative dimension d, and assume that Xη is quasi-smooth over K. Let ω be an X-bounded volume form on Xη . Let Y → X be a log resolution of X, with  Ni E i . Y= i∈I

For every i ∈ I, we denote by μi the order of ω along Ei . Then  L−μj T Nj  (J◦ )(L − 1)|J|−1 e(E Z(X,ω) (T ) = 1 − L−μj T Nj ∅=J⊂I

j∈J

in MXred [[T ]]. Proof. — Comparing terms in the desired formula, we see that it is enough to prove the following equation, for every positive integer n: (6.5.4.1)     − kj μj (J◦ )(L − 1)|J|−1 j∈J |ω(n)| = e(E L X(n)

∅=J⊂I

kj 1, j∈J j∈J

kj Nj =n

in MXred . Step 1: reduction to the case where all the strata EJ are regular. If Y is not of finite type over R, then the strata EJ are not necessarily regular. However, we can easily reduce to this situation. The motivic zeta function Z(Y,ω) (T ) is additive with respect to finite partitions of Yred into locally closed subsets {U1 , . . . , Ur }, in the following sense. For every in {1, . . . , r}, we denote by U the formal completion of Y along U . We abuse notation by writing ω for the restriction of ω to the generic fiber of U (which is an analytic domain in Yη ). Then it follows from proposition 5.2.12 that Z(Y,ω) (T ) =

r 

Z(U ,ω) (T )

=1

in MYred . Moreover, for every , we have that U is regular and (U )0 is a divisor with strict normal crossings, and the normalization of U ⊗R R(n) is canonically isomorphic to the base change to U of the normalization of Y ⊗R R(n). Thus the right-hand side of (6.5.4.1) is also additive with respect to the partition of Yred into locally closed subsets U . By choosing a partition {U1 , . . . , Ur } which refines the partition of Yred into the strata EJ◦ and such that every U is regular, we can thus reduce to the case where EJ is regular for every J ⊂ I.

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Step 2: the equation (6.5.4.1) holds when Y is n-adapted. In this case, the right-hand side of (6.5.4.1) simplifies to  ( ◦ )L−(N/ni )μi . e(E i i∈I, Ni |n

Denote by Z the normalization of Y ⊗R R(n). We have shown in proposition 6.4.3 that the R(n)-smooth locus Zsm of Z contains all the points of Z(R ), for every finite unramified extension R of R(n). We have also seen in proposition 6.4.3 that the scheme (Zsm )red is the disjoint union of the schemes ( ◦ with i ∈ I and Ni |n. In particular, by Step 1, (Zsm )red is smooth over E i k. Hence, if we denote by Z the dilatation of Z, then Z is a smooth formal R-scheme of finite type, the morphism h : Z → X(n) is a Néron smoothening, ( ◦ in Z is an affine bundle of rank d − dim(E ◦ ). and the inverse image of E i i red ( ◦ )Ld−dim(Ei◦ ) . ( ◦ ×Z Z ) = e(E This implies that e(E i i Thus, in order to finish the proof of Step 2, it is enough to show that ( ◦ ×Z Z is equal to the order of ω along each connected component of E i ◦ (n/Ni )μi + dim(Ei ) − d. Let π(n) be an n-th root of π in R(n). At every ( ◦ ×Z Z , the Jacobian ideal of the dilatation Z → Z is generated point of E i d−dim(Ei◦ ) by π(n) because we are blowing up a regular formal scheme along a regular center of codimension d + 1 − dim(Ei◦ ). Thus it is enough to show ( ◦ equals (n/Ni )μi . This follows from a simple that the order of ω(n) along E i computation in local coordinates. Replacing Y by its formal completion at any closed point of Ei◦ , we can reduce to the case where Y = Spf(R[[t0 , . . . , td ]]/(π − (t0 )Ni )). Then (t0 )1−Ni −μi (dπ ∧ ω) extends to a generator of Ωd+1 Y/k , by the definition of the numbers μi . Moreover,  k (k[ξ]/(ξ n − 1))) Z∼ = Spf(R [[t1 , . . . , td ]]⊗ and (π(n), t1 , . . . , td ) is a regular system of local parameters at every point of Zred . It follows that π(n)−(n/Ni )μi (dπ(n) ∧ ω(n)) is a generator for Ωd+1 Z/k , ◦ ( so that the order of ω(n) along E equals (n/Ni )μi , as required. i

Step 3: the equation (6.5.4.1) holds for all Y. By proposition 6.4.4, it suffices to show that the right-hand side of equation (6.5.4.1) does not change if we blow up Y at a center of the form EJ . This follows from a straightforward, but tedious, computation as in the proof of lemma 7.5 of Nicaise and Sebag (2007b). A cleaner way to write down the argument is to use the language of logarithmic geometry: see Bultot and Nicaise (2016). Corollary 6.5.5. — Let X be a formal R-scheme formally of finite type of pure relative dimension d, and assume that Xη is quasi-smooth over K. Let ω be a bounded volume form on Xη , and let Y be an snc-model for Xη such

§ 6. MOTIVIC ZETA FUNCTIONS OF FORMAL SCHEMES

that ω is Y-bounded. We write Y=



449

Ni E i .

i∈I

For every i ∈ I, we denote by μi the order of ω along Ei . Then   L−μj T Nj ( ◦ )(L − 1)|J|−1 Z(Xη ,ω) (T ) = e(E J 1 − L−μj T Nj ∅=J⊂I

j∈J

in Mk [[T ]]. Proof. — We can simply apply theorem 6.5.4 to the pair (Y, ω) and specialize the formula for the motivic zeta function Z(Y,ω) (T ) with respect to the forgetful morphism MYred → Mk . 6.6. Comparison with Denef and Loeser’s Motivic Zeta Function (6.6.1). — We can use theorem 6.5.4 to interpret the motivic zeta function of Denef and Loeser from §4.1 as a motivic zeta function of the form Z(X,ω) (T ), for a suitable choice of X and ω. Let X be a connected smooth separated k-scheme of finite type, and let f : X → Spec k[π] be a dominant morphism of k-schemes. We denote by X0 the closed subscheme of X defined by f = 0, and by Zf (T ) ∈ MX0 [[T ]] the motivic zeta function of f . The formal π-adic completion of X is a formal scheme X, flat, and of finite type over R = k[[π]], such that the generic fiber Xη is quasi-smooth. Working locally on X, we can assume that there exists a volume form ϕ on X over k. This form induces a volume form on X over k, which we still denote by ϕ. Using the isomorphism (6.5.2.1), we see that there exists a unique volume form ω on Xη over K such that (dπ/π) ∧ ω = ϕ. We call ω the logarithmic Gelfand–Leray form associated with ϕ, and denote it by ϕ/(dπ/π). Theorem 6.6.2 (Comparison Theorem, Nicaise and Sebag 2007b, theorem 9.10) We have Zf (T ) = Ldim(X)−1 Z(X,ω) (T ) in MX0 [[T ]]. Proof. — Let h : Y → X be a log resolution for the pair (X, X0 ). We denote by Y0 the closed subscheme of Y defined by f ◦ h. Then we can write   Y0 = Ni Ei , KY /X = (νi − 1)Ei . i∈I

i∈I

The formal π-adic completion  h : Y → X of h is a resolution of singularities for X. Thus, using the explicit formulas for Zf (T ) in terms of the log resolution h and for Z(X,ω) (T ) in terms of the strict normal crossings model Y, we see that it suffices to show that the order of ω along Ei is equal to νi . This is a direct consequence of the definition of the logarithmic Gelfand–Leray form ω.

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(6.6.3) The Analytic Milnor Fiber. — The comparison statement in theorem 6.6.2 becomes particularly striking when we formulate a local variant of the result. Let x be a closed point of X0 . The formal completion of f at x is a flat formal R-scheme formally of finite type. Its generic fiber is called the analytic Milnor fiber of f at x. We denote it by Fx . The name “analytic Milnor fiber” is chosen because of the analogy with the classical Milnor fibration in complex singularity theory (see §1/3.4.1). The formal germ Spf(OX,x ) plays the role of a small open ball around x in X, and Spf(R) plays the role of a small open disk around the origin of the complex plane. We can make the analogy even more explicit: if we denote by kx the residue field of X at x, there exists an isomorphism of k-algebras OX,x ∼ = kx [[z1 , . . . , zd ]], which allows us to identify Fx with the closed analytic subspace of the d-dimensional unit polydisk over kx ((π)) defined by the equation f = π (see example A/3.5.11). Theorem 6.6.4. — Assume that k is the field of complex numbers. Let x be a closed point on X0 , and denote by Fx the topological Milnor fiber of f at x. Then, for every integer i  0 and every prime number , there exists a canonical isomorphism s , Q ) → H i (Fx , Q ) H i (Fx ⊗K K e´t

sing

such that the action of the canonical topological generator of Gal(K s /K) ∼ = μ (C) on the source corresponds to the monodromy transformation on the target. Proof. — As explained in the proof of Nicaise and Sebag (2007b, 9.2), this is a combination of comparison theorems by Deligne and Berkovich. The singular cohomology of Fx is computed by the stalk at x of the complex analytic nearby cycles complex associated with f , and Deligne has shown that we get the same result by working with the -adic nearby cycles instead (SGA VII2 , XIV.2.8). It was proven by Berkovich that the -adic nearby cycles complex of f coincides with that of the formal π-adic completion X and that the stalk at x of the latter complex computes the -adic cohomology of Fx (Berkovich 1996a, 3.5). Theorem 6.6.5. — For every closed point x on X0 , we have Zf,x (T ) = Ldim(X)−1 Z(Fx ,ω) (T ) in Mk [[T ]]. Proof. — It suffices to apply the base-change morphism to both sides of the equality in theorem 6.6.2; the result then follows from proposition 5.2.12. (6.6.6). — Thus from the analytic Milnor fiber Fx , we can read off both the motivic zeta function of f at x and the local monodromy eigenvalues of f at x, the invariants that are related by Igusa’s monodromy conjecture. This makes non-Archimedean geometry a natural framework for the study of this conjecture; we refer to Nicaise and Xu (2016) for closely related results. Combining theorems 6.1.4, 6.6.4, and 6.6.5, one sees that the cohomological

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interpretation of the motivic zeta function in theorem 6.1.4 specializes to Denef and Loeser’s theorem on the Lefschetz numbers of the local monodromy (theorem 4.3.6). 6.7. Motivic Zeta Functions of Calabi–Yau Varieties (6.7.1). — In view of theorem 6.6.2, it is natural to ask if we can formulate a generalization of the monodromy conjecture for other motivic zeta functions considered in definition 6.1.3. A natural candidate is the following: let X be a geometrically connected, smooth, and proper K-variety with trivial canonical line bundle, and let ω be a volume form on X. We write Z(X,ω) (T ) for the motivic zeta function associated with the analytification of the pair (X, ω). This is an element of Mk [[T ]], and it is rational by corollary 6.5.5. Then we can ask whether its poles are related to the monodromy action on the -adic cohomology of X. Definition 6.7.2. — We say that X satisfies the monodromy property if there exists a finite set S of rational numbers such that Z(X,ω) (T ) belongs to the subring

 1 Mk T, 1 − La T b (a,b)∈Z×Z>0 , a/b∈S of Mk [[T ]] and such that, for every s ∈ S, every prime number , every embedding Q → C, and every monodromy operator σ of K, the number exp(2πis) is an eigenvalue of σ on H i (X ×K K s , Q ) for some i  0. (6.7.3). — It was shown by Halle and Nicaise (2011) that every abelian variety over K has the monodromy property; more precisely, it has a unique pole, which (after a suitable normalization of ω) coincides with Chai’s base-change conductor of X and gives rise to a monodromy eigenvalue on H dim(X) (X ×K K s , Q ) as in definition 6.7.2. The proof relies on a careful study of the basechange properties of Néron models of abelian varieties. The result was later extended to Calabi–Yau varieties that admit a particular type of semistable model, a so-called equivariant Kulikov model, in Halle and Nicaise (2017).

§ 7. MOTIVIC SERRE INVARIANTS OF ALGEBRAIC VARIETIES In this section we will generalize the construction of the motivic integral of a volume form and the motivic Serre invariant to the algebraic setting, relaxing the conditions on the discrete valuation ring R. In particular, we will prove that one can use algebraic Néron smoothenings and weak Néron models

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to compute motivic integrals and motivic Serre invariants of analytifications of smooth algebraic varieties. We will also explain how one can extend the definition of the motivic Serre invariant to arbitrary algebraic varieties over K, assuming that K has characteristic zero. For the remainder of the section, we assume that R is a (not necessarily complete) discrete valuation ring with quotient field K. We denote by m the maximal ideal of R and by vK : K × → Z the discrete valuation on K. We choose a strict Henselization Rsh of R. This is an unramified extension of R whose residue field k s is a separable closure of k. We denote by K sh the quotient field of Rsh and by vK sh the discrete valuation on K sh extending  and K  for the completions of R and K. For every R-scheme vK . We write R X , we set X0 = X ⊗R k and XK = X ⊗R K.

7.1. Weak Néron Models of Algebraic Varieties (7.1.1). — We start with a few reminders on weak Néron models of schemes taken from Bosch et al. (1990). Let X be a separated K-scheme of finite type, and let E be a subset of X(K sh ). If X is affine, then we say that E is bounded in X if we can find a closed immersion X → AnK for some positive integer n, and an integer M such that for every point (x1 , . . . , xn ) in E ⊂ (K sh )n , we have vK sh (xi )  M for all i in {1, . . . , n}. In other words, the set E must be contained in a sufficiently large ball around the origin in (K sh )n . It is not difficult to show that this property does not depend on the choice of the closed immersion X → AnK . If X is arbitrary, then we say that E is bounded in X if we can find an affine open cover {X1 , . . . , Xr } of X and a decomposition E = E1 ∪ . . . ∪ Er such that Ei is contained in Xi (K sh ) and bounded in Xi , for every i in {1, . . . , r}; if X is affine, this definition is equivalent to the previous one. We say that X is bounded if X(K sh ) is bounded in X. One can show that every proper K-variety is bounded (Bosch et al. 1990, 1.1.6). If R is excellent and U is an open subscheme of X such that X U does not contain a K sh -valued point, then a subset E of U is bounded in U if and only if it is bounded in X. This follows from Bosch et al. (1990, 1.1.9) and the fact that Rsh is still excellent. In particular, if R is excellent and X has a compactification X such that the boundary X X does not contain a K sh -valued point, then X is bounded. A typical example of a K-scheme that is not bounded is the affine space AnK , for any n > 0. (7.1.2). — If R is complete, then it is not hard to see that X is bounded if and only if its analytification X an contains a compact analytic domain U such that the map U (K  ) → X(K  ) is a bijection for every finite unramified extension K  of K (Nicaise 2011b, 4.3). More generally, if we only assume that R is excellent or that X is smooth over K, then X is bounded if and only  an contains such a compact analytic domain U (Nicaise 2011b, if (X ×K K) 4.3).

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(7.1.3). — A model for X is a flat separated R-scheme of finite type X , endowed with an isomorphism XK → X. A morphism of models is a morphism of R-schemes that induces the identity on X. If X is a model of X, then an admissible blow-up is the blow-up X  → X of X at an ideal containing a power of the maximal ideal m of R. This is a morphism of models of X. Proposition 7.1.4. — The K-scheme X is bounded if and only if it has a model X such that the natural map X (Rsh ) → X(K sh ) is bijective. Proof. — The “if” part of the statement is easy: we choose an affine open cover of X and closed embeddings of the pieces of the cover in some affine space AnR . Passing to the generic fibers, we find an affine open cover of X such that the set of K sh -valued points on X is contained in the union of the closed balls of radius one in the pieces of the cover. The “only if” part is more delicate; see Bosch et al. (1990, 3.5.7). (7.1.5). — Let Y be a flat separated R-scheme of finite type with smooth generic fiber YK . A Néron smoothening of Y is a morphism of separated R-schemes of finite type h : X → Y such that hK : XK → YK is an isomorphism, X is smooth over R, and the map X (Rsh ) → Y (Rsh ) is a bijection. The latter condition is equivalent to the property that X (R ) → Y (R ) is a bijection for every finite unramified extension R of R, because Rsh can be written as the union of these extensions and X and Y are of finite type over R. If X is a smooth separated K-scheme of finite type, then a weak Néron model of X is a model X of X such that X is smooth over R and every K sh -valued point on X extends to an Rsh -valued point on X (such an extension is automatically unique, by the valuative criterion of separatedness). Again, this is equivalent to saying that every K  -point on X extends to an R -point on X , for every finite unramified extension R of R with quotient field K  . Remark 7.1.6. — Our terminology deviates slightly from the one in Bosch et al. (1990): they define a Néron smoothening to be a proper morphism of R-schemes X → Y whose restriction to the R-smooth locus of X is a Néron smoothening in our sense. This does not make much of a difference since, by Nagata’s embedding theorem, a Néron smoothening according to our terminology can be compactified to a Néron smoothening in the sense of Bosch et al. (1990). Our definition is more convenient for the applications in this chapter. Example 7.1.7. — If X(K sh ) is empty, then X is a weak Néron model for itself (with empty special fiber). Note that this is the only weak Néron model of X: if X is a weak Néron model of X, then the k s -points are dense in X0 because X0 is smooth over k (Bosch et al. 1990, 2.2/13), and any k s -point of the special fiber lifts to an Rsh -point on the weak Néron model, and thus a

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K sh -point on X, by the Henselian property of Rsh (Bosch et al. 1990, 2.3.5). Thus X0 is empty. Proposition 7.1.8. — If Y is a regular R-scheme of finite type with smooth generic fiber YK , then the open immersion Ysm → Y of the smooth locus of Y into Y is a Néron smoothening. Proof. — Let R be a finite unramified extension of R. We must show that every R -point y on Y is contained in Ysm . Since R is étale over R, the scheme Y ⊗R R is again regular. Moreover, (Y ⊗R R )sm = Ysm ⊗R R by flat descent of smoothness (ÉGA IV4 , 17.7.2). Thus, we may assume that R = R. Then it follows from Bosch et al. (1990, 3.1/2) that the completed local ring of Y at y0 ∈ Y (k) is isomorphic to a formal power series ring R[[t1 , . . . , td ]]. Hence, Y is smooth over R at y0 , so that y lies in Ysm . Theorem 7.1.9 (Néron; Bosch-Lütkebohmert-Raynaud) a) Let Y be a flat separated R-scheme of finite type with smooth generic fiber YK . Then there exists a composition of admissible blow-ups h : Y  → Y  → Y of h to the R-smooth locus of Y  is a such that the restriction Ysm Néron smoothening of Y . b) Let X be a smooth separated K-scheme of finite type. Then X has a weak Néron model if and only if X is bounded. Proof. — The first part of the statement is (Bosch et al. 1990, 3.1.3). The idea of the proof is to attach to each Rsh -valued point y on Y a nonnegative integer δ, called Néron’s measure for the defect of smoothness, which measures the lack of smoothness of the morphism Y → Spec(R) at y0 ∈ Y (k s ). One shows that this invariant vanishes if and only if Y is smooth over R at y0 , that δ is bounded on Y , and that its maximal value can be improved by means of well-chosen admissible blow-ups. Note that an admissible blow-up induces a bijection between the sets of Rsh -valued points on source and target, by the valuative criterion of properness. After a finite number of such admissible blow-ups, one obtains a model Y  of R such that δ = 0 at every  (Rsh ). point of Y  (Rsh ); thus Y  (Rsh ) = Ysm To prove the second statement, one first chooses a model X of X such that the natural map X (Rsh ) → X(K sh ) is a bijection. Such a model exists by proposition 7.1.4. Then one takes a Néron smoothening of X ; the result is a weak Néron model of X. (7.1.10). — Now we compare the notions of Néron smoothening and weak Néron model with the formal setting. For every separated R-scheme of finite  its formal m-adic completion; this is a separated type X , we denote by X  Likewise, we write  formal scheme of finite type over R. h for the completion of η is in a canonical a morphism of R-schemes h. Recall that the generic fiber X  an . way a compact analytic domain in the analytification (X ×R K) Proposition 7.1.11. — a) Let Y be a flat separated R-scheme of finite type with smooth generic fiber YK . If h : X → Y is a Néron smoothening

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→ Y  is a Néron smoothening of Y , then the formal m-adic completion  h:X . of Y b) Let X be a bounded smooth separated K-scheme of finite type, and let  X be a weak Néron model of X. Then the formal m-adic completion X of X , endowed with the canonical immersion η → (X ×K K)  an , X  an . is a weak Néron model for (X ×K K)  is smooth over R  because Proof. — We start with (a). The formal scheme X  X is smooth over R. The restriction of h to the generic fibers fits into the commutative diagram Xη

(X ×R K)an



(hη )an



(Y ×R K)an

where the lower horizontal map is an isomorphism and the vertical maps are analytic domain immersions; thus  hη is an analytic domain immersion, too.     and the Since Y (R ) = Y (R ) for every finite unramified extension R of R analogous property holds for X , the only nontrivial property to prove is that  → Y ×R R  X ×R R is still a Néron smoothening; this follows from (Bosch et al. 1990, 3.6.6). Point (b) is proven analogously and can be found in Nicaise (2011b, 4.9).

7.2. Motivic Integrals and Motivic Serre Invariants for Smooth Algebraic Varieties Theorem 7.2.1. — We assume that the residue field k of R is perfect. a) Let Y be a flat separated R-scheme of finite type with smooth generic fiber YK , and let X → Y be a Néron smoothening of Y . Then the element e(X0 ) ∈ MYR0 /(L − 1) only depends on X , and not on the choice of the Néron smoothening X → Y. b) Let X be a bounded smooth separated K-scheme of finite type, and let X be a weak Néron model of X. Then the element e(X0 ) ∈ MkR /(L − 1) only depends on X, and not on the choice of the weak Néron model X .

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Proof. — These are immediate consequences of theorem 5.3.1 and proposition 7.1.11, since the special fibers of an R-scheme and of its formal m-adic completion are canonically isomorphic. Definition 7.2.2. — We assume that k is perfect. a) Let Y be a flat separated R-scheme of finite type with smooth generic fiber YK . Then we define the motivic Serre invariant S(Y ) of Y as S(Y ) = e(X0 )

∈ MYR0 /(L − 1)

where X → Y is any Néron smoothening. b) Let X be a bounded smooth separated K-variety of finite type. Then we define the motivic Serre invariant S(X) of X as S(X) = e(X0 )

∈ MkR /(L − 1)

where X is any weak Néron model of X. (7.2.3). — By theorem 7.2.1, these definitions are independent of the choice ) and S(X) = of X . In fact, by proposition 7.1.11, we have S(Y ) = S(Y an  ). If Y is a flat separated R-scheme of finite type with smooth S((X ×K K) generic fiber YK , then it follows immediately from the definitions that S(YK ) is the image of S(Y ) under the forgetful morphism MYR0 /(L−1) → MkR /(L− 1). Example 7.2.4. — If X is a smooth and proper K-scheme that has a smooth and proper model X over R, then X is a weak Néron model of X and S(X) = e(X0 ). For instance, we have S(P1K ) = e(P1k ) = 2 ∈ MkR /(L − 1). (7.2.5). — Now we extend the definition of the motivic integral of a volume form to the algebraic case. First, we need an auxiliary definition. Let X be a smooth R-scheme of finite type of pure relative dimension d, and let ω be a volume form on XK . Let π be a uniformizer in R. Let C be a connected component of X0 , and denote its generic point by ξ. Then the local ring of X at ξ is a discrete valuation ring, and π is a uniformizer in this ring, by smoothness of X . The stalk ΩdX /R,ξ of ΩdX /R at ξ is a free OX ,ξ -module of rank one, and the volume form ω defines a nonzero element of ΩdX /R,ξ ⊗R K. Thus there exists a unique integer such that π − ω extends to a generator of ΩdX /R,ξ . This integer is independent of the choice of π; we denote it by ordC (ω) and call it the order of ω along C. It measures the order of the zero or pole of ω along C. Theorem 7.2.6. — We assume that k is perfect. a) Let Y be a flat separated R-scheme of finite type of pure relative dimension d, with smooth generic fiber YK . Let ω be a volume form on YK ,

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and let X → Y be a Néron smoothening of Y . Then the element  L−d e(C)L− ordC (ω) ∈ MYR0 C∈π0 (X0 )

only depends on Y and ω, and not on the choice of the Néron smoothening X →Y. b) Let X be a bounded smooth separated K-scheme of finite type of pure dimension d, and let ω be a volume form on X. Let X be a weak Néron model of X. Then the element  e(C)L− ordC (ω) ∈ MkR L−d C∈π0 (X0 )

only depends on X and ω, and not on the choice of the weak Néron model X . Proof. — The analytification of the volume form ω is a volume form on η again by ω. If we identify the (YK )an . We denote its restriction to X , it is straightforward to check that the order of ω special fibers of X and X along a connected component does not change under formal completion. Thus the first assertion is again an immediate consequence of theorem 5.3.1 and proposition 7.1.11. The proof of the second assertion is completely similar. Definition 7.2.7. — We assume that k is perfect. a) Let Y be a flat separated R-scheme of finite type of pure relative dimension d, with smooth generic fiber YK . If ω is a volume form on YK , then we define the motivic integral of ω on Y by   |ω| = L−d e(C)L− ordC (ω) ∈ MYR0 Y

C∈π0 (X0 )

where X → Y is any Néron smoothening. b) Let X be a bounded smooth separated K-variety of finite type, of pure dimension d. If ω is a volume form on X, then we define the motivic integral of ω on X as   |ω| = L−d e(C)L− ordC (ω) ∈ MkR X

C∈π0 (X0 )

where X is any weak Néron model of X. (7.2.8). — These definitions are independent of the choice of X , by theorem 7.2.6. Note that we find the motivic Serre invariants of Y and X by reducing the motivic integrals of ω modulo L − 1, just like in the formalη induced by ω again by analytic setting. If we denote the volume form on Y the same symbol, then   |ω| = |ω|, Y Y  an . and the analogous statement holds for X and (X ×K K)

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7.3. Motivic Serre Invariants of Open and Singular Varieties (7.3.1). — We have introduced in definition 7.2.2 the motivic Serre invariant of a bounded smooth separated K-scheme of finite type; this is the only case where X admits a weak Néron model, by theorem 7.1.9. Nevertheless, if K has characteristic zero, the construction of the motivic Serre invariant was extended to arbitrary K-schemes of finite type in Nicaise (2011b, 5.4), by means of the following theorem: Theorem 7.3.2. — We assume that K has characteristic zero and that k is perfect. There exists a unique ring morphism S : MK → MkR /(L − 1) such that S(e(X)) = S(X) for every bounded smooth separated K-scheme of finite type. This morphism satisfies S(L) = 1. Moreover, if U is a compact  an such that U is quasi-smooth over K and analytic domain in (X ×K K)   the map U (K ) → X(K ) is a bijection for every finite unramified extension  then S(e(X)) = S(U ). In particular, S(e(X)) = 0 if X has no K  of K, sh K -valued points. Proof. — Since K has characteristic zero, we can use Bittner’s presentation of the Grothendieck ring in terms of smooth and proper K-varieties and blow-up relations—see remark 2/5.1.5. Every proper K-variety is bounded, so that uniqueness of S is clear. To prove the existence of a ring morphism S : MK → MkR /(L − 1) mapping the class of each smooth and proper K-variety X to S(X), we must show that the motivic Serre invariants of smooth and proper K-varieties satisfy the blow-up relations. This was proven in Nicaise (2011b, 5.3), using a generalization of the Néron smoothening procedure to pairs of varieties (Nicaise 2011b, §3). It was shown there that, for every smooth and proper K-variety X and every smooth closed subvariety Z of X, we can find a weak Néron model X of X such that the schematic closure Z of Z in X is a weak Néron model for Z. If we denote by X  the blow-up of X along Z, and by X  the blow-up of X along Z , then X  is a weak Néron model for X  , and the exceptional divisor E in X  is a weak Néron model for the exceptional divisor E in X  . This immediately implies that the motivic Serre invariant satisfies the blow-up relation S(X  ) − S(E) = S(X) − S(Z). Once we have such a ring morphism S, we can directly compute that S(L) = S(P1K ) − S(Spec(K)) = e(P1k ) − 1 = 1

∈ MkR /(L − 1)

by additivity. To prove the remainder of the statement, one can argue as follows (we refer to Nicaise (2011b, 5.3) for details). Suppose that X and U are as in the statement, and denote by Xsm the K-smooth locus of X. Then

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 because analytification U is contained in the analytification of Xsm ×K K preserves the set of smooth points. The existence of U implies that Xsm is bounded and that X Xsm does not contain any K sh -valued points. We have S(Xsm ) = S(U ) by §7.2.3, since every formal weak Néron model of U is also  an . a formal weak Néron model of (Xsm ×K K) We can partition X Xsm into smooth subvarieties. Therefore, it now suffices to prove the following property: if Y is a bounded smooth separated K-scheme of finite type, then S(e(Y )) = S(Y ) (recall that Y is automatically bounded if Y (K sh ) is empty, and that any weak Néron model of Y has empty special fiber, so that S(Y ) = 0). To prove this property, one takes any smooth compactification of X and shows that the boundary does not contain any K sh -points; then one can conclude the argument by induction on the dimension of X. Definition 7.3.3. — Assume that k is perfect and K has characteristic zero. Then we define the motivic Serre invariant of any K-scheme of finite type X by S(X) = S(e(X)). Example 7.3.4. — Assume that R is strictly Henselian. If X is a K-scheme of finite type and X(K) is finite, then S(X) is equal to the cardinality of X(K). Note that an isolated point of X(K) is either a zero-dimensional component of X or a singular point of X, because of the following fact: if X is a connected smooth K-scheme of finite type, then X(K) is either empty  with or dense in X with respect to the Zariski topology (and even in X(K) respect to the valuation topology). This follows from the implicit function theorem for Henselian fields. (7.3.5). — Theorem 7.3.2 is interesting because it gives a sufficient criterion for the existence of a rational point on an algebraic variety over a strictly Henselian field of characteristic zero, namely, the nonvanishing of the motivic Serre invariant. This criterion has been applied in Esnault and Nicaise (2011) to prove that, if K has characteristic zero, k is algebraically closed and is a prime number that is invertible in k; then every action of a finite -group on an affine space Am K has a rational fixed point. This result was motivated by a question of Serre, asking whether the action of a finite -group on an affine space Am F always has a rational fixed point, for any base field F of characteristic different from . This general question appears to be wide open for m  3, but other interesting special cases were solved by Haution (2017). We will now explain that, under a suitable tameness assumption (in particular, if k has characteristic zero), the motivic Serre invariant has a cohomological interpretation in terms of the monodromy action on the -adic cohomology of the variety.

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7.4. The Trace Formula (7.4.1). — The aim of this section is to establish a trace formula that gives a cohomological interpretation of the motivic Serre invariant (after specialization with respect to the Euler characteristic), in analogy with theorem 5.5.2. Let R be a Henselian discrete valuation ring with quotient field K and algebraically closed residue field k. We denote by p the characteristic exponent of k, and we fix a separable closure K s of K. We denote by K t the tame closure of K in K s , that is, the union of all the finite extensions of K in K s of degree prime to p. The absolute Galois group I = Gal(K s /K) is called the inertia group or geometric monodromy group of K. It fits into a short exact sequence of profinite groups: 1 → P = Gal(K s /K t ) → I → I t = Gal(K t /K) → 1. The group P is a pro-p-group called the wild inertia group of K. It is trivial when p = 1 but difficult to understand in general. The quotient I t of I is called the tame inertia group of K. It can be explicitly described in the following way. For every positive integer d prime to p, the field K has a unique extension K(d) of degree d in K s . It is obtained by joining a d-th root π(d) of any uniformizer π to K. The field K(d) is Galois over K, and if we denote by m(d) the maximal ideal of the valuation ring of K(d), then the map σ(π(d)) mod m(d) Gal(K(d)/K) → μd (k) : σ → π(d) is an isomorphism. It follows that the tame inertia group I t is canonically isomorphic to the projective limit μ (k) = lim μd (k) ←− (d,p)=1

where d runs through the set of positive integers prime to p, ordered by divisibility, and the transition morphisms are given by 

μd (k) → μd (k) : x → xd /d whenever d is a multiple of d. In particular, the group I t is procyclic. Every topological generator of I t will be called a tame monodromy operator; if p = 1 then I t = I, and we will simply speak of a monodromy operator. The choice of such a topological generator amounts to choosing a primitive d-th root d /d of unity ξd in k for every d > 0 prime to p in such a way that ξd = ξd whenever d is a multiple of d. (7.4.2). — Let X be a bounded smooth separated K-scheme of finite type. We have explained above that one can view the motivic Serre invariant S(X) as a measure for the set of rational points on X (observe that unramified points coincide with rational points by our assumption that k is algebraically closed). We extract a numerical invariant from S(X) by means of the following definition.

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Definition 7.4.3. — Let X be a bounded smooth separated K-scheme of finite type, and let X be a weak Néron model of X. Then we define the rational volume s(X) of X by s(X) = Eu(X0 )

∈ Z.

This definition does not depend on the choice of a weak Néron model X , because the rational volume s(X) is the image of the motivic Serre invariant under the Euler characteristic specialization Eu : MkR /(L − 1) → Z. We want to establish a cohomological interpretation for the rational volume s(X) in analogy with Grothendieck’s trace formula for varieties over finite fields. Recall that, for every scheme of finite type Y over a finite field F of cardinality q, Grothendieck’s trace formula states that the number of rational points on Y admits the following cohomological interpretation: 

2 dim(Y )

Card(Y (F )) =

i (−1)i Tr(Frob | Hét,c (Y ⊗F F s , Q )),

i=0 s

where F is a separable closure of F , Frob is the geometric Frobenius automorphism a → a1/q of F s over F , and is a prime different from the characteristic of F . If we replace Y by the K-scheme X, the role of Card(Y (F )) should be played by the rational volume s(X). We also need a Galois operator to replace the geometric Frobenius operator Frob in the right-hand side. The operator Frob is a topological generator of the absolute Galois group Gal(F s /F ) of F . The inertia group I = Gal(K s /K) is not procyclic if p > 1, but if we consider the tame closure K t rather than the separable closure K s , we still get a procyclic Galois group I t = Gal(K t /K). Unfortunately, this comes with a price: the -adic cohomology of X over K t only sees a part of the -adic cohomology of X over K s , namely, the part that is invariant under the action of the wild inertia group P . Thus we will need to impose some tameness conditions on X in order to obtain our trace formula for the rational volume s(X). (7.4.4). — Let Y be a regular flat R-scheme of finite type such that Y0 is a strict normal crossings divisor. If p = 1, we set WY = ∅. If p > 1, then we denote by WY the open subscheme of (Y0 )red consisting of the points x that are contained in a unique irreducible component E of Y0 and such that E has multiplicity pa in Y0 , for some a > 0. We will call WY the wild locus of Y0 . Theorem 7.4.5 (Trace Formula, Nicaise 2013, theorem 4.2.1) Let X be a smooth and proper K-scheme. Assume that X has a regular proper R-model Y whose special fiber Y0 is a strict normal crossings divisor.

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Then we have 

2 dim(X)

s(X) =

i (−1)i Tr(σ | Hét (X ⊗K K t , Q )) − Eu(WY )

i=0

for every prime different from p and every tame monodromy operator σ. In particular, if the multiplicity of each component in Y0 is prime to p, then 

2 dim(X)

s(X) =

i (−1)i Tr(σ | Hét (X ⊗K K t , Q )).

i=0

Proof. — Since Ysm is a weak Néron model for X by proposition 7.1.8, we t (Q ) the complex of tame -adic have s(X) = Eu((Y0 )sm ). We denote by RψY nearby cycles associated with Y , as defined in (SGA VII1 , exposé I). This is a bounded complex of constructible -adic sheaves on Y0 , equipped with a continuous action of the tame inertia group I t , that computes the -adic cohomology of X ⊗K K t via the spectral sequence in (SGA VII1 , exposé I, 2.7.3). In particular, we have 

2 dim(X) i (−1)i Tr(σ | Hét (X ⊗K K t , Q ))

i=0

=



a t (−1)a+b Tr(σ | Hét (Y0 , Rb ψY (Q )).

a,b0

The right-hand side of this expression can be computed by integrating the constructible function  t (−1)b Tr(σ | Rb ψY (Q )x ) x → b0

on Y0 (k) with respect to the Euler characteristic (see theorem 2.6.2 in Nicaise (2013)). Thus it suffices to prove that this function takes the value 1 if x lies in the smooth locus (Y0 )sm or the wild locus WY , and the value 0 otherwise. Let x be a point of Y0 , let r be the number of irreducible components of Y0 passing through x, and let m be the greatest common divisor of the multiplicities of these components. We denote by m the prime-to-p part of m. By Grothendieck’s local description of the tame -adic nearby cycles (SGA VII1 , exposé I, 3.3), we have b  , Qr−1 Rb ψ t (Q )x ∼ = (Q )A ⊗ Y



for every b  0, where σ acts on A by a transitive permutation, and trivially on the second factor of the tensor product. Thus the expression  t (−1)b Tr(σ | Rb ψY (Q )x ) b0

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463

vanishes, unless r = m = 1; in that case, this expression equals one. But r = m = 1 means precisely that x lies on a unique irreducible component of Y0 and the multiplicity of this component is a power of p; equivalently, x lies in (Y0 )sm (if the multiplicity is one), or x lies in the wild locus WY (if the multiplicity is a power of p with positive exponent). This concludes the proof. (7.4.6). — Theorem 7.4.5 implies, in particular, that Eu(WY ) only depends on X, and not on the choice of the model Y . It would be very interesting to obtain a more intrinsic characterization of the invariant Eu(WY ) and understand its precise relation with other measures of wild ramification for X. In particular, it is natural to ask for more general conditions that guarantee that Eu(WY ) vanishes. An interesting result in this direction is the following theorem of Smeets (2017): if X has a K t -rational point and Y is log smooth over R (with respect to the divisorial log structure induced by Y0 , and the standard log structure on Spec(R)), then Eu(WY ) = 0. The log smoothness condition on Y includes the case where Y is a strict normal crossings model with multiplicities prime to p, and it implies that the wild inertia P acts trivially on the -adic cohomology spaces of X (Nakayama 1998, 0.1.1). Question 7.4.7 (Nicaise 2011b). — Let X be a smooth and proper Kvariety such that X has a K t -rational point and the wild inertia P acts trivially on the -adic cohomology spaces i (X ⊗K K s , Q ), Hét

for all i  0. Is it true that 

2 dim(X)

s(X) =

i (−1)i Tr(σ | Hét (X ⊗K K t , Q ))?

i=0

An affirmative answer is known when X is a curve (Nicaise 2011b, §7) or an abelian variety (Halle and Nicaise 2016, 9.1.2.8), but the question is wide open in general. The condition that X has a K t -rational point cannot be omitted; see Nicaise (2011b, §7) for a counterexample. (7.4.8). — The situation becomes much more transparent in the case where k has characteristic zero, that is, p = 1. In this case, K t = K s and I t coincides with the full inertia group I. Moreover, we have defined the motivic Serre invariant S(X) for arbitrary K-schemes of finite type X in definition 7.3.3. Hence, we can also define the rational volume s(X) of X to be the image of S(X) under the Euler characteristic specialization: Eu : MkR /(L − 1) → Z. Theorem 7.4.9. — Assume that k has characteristic zero. Let X be a separated K-scheme of finite type. Then we have 

2 dim(X)

s(X) =

i=0

i (−1)i Tr(σ | Hét,c (X ⊗K K s , Q ))

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for every prime and every monodromy operator σ. Proof. — Since both sides of the equality are additive with respect to finite partitions of X into subvarieties, it suffices to consider the case where X is smooth and proper over K. This case follows immediately from theorem 7.4.5 and resolution of singularities (the ring R is excellent). (7.4.10). — Theorem 7.4.9 was reproven by Hrushovski and Loeser (2015), using an entirely different method; in particular, they avoid the use of resolution of singularities. Instead, they use techniques from model theory to decompose X ⊗K K s into so-called semi-algebraic pieces of an elementary form, for which the trace formula can be interpreted as a Lefschetz fixedpoint formula. The result then follows by additivity.

APPENDIX

§ 1. CONSTRUCTIBILITY IN ALGEBRAIC GEOMETRY 1.1. Constructible Subsets of a Scheme In this appendix, we recall some definitions and results concerning constructible subsets of a (general) scheme. This paragraph is based on (ÉGA III1 , 0, §9), (ÉGA IV1 , §1.8, §1.9), and (ÉGA IV3 , §8, §9). However, we follow the terminology of (ÉGA Isv ). (1.1.1). — Let X be a topological space. We say that a subset Z of X is retrocompact if the inclusion from Z to X is a quasi-compact morphism, that is, if Z ∩ U is quasi-compact for every quasi-compact open set U of X. If X is a scheme, a subset Z is retrocompact if and only if Z ∩ U is quasi-compact for every affine open subscheme U of X. The space X is retrocompact; the union of two retrocompact subsets of X is retrocompact; the intersection of two retrocompact open subsets is retrocompact. If X is a noetherian topological space, then every subset of X is retrocompact. (1.1.2). — Let X be a topological space. One says that a subset C of X is globally constructible (in X) if it belongs to the smallest set of subsets of X which contains all retrocompact open subsets of X and is stable under finite intersection and complements and hence also under finite unions. Explicitly, C is globally constructible if and only if there exist finite families (U1 , . . . , Un ) and (V1 , . . . , Vn ) of retrocompact open sets of X such that n  C= Ui ∩ (X Vi ). i=1

One says that C is constructible if every point of X is contained in an affine open subscheme V of X such that C ∩ V is globally constructible in V . © Springer Science+Business Media, LLC, part of Springer Nature 2018 A. Chambert-Loir et al., Motivic Integration, Progress in Mathematics 325, https://doi.org/10.1007/978-1-4939-7887-8

465

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We define ConsX to be the set of all constructible subsets of X. Remark 1.1.3. — By definition, every globally constructible subset of a topological space X is a constructible set. The converse holds true if X is quasi-compact and if the topology of X admits a basis consisting of retrocompact open sets. Lemma 1.1.4. — Let X be a quasi-separated scheme. Let U be an open subset of X. a) If U is quasi-compact, then U is retrocompact. b) If X is quasi-compact, then U is retrocompact if and only if U is quasicompact. c) If X is quasi-compact, then every constructible set of X is globally constructible. Proof. — a) Let V be an affine open subscheme of X. If X is quasi-separated, then, by definition, the diagonal morphism ΔX : X → X ×Z X is quasicompact. So U ∩ V = Δ−1 X (U ×Z V ) is quasi-compact. That concludes the proof. b) Let (Xi )i∈I be a finite open covering of X by affine schemes. If U is retrocompact, then U ∩ Xi is quasi-compact for each i ∈ I. Hence U = U ∩ X is also quasi-compact. If U is quasi-compact and X quasii i∈I separated, for every affine open subscheme V of X, the open set U ∩V is quasicompact as a finite union of quasi-compact open sets. So U is retrocompact. c) Let C be a constructible subset of X. By the definition of constructibility and the quasi-compactness of X, there exist a finite open covering (Xi )i∈I by affine schemes and, for each i ∈ I, a finite number of quasi-compact open sets Uα,i , Vα,i of Xi (by b)) such that   C ∩ Xi = Uα,i ∩ (Xi Vα,i ) = Uα,i ∩ (X Vα,i ). α

But C = C ∩ i∈I Xi = i∈I α (Uα,i ∩ (X constructible in X.

α

Vα,i )). Hence C is globally

Proposition 1.1.5. — The set ConsX of the constructible subsets of a scheme X is a Boolean algebra, i.e., ConsX is stable under finite unions, finite intersections, and complements. Proof. — See (ÉGA III1 , 0III , §9.1). Remark 1.1.6. — Let X be a noetherian scheme. By Lemma 1.1.4, b), every open set of X is retrocompact. Consequently, the following properties are equivalent, for a subset C of X: (i) C is constructible; (ii) C is globally constructible in X; (iii) There exist locally closed subsets C1 , . . . , Cm of X such that C = C1 ∪ · · · ∪ Cm . Moreover, introducing the irreducible components of the locally closed subsets Ci in (iii), we see that they are equivalent to the following:

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467

(iv) There exist irreducible, locally closed, and pairwise disjoint subsets C1 , . . . , Cm of X such that C = C1 ∪ · · · ∪ Cm . 1.2. The Constructible Topology (1.2.1). — Let X be a scheme. A subset F of X is proconstructible if, for all x ∈ X, there exists an affine open subscheme U of X, containing x, such that F ∩ U is an intersection of constructible subsets of U . A subset E of X is indconstructible if, for all x ∈ X, there exists an affine open subscheme U of X, containing x, such that F ∩ U is a union of constructible subsets of U . The family of the indconstructible subsets of X forms the open sets of a topology on X that we call constructible topology. The closed sets of this topology are exactly the proconstructible subsets of X, by (ÉGA IV1 , 1.9.13). So, by the definition of indconstructibility, this topology is generated by the constructible subsets of X, which are open and closed in X. Remark 1.2.2. — When X is a noetherian scheme, this topology is also generated by the (locally closed) subschemes of X. Remark 1.2.3. — The constructible topology is different, in general, from the discrete topology (even in the noetherian case). Consider, for example, X = Spec(Z). For the constructible topology, the prime ideal (p) in Z, with p a prime number, corresponds to an open subset of X; but the generic point η, corresponding to the ideal (0), is closed (since it corresponds to X p V (p), where p runs over the set of the prime numbers of Z) and not open in X. Indeed, if η is indconstructible, then {η} = i∈I Ci , where Ci is locally closed in X. So there exists i0 such that {η} = Ci0 . Since η is the generic point of X (for the Zariski topology), Ci0 contains necessarily a nonempty open subset of X (for the Zariski topology). This is a contradiction. Theorem 1.2.4. — a) Let X be a quasi-compact scheme. Then X is quasi-compact for the constructible topology. In particular, let F be a proconstructible subset of X, and let (Oi )i∈I be a family of indconstructible subsets of X such that F ⊂ i∈I Oi ; then there exists a finite set J ⊂ I such that F ⊂ j∈J Oij . b) (Chevalley) Let f : X → Y be a morphism of schemes which is of finite type, and let C be a constructible subset of X. Then the subset f (C) of Y is a constructible subset of Y . If Y is noetherian, and if Z is a globally constructible subset of X, then f (Z) is a globally constructible subset of Y . c) Let f : X → Y be a morphism of schemes, and let C be a (resp. globally) constructible subset of Y . Then the subset f −1 (C) of X is a (resp. globally) constructible subset of X. Proof. — a) For the second assertion, see (ÉGA IV1 , 1.9.9), and the first assertion follows from it, see (ÉGA IV1 , 1.9.15). Assertion b) is Chevalley’s theorem. See (ÉGA IV1 , 1.8.4).

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For c), see (ÉGA IV1 , 1.8.2). 1.3. Constructible Subsets of Projective Limits The arc schemes and the Greenberg schemes used in motivic integration are non-noetherian in general but are naturally written as a projective limit of a sequence of noetherian schemes. We recall in this section results of (ÉGA IV1 , 8.3) that describe the constructible subsets of such projective limits. (1.3.1). — Let (I, ) be a filtrant ordered set, and let ((Xi )i∈I , (uij )ij ) be a projective system of noetherian schemes, such that for (i, j) ∈ I 2 , i  j, uij is an affine morphism of schemes. By (ÉGA IV3 , Proposition 8.2.3), the projective limit X = lim Xi exists in the category of schemes. Moreover, for ←− every i ∈ I, the canonical morphism of schemes ui : X → Xi is affine. In particular, X is quasi-compact and quasi-separated, since the Xi are quasicompact and quasi-separated. (1.3.2). — By theorem 1.2.4/c), the family

(ConsXi )i∈I , (u−1 ij )ij ) forms an inductive system of sets. Moreover, for every i ∈ I, the morphism ui : X → Xi induces a map u−1 i : ConsXi → ConsX , compatibly with the . We thus obtain a canonical map: maps u−1 ij v : lim ConsXi → ConsX . −→

(1.3.2.1)

i∈I

By restriction to the open, resp. closed constructible subsets, this map induces maps (1.3.2.2)

v  : lim OpenXi → OpenConsX , −→ i∈I

(1.3.2.3)



v : lim ClosedXi → ClosedConsX . −→ i∈I

Proposition 1.3.3. — Let ((Xi )i∈I , (uij )ij ) be a projective system of noetherian schemes, such that for (i, j) ∈ I 2 , i  j, uij is an affine morphism of schemes. Then the maps v, v  , v  defined above are bijections. Moreover, every constructible subset of X is globally constructible. Proof. — Let i ∈ I. Since Xi is noetherian, constructible and globally constructible subsets of Xi coincide. By théorème 8.3.11 of (ÉGA IV3 ), the assertion holds when “constructible” is replaced with “globally constructible.” In particular, the image of v lies in the set of globally constructible subsets of X. To conclude the proof of the proposition, it suffices to show that every constructible subset of X is globally constructible. Let thus A be a constructible subset of X, and let (Us )s be an open cover of X such that A ∩ Us is globally constructible in Us for every s. Since X is quasi-compact, we may assume that this cover is finite; by definition of the topology on X,

§ 2. BIRATIONAL GEOMETRY

469

we may then assume that there exist an element i ∈ I and a finite open cover (Us ) of Xi such that Us = vi−1 (Us ) for every j. Fix an index s. Since −1 vi−1 (Us ) = limji vij (Ui ), the first part of the proof implies that there exist ←− −1 an element j ∈ I such that j  i and a constructible subset As of vij (Ui ) −1 and A ∩ Us = vj (As ). Then A = As is a constructible subset of Xj such that A = vj−1 (A ). This proves that A is globally constructible. If A is closed −1 (resp. open), we may moreover take As to be closed (resp. open) in vij (Ui ),  so that A is closed (resp. open) in Xj . This concludes the proof.

§ 2. BIRATIONAL GEOMETRY We give a recollection of the main notion of birational geometry that we use in the book. 2.1. Blow-Ups Definition 2.1.1. — Let X be a scheme. Let Y be a closed subscheme of X, and let IY ⊂ OX be its sheaf The blow-up of X along Y is ! of ideals. n I the X-scheme BlY (X) = Proj n∈N Y ; the projection p : BlY (X) → X is called the blowing-up of X along Y , and the subscheme Y is called its center. Let us retain the notation of Definition 2.1.1. (2.1.2). — By construction, the tautological line bundle O(1) on BlY (X) is endowed with an isomorphism to p∗ IY ; in particular, p∗ IY is an invertible sheaf, and E = p−1 (Y ) is a Cartier divisor in BlY (X). It is called the exceptional divisor. In fact, the blowing-up is the universal morphism to X that makes IY a line bundle. In view of the isomorphisms IYn ⊗ (OX /IY )  IYn /IYn+1 , one has an isomorphism E  Proj



IYn /IYn+1



n∈N

of E with the projectivized normal cone of Y . Assume that Y is locally defined by a regular sequence of length r in X, then IY /IY2 is locally free of rank r as an OY -module, and the canonical morphism 

Sym• IY /IY2 → IYn /IYn+1 n∈N

is an isomorphism of graded algebras. In this case, E = p−1 (Y ) is a projective bundle of rank r − 1 on Y .

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APPENDIX

(2.1.3). — Above X Y , the ideal sheaf IY is invertible (generated by 1); hence, the blowing-up map p induces an isomorphism from BlY (X) E to X Y . In particular, if Y is nowhere dense, then p is a birational morphism. (2.1.4). — Let Z be a closed subscheme of X. The Zariski closure Z˜ of p−1 (Z Y ) in BlY (X) is called the strict transform of Z in BlY (X). The projection Z˜ → Z identifies with the blowing-up of Z along Y ∩ Z. Example 2.1.5. — The following example is both the simplest one and is the source of any property of blow-ups. Let X = Ank = Spec(k[T1 , . . . , Tn ]) be the affine space over a field k, and let Y = V (T1 , . . . , Tn ) be the origin in X. Inside X ×k Pn−1 = k Spec(k[T1 , . . . , Tn ]) ×k Proj(k[S1 , . . . , Sn ]), let B be the closed subscheme defined by the equations Ti Sj − Sj Ti , for 1  i < j  n. (Note that they are homogeneous of degree 1 in S1 , . . . , Sn .) Then the projection p : B → X is the blowing-up of X along Y ; it is projective. . The exceptional The fibers of p identify with closed subschemes of Pn−1 k , but for x = (x , . . . , x ) =  0, the fiber p−1 (x) divisor E = p−1 (Y ) is Pn−1 1 n k n−1 is the single point of Pk with homogeneous coordinates [x1 : . . . : xn ]. Proposition 2.1.6. — Let k be a field, and let us assume that X is ksmooth and Y is a smooth closed subscheme of Y . Then BlY (X) is k-smooth. 2.2. Resolution of Singularities (2.2.1). — Let k be a field and let X be an integral k-variety. Let D be a divisor (closed, purely 1-codimensional subscheme) in X, and let (Di )i∈I be the family  of its (reduced) irreducible components. For every subset J of I, let DJ = i∈J Di . One says that D has strict normal crossings if, for every subset J of I, the subscheme DJ of X is smooth and purely Card(J)-codimensional, in other words, if the irreducible components of D are smooth and meet transversally. (2.2.2). — A resolution of singularities of X is a proper birational morphism p : Y → X such that Y is smooth. Let p : Y → X be a resolution of singularities of X. There exists a largest subscheme U of X above which p is an isomorphism; the complementary subset E is a closed subscheme of X called the exceptional locus of p. (2.2.3). — Let Z be a closed subscheme of X, for example, an effective divisor in X. One says that p is a log resolution of the pair (X, Z), if its exceptional locus E is a divisor, as well as p−1 (Z), and if the divisor E + p−1 (Z) on Y has strict normal crossings. Theorem 2.2.4 (Hironaka 1964). — Let k be a field of characteristic zero. Let X be an integral k-variety and let Z be a closed subscheme of X. Then there exists a log resolution p : Y → X of the pair (X, Z) which is an

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471

isomorphism outside of Xsing ∪ Z. Moreover, the morphism p can be taken as a composition of blowing-ups with smooth centers. More recent versions show that p can be chosen so as to commute with smooth morphisms. Even more recently, Temkin (2012) has proved a functorial version of Theorem 2.2.4 valid for arbitrary quasi-excellent Q-schemes. See also (Kollár 2007) for a comprehensive survey, additional references, as well as a relatively short proof. Resolution of singularities is yet unknown in positive characteristic, but partial results are available in small dimension. This dates from the nineteenth century in Dimension 1, is due to Abhyankar (1998); Lipman (1978) in Dimension  2, and has been recently proved by Cossart and Piltant (2014) in Dimension 3. (2.2.5). — We will say that a field k allows resolution of singularities (for varieties of dimension  n) if the conclusion of Theorem 2.2.4 holds for every pair (X, Z), whenever X is a k-variety (of dimension  n). 2.3. Weak Factorization Theorem (2.3.1). — Let k be a field, and let X, Y be integral k-varieties. A rational map ϕ : X  Y is the datum of a morphism ϕU : U → Y defined over a dense open subscheme U of X, called an open subscheme of definition. There exists a largest open subscheme of definition, the domain of ϕ. We identify two rational maps when they coincide over a dense common open subscheme of their domains of definition; rational maps X  Y thus correspond to k(X)-points of Y . Any rational map ϕ has a graph Γϕ which is the smallest closed subscheme of X ×k Y containing the graph of a morphism ϕU : U → Y defining ϕ. One says that ϕ is dominant if ϕU is dominant for some (equivalently, any) open subscheme of definition U . This means that the projection from Γϕ to Y is dominant or, equivalently, that the image of the associated k(X)-point of Y is the generic point of Y . Dominant rational maps can be composed naturally and give rise to the rational category. One says that ϕ is birational if it is invertible in the rational category; this means that it is dominant and induces an isomorphism from k(X) to k(Y ). One says that ϕ is proper if the two projections from Γϕ to X and Y are proper. This is automatic if X and Y are themselves proper over k. Example 2.3.2. — Let X be an integral k-variety, and let Y be a strict closed subscheme of X. Then, the blowing-up p : BlY (X) → X of X along Y is a proper birational morphism. Symmetrically, the rational map p−1 : X  BlY (X) is a proper birational morphism; one often says that p−1 is a blowing-down. Theorem 2.3.3. — Let k be a field of characteristic zero. Let X, Y be smooth k-varieties, and let ϕ : X  Y be a proper birational map.

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a) There exist a sequence (V0 , . . . , Vm ) of smooth k-varieties such that V0 = X, Vm = Y , and, for every i ∈ {1, . . . , m}, a proper rational map ϕi : Vi−1  Vi such that either ϕi or its inverse is is a blowing-up along a smooth center, and such that ϕ = ϕm ◦ · · · ◦ ϕ1 . Such a sequence can be chosen so as to satisfy the following additional requirements: b) There exists an integer p such that Vi  X is defined everywhere for i  p, and Vi  Y is defined everywhere for i > p. c) Let U ⊂ X and V ⊂ Y be dense open subschemes such that ϕ is induced by an isomorphism from U to V ; then there exists a dense open subscheme Ui of ϕi , with U0 = U and V0 = V , such that, for every i, ϕi induces an isomorphism from Ui−1 to Ui . d) Assume moreover that D = X U (resp. E = Y V ) has a strict normal crossings divisor. Then one may assume that the inverse image of D (resp. of E) in Vi has strict normal crossings. e) Let S be a k-variety. Assume that X and Y are S-schemes and that ϕ is defined by an S-morphism (over some open subscheme of definition). Then one may assume that every Vi is an S-scheme and that the morphisms ϕi are rational maps of S-schemes. This is the weak factorization theorem of Abramovich et al. (2002); Włodarczyk (2003). It shows in particular that birational morphisms between smooth proper k-varieties are compositions of blowing-ups along smooth centers and their inverses. The case of surfaces goes back to the nineteenth century and holds in any characteristic: any birational map between smooth projective surfaces over a field k is a composition of blowing-up and blowingdowns along points. The adjective weak is a reference to the strong factorization conjecture, which states that one can even assume that ϕ1 , . . . , ϕp are blowing-ups and ϕp+1 , . . . , ϕm are blowing-downs. This statement is known to hold for surfaces but is yet unproven in dimension  3. 2.4. Canonical Divisors and Resolutions (2.4.1) Canonical Divisors. — Let k be a perfect field, and let X be a normal integral k-variety. Let U = Xsm be the smooth open subset of X; since k is perfect, U coincides with the regular locus of X, and U is dense in X. In fact, the assumption that X is normal implies that codim(X U )  2; as a consequence, the restriction map Div(X) → Div(U ) is an isomorphism of abelian groups. Let d = dim(X); the restriction to U of the canonical sheaf ΩdX is locally free. One says that a divisor D on X is a canonical divisor if there exists a nonzero meromorphic d-form ω on U such that div(ω) = D|U . Since codim(X U, X)  2, canonical divisors exist, and two canonical divisors on X differ by a principal divisor.

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(2.4.2). — Let KX be a canonical divisor on X. One says that X is Gorenstein if KX is a Cartier divisor. More generally, one says that X is QGorenstein if KX is Q-Cartier, that is, if there exists an integer m  1 such that mKX is a Cartier divisor. Then every canonical divisor is Cartier (resp. is Q-Cartier). (2.4.3) Relative Canonical Divisors. — From now on, we assume that X is Q-Gorenstein. Let KX be a canonical divisor on X. Let X  be a normal integral scheme, and let p : X  → X be a birational morphism; let V be the maximal open subset of X over which p is an isomorphism; since X is normal, one has codim(X V, X)  2. Let KX  be a canonical divisor on X  . Since the divisor KX is Q-Cartier and p is dominant, one may consider the pull-back p∗ KX of KX in DivQ (X  ). Let us recall its definition: let U be an open subset of X, u be a rational function on U , and m be a positive integer such that mKX |U = div(u); then p∗ KX |p−1 (U ) = div(p∗ u). The Q-divisor KX  − p∗ KX on X  is called a relative canonical divisor. It is a priori defined up to a principal divisor on X  . However, there is a canonical representative KX  /X . Let indeed U be the smooth locus of X, and let ω be a nonzero meromorphic form of maximal degree on U ; then p∗ ω is a nonzero form of maximal degree on p−1 (U ). Let U  be the smooth locus of X  ; there is a unique meromorphic form ω  of maximal degree on U  whose restriction to U  ∩ p−1 (U ) coincides with p∗ (ω). We set KX  /X = div(ω  ) − p∗ (div(ω)). Let ω1 be any other nonzero meromorphic form of maximal degree on U ; there exists a unique rational function f ∈ k(X)× such that ω1 = f ω. The previous construction leads to ω1 = (p∗ f )ω  and div(ω1 ) − p∗ div(ω1 ) = div((p∗ f )ω  ) − p∗ (div(f ω)) = div(p∗ f ) + div(ω  ) − div(p∗ f ) − p∗ (div(ω)) = KX  /X , so that the divisor KX  /X is a well-defined relative canonical divisor, independently of any choice. This divisor KX  /X is called the relative canonical divisor of X  /X; by construction, its support is contained in the exceptional locus Exc(p) of p. One says that the morphism p : X  → X is crepant if KX  /X = 0. Lemma 2.4.4. — Let X and X  be smooth integral k-schemes of finite type, and let p : X  → X be a birational morphism. Then the morphism p induces an injective morphism p∗ ΩdX → ΩdX  of line bundles whose image is ΩdX  (−KX  /X ). In particular, the relative canonical divisor is an effective Cartier divisor, and its support is precisely the exceptional locus of p. Proof. — The injectivity of this morphism follows from the fact that p is generically smooth—it is generically an isomorphism. Let E be the unique Cartier divisor on X  such that p∗ ΩdX maps to ΩdX  (−E). Let U = X  E so

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that the morphism p|U is étale. By Zariski’s main theorem, see (ÉGA III1 , corollaire 4.4.5); the morphism p|U is an open immersion, hence Exc(p) ⊂ E. On the other hand, p is not a local isomorphism at any point of E, so that E ⊂ Exc(p), and finally |E| = Exc(p). Since p∗ ΩdX  ΩdX  (−E), the divisor E is linearly equivalent to KX  /X , hence equal to KX  /X . This concludes the proof. Example 2.4.5. — Let X be a smooth integral k-variety, and let Z be a smooth irreducible subset of codimension r in X. Let X  be the blow-up of X along Z, and let p : X  → X be the canonical map; let E = p−1 (Z) be the exceptional divisor. One has KX  /X = (r − 1)E. (2.4.6) Discrepancies. — Let X be an integral k-variety; let us assume that X is Q-Gorenstein. Let p : X  → X be a proper birational morphism such that X  is smooth and Exc(p) is a divisor with strict normal crossings; let KX  /X be the relative canonical divisor of p. Let (Ei )i∈I be the family of irreducible components of Exc(p); there exists a unique family (νi )i∈I of rational numbers such that  ν i Ei . KX  /X = i∈I

The rational number νi is called the discrepancy of KX  /X along Ei . One says that the singularities of X are canonical (resp. log terminal), or that X is canonical (resp. log terminal), if one has νi  0 (resp. νi > −1) for every i ∈ I. One may prove, see Kollár and Mori (1998, corollary 2.31), that this property is independent of the choice of the resolution p, but this will also follow from results below. 2.5. K-equivalence Definition 2.5.1. — Let k be field. We say that two Q-Gorenstein kvarieties X and Y are K-equivalent if there exist a smooth k-variety Z and two proper birational morphisms (2.5.1.1)

Z f

X

g

Y

such that the relative canonical divisors KZ/X and KZ/Y are equal. Such a diagram is called a K-equivalence between X and Y . Remark 2.5.2. — By definition, two K-equivalent varieties are birational. Conversely, let X and Y be two Q-Gorenstein k-varieties which are birational. Assume that resolution of singularities holds for k-varieties of dimension  dim(X).

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Let h : X  Y be a birational map. Let U be a dense open subscheme of X on which h is defined, and let Z  ⊂ X ×k Y be the closure of its graph; it does not depend on the choice of U ; denote by f  , g  the projections of Z  to X, Y . Let p : Z → Z  be a resolution of singularities; define f = f  ◦ p and g = g  ◦ p. We thus obtain a diagram as in (2.5.1.1). If X and Y are proper or, more generally, if the birational map h is proper, then the morphisms f and g are proper. However, the condition KZ/X = KZ/Y might not hold in general. Proposition 2.5.3. — Let k be a field, and let X, Y , and Z be connected, Q-Gorenstein and proper k-varieties. Let f : Z → X and g : Z → Y be proper birational k-morphisms such that the line bundles f ∗ ωX and g ∗ ωY are numerically equivalent. If that resolution of singularities holds for varieties of dimension  dim(X), then the relative canonical divisors KZ/X and KZ/Y are equal. In particular, the line bundles f ∗ ωX and g ∗ ωY are isomorphic, and the diagram f g −Z− → Y is a K-equivalence. X← Proof. — Recall that the relative canonical divisors KZ/X and KZ/Y are the divisors defined by the Jacobian ideals of f and g. In particular, KZ/X −1 belongs to the divisor class of ωZ ⊗ f ∗ ωX , and KZ/Y belongs to the di∗ −1 visor class of ωZ ⊗ g ωY . Thus our assumption implies that the divisor D = KZ/Y − KZ/X is numerically trivial. Since each prime divisor in KZ/X is exceptional with respect to the morphism f , we have f∗ KZ/X = 0. Consequently, f∗ D = f∗ KZ/Y is both effective and numerically trivial; hence, f∗ D = 0. Applying lemma 3.39 of Kollár and Mori (1998) to the morphism f and to the f -nef line bundles ±D, this implies that D and −D are effective; hence, D = 0. Example 2.5.4. — Let k be a field. We say that a Q-Gorenstein proper variety is a Calabi–Yau variety if its canonical class vanishes modulo numerical equivalence. Since proposition 2.5.3 applies when X and X  have trivial canonical class, Remark 2.5.2 implies that birational Calabi–Yau varieties are K-equivalent. Example 2.5.5. — Let X be a Q-Gorenstein proper complex variety. Let fi : Xi → X, i ∈ {1, 2} be two crepant resolutions of singularities of X, i.e., which verify KXi − fi∗ KX = 0, for every i ∈ {1, 2}. It follows from Proposition 2.5.3 that X1 and X2 are K-equivalent. Remark 2.5.6. — The minimal model program furnishes other examples of K-equivalences. Recall that an integral proper k-variety is said to be minimal if it has terminal singularities and if its canonical divisor is numerically effective. The minimal model program predicts that every integral proper k-variety is birational to a minimal variety. This is known in Dimension 2; in fact, every

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integral projective k-surface is birational to a unique minimal variety, which is moreover smooth. The results of Birkar et al. (2010) imply that for every projective integral Q-Gorenstein k-variety X, there exist integral k-varieties X  with terminal singularities, endowed with a projective proper birational morphism to X which are “minimal over X,” that is to say, such that KX  /X is f -nef. (Explicitly, this means that for every closed irreducible curve C ⊂ X  such that f (C) is a point, one has KX  /X · C  0.) Wang (1998, Theorem 1.4, variant 1.11) has proved that two models of a projective integral k-variety X which are minimal over X are K-equivalent.

2.6. A Birational Cancellation Lemma Definition 2.6.1. — Let X and Y be irreducible k-varieties. We say that X and Y are stably birational if there exist m, n ∈ N such that X ×k Pm k and Y ×k Pnk are birational. Remark 2.6.2. — Obviously, two birational varieties are stably birational. The question whether two stably birational complex varieties of the same dimension are rational had been put forward by O. Zariski; see Segre (1950). However, by Beauville et al. (1985, §3, example 3), the hypersurface V in A4C defined by the equation y 2 + (t4 + 1)(t6 + t4 + 1)z 2 = 2x3 + 3t2 x2 + t4 + 1. is not rational (is not birational to P3 ), but V ×C A3C is rational. Consequently, P3 and V are stably birational but not birational. This example also shows that one cannot take X = V , Y = W = Z = A3 in Theorem 2.6.3 below. Theorem 2.6.3 (Liu and Sebag 2010). — Let k be a field, and let X and Y be integral k-varieties of the same dimension such that X and Y are not both uniruled. Consider two geometrically integral, rationally chain connected k-varieties W and Z and a birational map f : X ×k W Y ×k Z. Then neither X nor Y is uniruled, and there exists a unique birational map g : X  Y such that the diagram X ×k W

f

Y ×k Z

p1

X

q1 g

Y

commutes (the vertical arrows are the projection morphisms). The general idea of the proof is the following. Because of the assumption on X, Y , we can show that the action of the given birational map is constant on the second factors and thus cancellable. That procedure gives rise the birational map g.

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We present below another argument, more conceptual, based on the theory of maximal rationally connected (MRC) fibrations (see Kollár 1996, IV.5.1 and IV.5.4) but over fields of characteristic zero. For the general case, we refer to (Liu and Sebag 2010, Theorem 2). Proof. — Since the question is symmetric in X and Y , we may assume that X is not uniruled. As the question only depends on X, Y , Z, and W up to birational equivalence and k has characteristic zero, we can suppose that all varieties are projective and smooth, by resolution of singularities. Let π : X  R(X),

θ : Y  R(Y )

be the maximal rationally connected (MRC) fibrations of X and Y , respectively. Then the MRC fibrations of X ×k W and Y ×k Z are X ×k W → X  R(X) and Y ×k Z → Y  R(Y ). Indeed, let T be the MRC fibration of X ×k W . Since X ×k W is dominant, Kollár (1996, Chapter IV, Theorem 5.5) implies that there exists a dominant map T  R(X). By Kollár (1996, chapter IV, (5.1.2)), we have to verify that the fibers of X ×k W  R(X) are rationally chain connected. By Kollár (1996, chapter IV, (5.1.2) and definition 5.3), the claim is implied by the assumption on W and the property of R(X), because of the assumption on the field k. By Kollár (1996, IV.5.5), f induces a birational map g  : R(X)  R(Y ). Since X is not uniruled, π : X  R(X) is birational. Thus, it implies in particular that X has the same dimension as R(X), which is also the dimension of R(Y ). Thus Y and R(Y ) have the same dimension. It follows that Y cannot be uniruled, since otherwise R(Y ) would have smaller dimension than Y . Thus θ : Y  R(Y ) is birational, and g = θ−1 ◦ g  ◦ π is a birational map from X to Y that satisfies the conditions of the lemma. Uniqueness of g is obvious. Corollary 2.6.4. — Let k be a field of characteristic zero, and let X and Y be stably birational integral k-varieties such that X is not uniruled and dim(Y )  dim(X). Then X and Y are birational. Proof. — Applying Theorem 2.6.3 to X and Y ×k Pnk , with n = dim(X) − dim(Y ), we find that X and Y ×k Pnk are birational. Since X is not uniruled, this implies that n = 0. Corollary 2.6.5. — Let k be an algebraically closed field of characteristic zero, and let X, Y be integral k-varieties of the same dimension d  2. If X and Y are stably birational, then X and Y are birational. Proof. — Up to replacing X and Y by a smooth projective model, we may assume that X and Y are connected smooth projective k-varieties. The case d = 0 is trivial.

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Let us assume that d = 1. For the case d = 1, it suffices to note that by Lüroth’s theorem, X is not uniruled unless X ∼ = P1k . Consequently, the result follows from Corollary 2.6.4. Let us assume that d = 2. By corollary 2.6.4, we may suppose that X and Y are uniruled and thus of Kodaira dimension −∞. Looking at the Enriques classification of surfaces, we see that there exist smooth, projective, connected k-curves C and D such that X is birational to C ×k P1k and Y is birational to D ×k P1k . Then C and D are stably birational and thus isomorphic by the case d = 1. Hence X and Y are birational.

§ 3. FORMAL AND NON-ARCHIMEDEAN GEOMETRY The aim of this section is to provide a quick introduction to formal schemes and non-Archimedean analytic spaces, at the level needed to read the chapters on motivic integration on formal schemes and analytic spaces. In particular, we need to deal with formal schemes that are not adic and not noetherian, but only at the most basic level; so we will not develop this theory any further. Likewise, we will say very little about the foundations of non-Archimedean geometry, since we will mostly work with non-Archimedean analytic spaces in terms of their formal models. For a more thorough introduction to formal and non-Archimedean geometry, we recommend (ÉGA I, §10), (Fantechi et al. 2005, Ch.8), Abbes (2010), Bosch (2014), and Temkin (2015). 3.1. Formal Schemes The language of formal schemes was developed by Grothendieck to analyze infinitesimal structures in algebraic geometry. It has proven to be extremely useful in various context, in particular in deformation theory and more general moduli problems. See, for instance, (Fantechi et al. 2005, §8.5) for a taste of such applications. The main difference with the language of schemes is that the algebraic building blocks are topological rings. (3.1.1) Admissible and Adic Topological Rings. — Let A be a ring, endowed with a topology. We say that A is a topological ring if addition and multiplication are continuous. A topological A-algebra is a topological ring B endowed with a continuous ring morphism A → B. The topology on a topological ring A is called linear if the zero element has a basis of neighborhoods that are ideals. Note that an ideal of A with nonempty interior is automatically open, since every translation is a homeomorphism on A. We say that the topological ring A is pre-admissible if there exists an ideal I in A such that I is open and such that the powers I n tend to zero as n → ∞; this means that for every open neighborhood V of 0 in A, there

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exists a positive integer n0 such that the ideal I n is contained in V for all n  n0 . Such an ideal I is called an ideal of definition. A pre-admissible topological ring is called admissible if it is separated and complete. (3.1.2). — The most important class of topological rings in formal geometry are the adic rings. A pre-admissible topological ring A is called pre-adic if it has an ideal of definition I such that I n is open for every n > 0; this implies that the ideals I n form a basis of open neighborhoods of 0 in A. If this holds for one ideal of definition I, then every ideal of definition has the same property. We say that A is adic if, moreover, A is separated and complete. This is equivalent to saying that the natural morphism (A/I n ) A → lim ←− n>0

is an isomorphism of topological rings, where A/I n carries the discrete topology for every n. In that case, we call the topology on A the I-adic topology. If A is an adic topological ring with ideal of definition I and J is an ideal in A, then J is an ideal of definition if and only if there exist integers m, n > 0 such that J m ⊂ I n ⊂ J. Example 3.1.3. — Let A be a ring of characteristic p > 0, for some prime p. Then the ring of Witt vectors W (A) is an admissible topological ring. It is adic if A = Ap , but not in general. (3.1.4) The Category of Formal Schemes. — Let X be a scheme. By putting the discrete topology on the sheaf of regular functions OX , we obtain a presheaf of topological rings, which is not a sheaf when X is not quasicompact (since the product topology on an infinite product of discrete spaces is not the discrete topology). Passing to the associated sheaf, we obtain a topologically ringed space X top whose underlying ringed space is X and whose rings of sections are discrete on every quasi-compact open subset of X. This construction gives rise to a full embedding of the category of schemes into the category of locally topologically ringed spaces. From now on, we will view every scheme as a topologically ringed space in this way. (3.1.5). — Let A be an admissible topological ring. Then one can associate with A its formal spectrum Spf(A), which is a locally ringed space in topological rings. Its underlying topological space is the set of open prime ideals in A, endowed with the Zariski topology (the topology induced by the Zariski topology on Spec(A)). Note that for every ideal of definition I in A, the morphism Spec(A/I) → Spec(A) is a homeomorphism onto Spf(A), because every open prime ideal of A contains I n for sufficiently large n and thus I.

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The structure sheaf on Spf(A) is characterized by the following property: for every open subset U of Spf(A), we have a natural isomorphism of topological rings OSpec(A/I) (U ) OSpf(A) (U ) ∼ = lim ←− I

where I runs through a fundamental system of ideals of definition in A, ordered by inclusion, and OSpec(A/I) (U ) carries the discrete topology. In particular, OSpf(A) (Spf(A)) = A, and we have a natural morphism of locally topologically ringed spaces ι : Spf(A) → Spec(A). If P is an open prime ideal of A, then the local ring OSpf(A),P is not complete, in general, but its separated completion with respect to the maximal ideal is isomorphic to the separated completion of the local ring AP . This implies that ι is flat if A is noetherian. Example 3.1.6. — Let A be a local adic topological ring whose maximal ideal is an ideal of definition. Then the formal spectrum Spf(A) consists of a unique point. Thus the underlying topological space of Spf(A) contains very little information about A, but we can recover the topological ring A by looking at the global sections of Spf(A). (3.1.7). — A formal scheme is a topologically ringed space X that is locally of the form Spf(A), with A an admissible topological ring. We say that X is affine if it is isomorphic to Spf(A) for some admissible topological ring A. If U is an open subspace of a formal scheme X, and we denote by OU the restriction of OX to U, then the topologically ringed space (U, OU ) is again a formal scheme. We call such a pair an open formal subscheme of X. (3.1.8). — A morphism of formal schemes Y → X is a morphism of locally ringed spaces in topological rings. Thus the formal schemes form a full subcategory (For) of the category of locally ringed spaces in topological rings. If X is affine, then the correspondence f : Y → X) → (f  ,

OX (X) → OY (Y))

defines a bijection between the set of morphisms of formal schemes Y → X and the set of continuous ring morphisms OX (X) → OY (Y) (ÉGA I, 10.4.6). The category (For) has fibered products (ÉGA I, 10.7.3). If A, B, and C are admissible topological rings and A → B and A → C are continuous ring morphisms, then the fibered product of Spf(B) and Spf(C) over Spf(A)  AC is given by the formal spectrum of the completed tensor product B ⊗ (ÉGA I, 0.7.7.5). A morphism of formal schemes Y → X is called separated if the image of the diagonal morphism Y → Y ×X Y is closed. A formal scheme X is called separated if the unique morphism X → Spec(Z) is separated. (3.1.9) Locally Noetherian Formal Schemes. — A formal scheme X is called adic (resp. locally noetherian) if it can be covered by affine open formal subschemes U such that OX (U) is an adic topological ring (resp. a noetherian

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adic topological ring). We say that X is noetherian if it is locally noetherian and its underlying topological space is quasi-compact. One can prove that for every affine open formal subscheme U of a locally noetherian formal scheme X, the topological ring OX (U) is adic and noetherian (ÉGA I, 10.6.5). All the local rings of a locally noetherian formal scheme are noetherian. (3.1.10). — For every locally noetherian formal scheme X, we define the dimension dim(X) of X as the supremum of the Krull dimensions of its local rings. If X is affine, then the dimension of X is equal to the Krull dimension of OX (X), because the local ring of X at a point x and the localization of O(X) at the open prime ideal corresponding to x have the same completion, and completing a noetherian local ring preserves the dimension. (3.1.11). — Let A be a noetherian adic topological ring. An ideal of definition of X = Spf(A) is an ideal sheaf I on X for which there exists an ideal of definition I in A such that the ideal I (U) is generated by the image of I in OX (U), for every affine open formal subscheme U of X. If X is any locally noetherian formal scheme, then an ideal of definition of X is an ideal sheaf I whose restriction to each affine open formal subscheme U is an ideal of definition of U. (3.1.12). — If X is a locally noetherian formal scheme and I is an ideal of definition of X, then the locally ringed space V (I n ) = (|X|, OX /I n ) is a scheme, for every integer n > 0, and the natural morphism of topologically ringed spaces X∼ V (I n ) = lim −→ n

is an isomorphism. In practice, it is often convenient to describe a locally noetherian scheme X in terms of the schemes V (I n ). (3.1.13). — Every locally noetherian formal scheme X has a largest ideal of definition I , which is equal to the radical of each ideal of definition of X. The scheme V (I ) is reduced and is called the reduction of X and denoted by Xred . If J is any ideal of definition of X and f : Y → X is a morphism of locally noetherian formal schemes, then J OY must be contained in an ideal of definition of Y, by continuity of the morphism OX → f∗ OY . Thus the correspondence X → Xred gives rise to a functor (·)red from the category of locally noetherian formal schemes to the category of reduced schemes. A morphism of locally noetherian formal schemes f : Y → X is separated if and only if fred is a separated morphism of schemes. Example 3.1.14. — If X is a locally noetherian scheme, then the associated topologically ringed space is a locally noetherian formal scheme. Its

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largest ideal of definition is the nilradical of X, so that Xred is the maximal reduced closed subscheme of X. (3.1.15). — A morphism of locally noetherian formal schemes Y → X is called adic if there exists an ideal of definition I of X such that J = I OY is an ideal of definition of Y; then this property holds for every ideal of definition of X. If X is a locally noetherian formal scheme, then an X-adic formal scheme is a locally noetherian formal scheme Y endowed with an adic morphism Y → X. If X = Spf(A), then we also speak of A-adic formal schemes instead of X-adic formal schemes. Example 3.1.16. — Let X be a locally noetherian formal scheme (for instance, a locally noetherian scheme), and let Z be a subscheme of Xred . Then * of X along Z is defined as follows. First, we choose the formal completion X/Z an open formal subscheme U of X containing Z such that Z is closed in X. We denote by I the defining ideal of Z in U, and we set * = lim V (I n ) X/Z −→ n>0

where the limit is taken in the category of topologically ringed spaces. This definition only depends on X and the underlying space of Z, and not on the choice of U or the schematic structure of Z. * is a locally noetherian formal scheme, The topologically ringed space X/Z with underlying topological space Z, and the morphism of locally ringed spaces * →X X/Z is called the completion morphism. Note that I O * is an ideal of definiX/Z * and that the reduction of X/Z * is the maximal reduced closed tion of X/Z subscheme Zred of Z. * should be viewed as an infinitesimal tube around The formal scheme X/Z Z in X; it can be used to study the infinitesimal structure of X around Z. If X is another locally noetherian formal scheme, Z  is a subscheme of  Xred , and f : X → X is a morphism of formal schemes such that f (Z  ) is contained in Z and then f induces a morphism of formal schemes *  /Z  → X/Z.  f: X It is adic if f −1 (Z) is open in Z  , but not in general. (3.1.17) Coherent Sheaves. — Let X be a locally noetherian formal scheme. Then by (ÉGA I, 10.11.1), the structure sheaf OX is coherent, and every ideal of definition of X is coherent. Coherent sheaves of OX -modules are defined in the usual way (ÉGA I, 0.5.3). They can be described in terms of coherent sheaves on schemes as follows (see (ÉGA I, §10.10 and §10.11) for details). Let I be an ideal of definition of

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X. For all integers n  m > 0, we denote by ιn the morphism of topologically ringed spaces ιn : V (I n ) → X and by ιmn the closed immersion of schemes ιmn : V (I m ) → V (I n ). If F is a coherent sheaf on X, then ι∗n F is a coherent sheaf on the scheme V (I n ). Conversely, suppose that we are given a projective system (Fn , fnm ) where Fn is a coherent sheaf on V (I n ), for every n > 0, and such that for all n  m > 0, the transition map fnm : Fn → Fm is OV (I n ) -linear and induces an isomorphism ι∗mn Fn → Fm . Then the projective limit of (Fn , fnm ) in the category of OX -modules is a coherent sheaf F , and ι∗n F is isomorphic to Fn for all n. If X = Spf(A) is a noetherian affine formal scheme and M is an A-module of finite type, then we can define a coherent sheaf M on X by first taking the coherent sheaf on Spec(A) associated with M and then pulling it back through the completion morphism Spf(A) → Spec(A). For every affine open formal subscheme U of X, we have M (U) = M ⊗A OX (U). In particular, M (X) = M . The correspondence M → M defines an equivalence of categories between the category of A-modules of finite type and the category of coherent sheaves on X = Spf(A). (3.1.18) Closed Formal Subschemes. — Let X be a locally noetherian formal scheme, and let J be a coherent ideal sheaf of X. If we denote by Y the support of the quotient sheaf OX /J and by OY the restriction of OX /J to Y, then the pair (Y, OY ) is again a locally noetherian formal scheme. Formal schemes that are constructed in this way are called closed formal subschemes of X. A closed (resp. open) immersion of locally noetherian formal schemes Y → X is a morphism of formal schemes that factors through an isomorphism onto a closed (resp. open) formal subscheme of X.

3.2. Morphisms of Finite Type and Morphisms Formally of Finite Type (3.2.1) Algebras of Convergent Power Series. — Let A be a noetherian adic topological ring with ideal of definition I, and let r be a positive integer. The A-algebra of convergent power series A{z1 , . . . , zr } in the variables z1 , . . . , zr is the sub-A-algebra of A[[z1 , . . . , zr ]] consisting of the power series whose coefficients tend to zero:  A{z1 , . . . , zr } = { aν z ν | aν → 0 as ν → ∞} ν∈Nr

484

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where we used the usual multi-index notation z ν = z1ν1 . . . zrνr and we set ν = ν1 + . . . + νr . We turn B = A{z1 , . . . , zr } into an adic topological A-algebra by endowing it with the IB-adic topology. Another way to describe this topological A-algebra is as a projective limit (A/I n )[z1 , . . . , zr ] A{z1 , . . . , zr } = lim ←− n>0

where (A/I n )[z1 , . . . , zr ] carries the discrete topology, since the ideals I n form a basis of open neighborhoods of 0 in A. The ring A{z1 , . . . , zr } is noetherian (ÉGA I, 0.7.5.4). The name “convergent power series” refers to the fact that for every element f in A{z1 , . . . , zr } and every a in Ar , the series f (a) converges in A. Some authors use the term “restricted power series” instead. A topological A-algebra is called topologically of finite type if it is isomorphic to a topological A-algebra of the form C = A{z1 , . . . , zr }/J, endowed with the IC-adic topology, where J is an ideal in A{z1 , . . . , zr }. (3.2.2) Morphisms of Finite Type. — We say that a morphism of locally noetherian formal schemes f : Y → X is locally of finite type if, for every point y of Y, we can find an affine open neighborhood U of f (y) in X and an affine open neighborhood V of y in f −1 (U) such that OY (V) is topologically of finite type over OX (U). This is equivalent to saying that f is adic and, for some ideal of definition I of X, the scheme V (I OY ) is locally of finite type over V (I ); then this property holds for all ideals of definition I of X. We say that f is of finite type if it is locally of finite type and quasicompact. The class of morphisms locally of finite type (resp. of finite type) is stable under composition and base change. Open and closed immersions of locally noetherian formal schemes are morphisms of finite type. Example 3.2.3. — Let A be a noetherian adic topological ring with ideal of definition I. For every A-scheme X locally of finite type, the completion of X along the closed subscheme X ×A (A/I) is a formal A-scheme locally of  It is called the I-adic completion of X. finite type, which we denote by X.  ×A (A/I n ) = X ×A (A/I n ) for every n > 0. By construction, we have X If X is affine, say, X = Spec(A[z1 , . . . , zr ]/(f1 , . . . , f )),  is given by then X  = Spf(A{z1 , . . . , zr }/(f1 , . . . , f )). X (3.2.4) Morphisms Formally of Finite Type. — One can relax the definition of a morphism of finite type and still obtain a class of morphisms with good properties. Let A be a noetherian adic topological ring with ideal of definition I. A topological A-algebra is called formally of finite type if it is isomorphic to a

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quotient of a topological A-algebra of the form B = A{z1 , . . . , zr }[[w1 , . . . , ws ]] endowed with the IB + (w1 , . . . , ws )-adic topology. (3.2.5). — Let X be a locally noetherian scheme. We say that a morphism of formal schemes f : Y → X is locally formally of finite type if, for every point y of Y, there exist an affine open neighborhood U of f (y) in Y and an affine open neighborhood V of y in f −1 (U) such that OY (V) is formally of finite type over OX (U). This implies, in particular, that Y is locally noetherian. It follows from Tarrío et al. (2007, 1.7) that a morphism of formal schemes f : Y → X is locally formally of finite type if and only if there exist an ideal of definition I of X and an ideal of definition J of Y such that I OY is contained in J and the scheme V (J ) is locally of finite type over V (I ); then this property holds for all ideals of definition I and J such that I OY is contained in J . Note that every morphism locally of finite type is locally formally of finite type and that a morphism locally formally of finite type is locally of finite type if and only if it is adic. We say that f is formally of finite type if it is locally formally of finite type and quasi-compact. The class of morphisms locally formally of finite type (resp. formally of finite type) is stable under composition and base change. These morphisms appear under various names in the literature: morphisms locally formally of finite type are called special in Berkovich (1996a), and morphisms formally of finite type are called morphisms of pseudo-finite type in Tarrío et al. (2007). Example 3.2.6. — Let A be a noetherian adic topological ring with ideal of definition I, let X be an A-scheme locally of finite type, and let Z be  of X along Z is a a subscheme of X ×A (A/I). Then the completion X/Z formal A-scheme locally formally of finite type, since its reduction is equal to Zred and thus locally of finite type over Spf(A)red = Spec((A/I)red ). The  is locally of finite type over A if Z is open in X ×A (A/I), formal scheme X/Z but not in general. It is separated if and only if Z is separated, and it is flat over A if and only if the scheme X is flat over A at every point of Z. If X is affine, say, X = Spec(A[z1 , . . . , zr ]/(f1 , . . . , f )), and Z is the zero locus of the ideal generated by I and (zq , . . . , zr ) for some  is given by integer q > 0, then X  = Spf(A{z1 , . . . , zq−1 }[[zq , . . . , zr ]]/(f1 , . . . , f )). X More generally, if X is a formal scheme locally formally of finite type over * is still locally A and Z is a subscheme of Xred , then the completion X/Z formally of finite type over A.

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3.3. Smoothness and Differentials In this book, the notion of smoothness of formal schemes and the sheaves of differentials play an important role. The theory can be developed analogously to the case of schemes, and we refer to (Tarrío et al. 2007) for a detailed account. (3.3.1) Infinitesimal Lifting Criteria. — One can define formally unramified, étale, and smooth morphisms of formal schemes much in the same way as for schemes, in terms of infinitesimal lifting criteria. Let f : Y → X be a morphism of locally noetherian formal schemes. Then we say that f is formally unramified (resp. formally étale, formally smooth) if it satisfies the following infinitesimal lifting criterion: for every affine scheme Z over X and every closed subscheme T of Z define by a square zero ideal, the map Hom(For/X) (Z, Y) → Hom(For/X) (T, Y) is injective (resp. bijective, surjective) (Tarrío et al. 2007, definition 2.1). (3.3.2). — A morphism of locally noetherian formal schemes f : Y → X is called unramified (resp. étale, smooth) if it is formally unramified (resp. formally étale, formally smooth) and locally formally of finite type (Tarrío et al. 2007, definition 2.6). Each of these classes of morphisms is stable under composition and base change (Tarrío et al. 2007, proposition 2.9). We say that f is unramified (resp. étale, smooth) at a point y of Y if there exists an open neighborhood U of y in Y such that the morphism U → X induced by f is unramified (resp. étale, smooth). Example 3.3.3. — the formal scheme

a) If A is a noetherian adic topological ring, then Spf (A{z1 , . . . , zr }[[w1 , . . . , ws ]])

is smooth over Spf(A) for all r, s  0. Beware that it is not of finite type over Spf(A) unless s = 0. Assume that A is local and that its maximal ideal m is an ideal of definition. We denote by k = A/m the residue field of A. Then a formal A-scheme formally of finite type X is smooth over Spf(A) at a k-rational point of Xred if and only if the completed local ring OX,x is isomorphic, as an A-algebra, to a power series ring A[[w1 , . . . , ws ]]. b) If X is a locally noetherian formal scheme and Z is a subscheme of Xred , then the completion morphism * →X X/Z is étale. (3.3.4) Modules of Differentials. — Analogously to the case of schemes, one can study differential properties of morphisms of formal schemes by means of modules of differentials.

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Let A be a noetherian adic topological ring, and let B be a topological A-algebra formally of finite type. The usual module of Kähler differentials ΩB/A (forgetting the topology on A and B) is not of finite type over B, in general. However, its completion is well behaved; it is defined by  B/A = lim (ΩB/A /J n ΩB/A ) Ω ←− n>0

where J is any ideal of definition in B. This is a B-module of finite type. We call it the module of continuous differentials of B over A. The map  B/A , b → db d: B → Ω is the universal continuous derivation of B over A into a complete B-module; see Tarrío et al. (2007, 1.10). Example 3.3.5. — Assume that B = A{z1 , . . . , zr }[[w1 , . . . , ws ]].  B/A , In this case, the elements dz1 , . . . , dzr , dw1 , . . . , dws form a basis of Ω which is thus free of rank r + s, and the differential of an element f ∈ B is computed formally: df =

r s   ∂f ∂f dzi + dwj . ∂z ∂w i j i=1 j=1

 B/A Let I be an ideal of B, and let C = B/I. Let N be the submodule of Ω generated by elements of the form df , with f ∈ I. The universal continuous  B/A /N )⊗A C of C derivation on B induces a continuous derivation d : C → (Ω over A which satisfies the universal property. This leads to the fundamental exact sequence of C-modules: (3.3.5.1)

 B/A ⊗B C → Ω  C/A → 0. I/I 2 → Ω

The corresponding complex (3.3.5.2)

 B/A ⊗B C → Ω  C/A → 0 0 → I/I 2 → Ω

will be called the fundamental complex associated with the pair (B, I). (3.3.6). — This definition of continuous differentials can be globalized to define the coherent sheaf of (continuous) differentials ΩY/X for an arbitrary morphism Y → X between locally noetherian schemes which is locally formally of finite type. In particular, if Y = Spf(B) → X = Spf(A) is a morphism of finite type of noetherian affine formal schemes, then the sheaf of differentials ΩY/X is the coherent sheaf on Y associated with the B-module  B/A . of finite type Ω These sheaves of differentials satisfy the usual calculus. Let f : Y → X and g : Z → Y be morphisms locally formally of finite type between locally noetherian formal schemes.

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a) There is a canonical exact sequence of OZ -modules: (3.3.6.1)

g ∗ ΩY/X → ΩZ/X → ΩZ/Y → 0.

b) Assume moreover that g is a closed immersion, defined by a coherent ideal sheaf J on Y, then there is a canonical fundamental exact sequence of coherent OZ -modules: (3.3.6.2)

J /J 2 → g ∗ ΩY/X → ΩZ/X → 0,

and a canonical fundamental complex of coherent OZ -modules associates with the immersion g: (3.3.6.3)

0 → J /J 2 → g ∗ ΩY/X → ΩZ/X → 0,

which generalize the exact sequences (3.3.5.1) and (3.3.5.2) in the affine case. (3.3.7). — One has the expected relations between infinitesimal lifting properties and modules of differentials; proofs can be found in section 4 of Tarrío et al. (2007). Let f : Y → X and g : Z → Y be morphisms locally formally of finite type between locally noetherian formal schemes. Then f is unramified if and only if ΩY/X = 0. If g is étale, then the map g ∗ ΩY/X → ΩZ/X is an isomorphism. If f is smooth, then f is flat, and ΩY/X is locally free. Moreover, if f is smooth, then g is smooth if and only if g ◦ f is smooth and the sequence of OZ -modules 0 → g ∗ ΩY/X → ΩZ/X → ΩZ/Y → 0 is exact and locally split. Finally, there is also a version of Zariski’s Jacobian criterion in this setting: if f is smooth and g is a closed immersion defined by a coherent ideal sheaf J on Y, then g ◦ f is smooth if and only if the sequence of OZ -modules 0 → J /J 2 → g ∗ ΩY/X → ΩZ/X → 0 is exact and locally split. 3.4. Formal Schemes over a Complete Discrete Valuation Ring (3.4.1). — For our purposes, the most important class of formal schemes is the following. Let R be a complete discrete valuation ring with maximal ideal m. We endow R with its m-adic topology; then R becomes an adic topological ring. A formal R-scheme of finite type (resp. formally of finite type) is a formal scheme X over Spf(R) such that the structural morphism X → Spf(R) is of finite type (resp. formally of finite type). Formal schemes formally of finite type over R are excellent; this is Proposition 7 of Valabrega (1975) when R has equal characteristic and Theorem 9 of Valabrega (1976) in the mixed characteristic case.

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(3.4.2). — For every integer n  0, we set Rn = R/mn+1 and Xn = X ⊗R Rn . If X is formally of finite type over R, then Xn is a formal Rn -scheme formally of finite type, for every n. The formal k-scheme X0 is called the special fiber of X. If X is of finite type over R, then Xn is a scheme of finite type over Rn for every n. Conversely, if (Xn )n>0 is a direct system of R-schemes of finite type such that mn+1 = 0 on Xn for every n  0 and such that the transition morphism fmn : Xm → Xn induces an isomorphism of Rm -schemes Xm ∼ = Xn ⊗Rn Rm for all 0  m  n, then the direct limit X = lim Xn −→ n>0

in the category of topologically ringed spaces is a formal R-scheme of finite type, and the natural morphism Xn → X induces an isomorphism Xn → Xn for every n  0. (3.4.3). — In the same way, giving a morphism f : Y → X of formal Rschemes of finite type amounts to giving a family of morphisms (fn : Yn → Xn )n0 such that fn is a morphism of Rn -schemes and all the squares Ym

fm

Xm

Yn

fn

Xn

are Cartesian. (3.4.4). — Let X be a formal scheme formally of finite type over R. We define the relative dimension of X to be the dimension of its special fiber X0 . More generally, for every point x ∈ X(k), the dimension of the special fiber X0 at the point x is called the relative dimension of X at x. We say that X is flat over R if the structural morphism X → Spf(R) is a flat morphism of locally ringed spaces; this is equivalent to saying that for every affine open formal subscheme U of X, the R-algebra OX (U) has no m-torsion. (3.4.5). — An immersion of formal R-schemes f : X → Y is called regular at x ∈ X if the kernel of the local homomorphism OY,f (x) → OX,x can be generated by the elements of a regular sequence in OY,f (x) . This property is stable under base change to arbitrary extensions of R, because the regularity of a sequence of elements in a ring is preserved under flat morphisms (Liu 2002, 6.3.10). We say that a formal R-scheme of finite type X is a local complete intersection at a point x ∈ X if x has an open neighborhood U in X such that there exist a smooth formal R-scheme Y and an immersion X → Y which is

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regular at x. We say that X is a local complete intersection if it is so at every point. Lemma 3.4.6. — Let X be a formal R-scheme of finite type, and let x ∈ X(k); let d be the relative dimension of X at x over R. If X is a local complete intersection at x, then there exist an integer  0, a regular sequence (f1 , . . . , f ) in R[[z1 , . . . , z+d ]], and an isomorphism of completed local rings 

→ OX,x . R[[z1 , . . . , z+d ]]/(f1 , . . . , f ) − Proof. — Up to replacing X by a formal affine open neighborhood of x, we may assume that there exists a regular immersion f : X → Y, where Y is a smooth formal R-scheme. Let y = f (x) and let r be the relative dimension of Y at y. The completed local ring of Y is isomorphic to R[[z1 , . . . , zr ]]. By flatness of the completion morphism OX,x → OX,x , the immersion f induces a surjective morphism R[[z1 , . . . , zr ]]→ OX,x , whose kernel I is generated by a regular sequence (f1 , . . . , f ). It then follows from (ÉGA IV2 , 7.1.4) that dim(OX,x ) = dim(OX,x ) = dim(R[[z1 , . . . , zr ]]) − = r − . This concludes the proof. (3.4.7) Rig-Irreducible Components of Formal Schemes. — Let X be a formal R-scheme formally of finite type. We want to define a notion of irreducible components of X. This is not obvious, because the topology of X reflects its geometry rather poorly. For instance, if X is the formal spectrum of a complete local ring, then its underlying topological space consists of a single point, but its ring of regular functions may have several minimal prime ideals. In order to solve this issue, we adopt a similar strategy as the one of Conrad (1999). ( → X be the normalization morphism constructed in Conrad Let h : X ( is the formal (1999, 2.1). If X is affine and N is the nilradical of O(X), then X spectrum of the integral closure of O(X)/N in its total ring of fractions. The general construction is then carried out by gluing. Since X is excellent, the ( → X is finite. normalization morphism h : X The rig-irreducible components of X are the closed formal subschemes of X defined by the coherent ideal sheaves of the form ker(OX → h∗ OC ) where C ( is a connected component of X. If X is affine, then its rig-irreducible components are simply the closed formal subschemes defined by the minimal prime ideals in OX (X). We say that X is rig-irreducible if it is nonempty and has a unique rigirreducible component.

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Remark 3.4.8. — The rig-irreducible components of X may display some unexpected behavior: the following example shows that a nonempty open subscheme of a rig-irreducible formal scheme is not necessarily rig-irreducible! Let π be a uniformizer in R and let X = Spf(R{x, y}/(π − xy)). Then X is normal (even regular) and connected. However, if we remove the point defined by the ideal (π, x, y), then we find an open formal subscheme of X which is disconnected and has two rig-irreducible components (defined by the equations x = 0 and y = 0, respectively). Proposition 3.4.9. — Let X be a formal R-scheme formally of finite type and let d be a nonnegative integer. Then the following are equivalent: a) Every rig-irreducible component of X has dimension d; b) The local ring at each closed point of X is equidimensional of dimension d. If X satisfies the equivalent conditions of Proposition 3.4.9, we say that X has pure dimension d. Proof. — Since normalization commutes with base change to an open formal subscheme, we may assume that X is affine, say, X = Spf(A). Then the rig-irreducible components of X are the closed formal subschemes defined by the minimal prime ideals of A. It follows from (ÉGA IV2 , 7.1.2) that the localization of A at an open prime ideal p is equidimensional if and only if the local ring of X at the point corresponding to p is equidimensional. Thus we may assume that A is integral, and it is enough to show that the Krull dimension of A is equal to the height of each maximal ideal. Every maximal ideal of A contains m, because A is m-adically complete. Thus, in order to prove the desired property, we may replace A by A ⊗R k, since this either reduces the dimension of A and the height of each maximal ideal by 1 (if m is nonzero on A) or leaves them invariant (if m = 0 on A). The ring A ⊗R k is an adic completion of a finitely generated k-algebra, so that the result follows from the analogous property for finitely generated k-algebras (ÉGA IV2 , 5.2.1). 3.5. Non-Archimedean Analytic Spaces (3.5.1). — Let R be a complete discrete valuation ring with residue field k and quotient field K. The valuation vK on K gives rise to an absolute value by setting |x| = exp(−vK (x)) for every element x of K × . Non-Archimedean analytic geometry is a theory of analytic spaces over the non-Archimedean field K. Naïvely mimicking the definitions and constructions over the complex numbers gives rise to the theory discussed in §1/1. Despite its interest, it is not satisfactory for the purpose of algebraic geometry, because the metric topology on a non-Archimedean field is totally disconnected. Historically,

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various theories have been proposed to solve this issue: rigid analytic spaces (Tate 1971), formal geometry up to admissible blowing-ups (Raynaud 1974; Abbes 2010; Bosch 2014), and, more recently, the theory of analytic spaces developed by Berkovich (1990, 1993) and the theory of adic spaces (Huber 1994). For the spaces that we will consider, all of these theories give rise to essentially equivalent categories. For practical reasons, and because they are closer to the standard intuition, we use in this book Berkovich’s theory (1) . We will give a concise overview of some important features of the theory, with an emphasis on the theory of formal models, which is a fundamental viewpoint for the applications in this volume. (3.5.2) Affinoid Algebras. — The basic building blocks of the theory are spectra of affinoid algebras. For every integer r > 0, we consider the Tate algebra Tr of convergent power series with coefficients in K, defined by Tr = K{z1 , . . . , zr }  = {f = aν z ν ∈ K[[z1 , . . . , zr ]] | |aν | → 0 as ν → ∞}, ν∈Nr

in which we the standard multi-index notation z ν = z1ν1 · . . . · zrνr and use r set ν = i=1 νi . The convergence condition in the definition implies in particular that the coefficient aν belongs to R when ν is sufficiently large. More precisely, the canonical morphism of K-algebras R{z1 , . . . , zr } ⊗R K → K{z1 , . . . , zr } is an isomorphism. The algebra Tr is a Banach algebra over K with respect to the Gauss norm f = max|aν |. ν

One can show that Tr is noetherian and that every ideal I of Tr is closed with respect to this Gauss norm, so that we can consider the residue norm on the quotient Tr /I. A Banach algebra A over K is called (strictly) affinoid if there exists a surjective morphism of K-algebras ϕ : Tr → A such that the norm on A is equivalent to the residue norm on Tr / ker(ϕ). (3.5.3) Berkovich Spectrum. — Let A be Banach algebra. The Berkovich spectrum of A is the set M (A) of all bounded multiplicative seminorms on A. In other words, it is the set of maps x : A → R0 such that – x(0) = 0 and x(a + b)  x(a) + x(b) for all elements a, b in A; – x(1) = 1 and x(ab) = x(a)x(b) for all elements a, b in A; – x(a) = |a| for every a ∈ K; – There exists a constant C > 0 such that x(a)  C a for every a in A.

(1) We

will only consider strictly K-analytic and Hausdorff analytic spaces.

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We endow M (A) with the topology of pointwise convergence, that is, the weakest topology such that the evaluation map x → x(a) from M (A) to R0 is continuous, for every a in A. One also defines a Grothendieck topology on M (A), called the G-topology, and a sheaf of analytic functions with respect to the G-topology. (3.5.4) K-analytic Spaces. — K-analytic spaces are defined by considering topological spaces endowed with a suitable notion of affinoid atlas, using the language of nets—see Berkovich (1993, §1). They are endowed both with a Grothendieck topology and with a usual topology. Every G-open of an analytic space inherits a natural structure of analytic space, and such a subspace is called an analytic domain. The embedding of an analytic domain in a K-analytic space is called an analytic domain immersion . A K-analytic space is called good if every point has an affinoid neighborhood. This is the category of spaces that was originally defined in Berkovich (1990), but non-good analytic spaces arise naturally, for instance, by considering generic fibers of certain formal R-schemes. See, for instance, Example 4.2.1.4 in Temkin (2015). To every separated K-scheme X of finite type, one can associate a Kanalytic space X an by a process of analytification. The space X an is good, and it is compact if and only if X is proper over K. (3.5.5) Analytic Spaces Versus Rigid Varieties. — Historically, the first fully developed theory of non-Archimedean analytic spaces was Tate’s theory of rigid varieties. The category of Hausdorff strictly K-analytic spaces admits a fully faithful embedding into the category of quasi-separated rigid K-varieties, and this embedding restricts to an equivalence between the category of paracompact Hausdorff strictly K-analytic spaces and the category of quasi-separated rigid K-varieties that have an admissible affinoid covering of finite type (Berkovich 1993, 1.6.1). In particular, we have an equivalence between the category of compact strictly K-analytic spaces and the category of quasi-compact quasi-separated rigid K-varieties. Under this embedding, strict affinoid domains in K-analytic spaces correspond to affinoid open subvarieties in rigid K-varieties, and strict compact analytic domains correspond to quasi-compact open rigid subvarieties (Berkovich 1993, 1.6.2). (3.5.6). — A little extra care is needed in dealing with the notion of smoothness: if f is a morphism of Hausdorff strictly K-analytic spaces, then f is quasi-smooth (or rig-smooth) if and only if the associated morphism of rigid K-varieties is smooth. This notion will be more useful for us than Berkovich’s definition of smooth morphisms from Berkovich (1993). A nice reference for the theory of quasi-smooth morphisms is chapter 5 of (Ducros 2018). If the source and target of f are good K-analytic spaces, the morphism f is smooth if and only if it is quasi-smooth and has no boundary. A typical example of a morphism that is quasi-smooth (even quasi-étale) but not smooth

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is the embedding of the closed unit disk in the analytification of the affine line over K; the existence of a boundary point prevents this embedding from being smooth. (3.5.7) Formal Models. — The comparison results for the categories of rigid K-varieties and K-analytic spaces allow us to apply results from the literature on rigid analytic geometry to K-analytic spaces, in particular, Raynaud’s description of the category of rigid K-varieties in terms of their formal models (Bosch and Lütkebohmert 1993). Let us briefly recall the main ingredients of this theory, translated into the language of K-analytic spaces and applying our convention that all Kanalytic spaces are assumed to be strictly K-analytic and Hausdorff. There is a generic fiber functor X → Xη from the category of formal Rschemes of finite type to the category of compact K-analytic spaces. If X is affine, say X = Spf(A), then Xη = M (A ⊗R K). If Y → X is an open (resp. closed) immersion of formal R-schemes of finite type, then Yη → Xη is an analytic domain (resp. closed) immersion of K-analytic spaces. If X is a flat formal R-scheme of finite type whose generic fiber Xη is good and quasi-smooth, then Xη is smooth if and only if X is proper over R. (3.5.8). — We call a formal R-scheme admissible if it is flat and of finite type (beware that, in Bosch and Lütkebohmert (1993), admissible means flat and locally of finite type). Let X be an admissible formal R-scheme with ideal of definition I . Let A be an open coherent sheaf of ideals on X, and let Y = V (A ) be the formal subscheme of X that it defines. For every n ∈ N, let Xn = X ⊗R Rn and Yn = Y ⊗R Rn . The formal R-scheme BlYn (Xn ) X = lim −→ n

is an admissible formal R-scheme, called the formal blow-up of X along Y; the canonical morphism X → X is called a formal blowing-up, and Y is called its center. The generic fiber functor maps admissible blowing-ups of admissible formal R-schemes to isomorphisms of K-analytic spaces. Thus, it induces a functor from the category of admissible formal R-schemes modulo admissible blowing-ups to the category of compact K-analytic spaces. The key result in Raynaud’s theory states that this is an equivalence of categories (Bosch and Lütkebohmert 1993, 4.1). In more explicit terms, every compact K-analytic space is isomorphic to the generic fiber of an admissible formal R-scheme (called an admissible formal model of the analytic space), and two morphisms of admissible formal R-schemes X → Y coincide if and only if the induced morphisms Xη → Yη are equal. Moreover, for every pair of admissible formal R- schemes X, Y and every morphism of K-analytic spaces f : Yη → Xη , there exists an admissible blowing-up Y → Y such that f extends to a morphism of formal R-schemes

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Y → X. In particular, the natural map X(R ) → Xη (K  ) is a bijection, for every finite extension R of R with quotient field K  . We will also use that, for every admissible formal R-scheme X and every compact analytic domain U in Xη , there exist an admissible blowing-up X → X and an open immersion U → X such that U is the image of Uη → Xη (Bosch and Lütkebohmert 1993, 4.4).  its formal (3.5.9). — Let X be an R-scheme of finite type and denote by X m-adic completion. There exists a natural morphism of K-analytic spaces η → (XK )an . This is an analytic domain immersion if X is separated over X R, and it is an isomorphism if X is proper over R. (3.5.10) The Generic Fiber of a Formal Scheme Formally of Finite Type. — In Berthelot (1996), the construction of the generic fiber functor for formal R-schemes of finite type is extended to formal R-schemes that are formally of finite type. A summary together with some additional results can also be found in de Jong (1995, §7). An equivalent construction is given in Berkovich (1996a) in the framework of K-analytic spaces. For our purposes, it is sufficient to recall the following properties. Let Y be a formal R-scheme formally of finite type, and let Yη be its generic fiber. This is a paracompact Hausdorff strictly K-analytic space. It is compact if X is of finite type over R, but not in general. For every finite extension R of R with quotient field K  , there is a natural bijection Y(R ) → Yη (K  ). Moreover, if X → Y is the dilatation of Y (defined in section 7/5.1.1), then the induced morphism Xη → Yη is an analytic domain immersion, and it induces a bijection Xη (K  ) → Yη (K  ) for every finite unramified extension K  of K (the latter property follows from the fact that X(R ) → Y(R ) is a bijection for every finite unramified extension R of R). If X is a K-analytic space, then a formal R-model of X is a formal Rscheme X formally of finite type endowed with an isomorphism of K-analytic spaces Xη → X. If X and X are formal R-models of X, then a morphism of formal Rmodels X → X is a morphism of formal R-schemes whose restriction to the generic fibers commutes with the isomorphisms to X. If such a morphism exists, we say that X dominates X. If X is flat over R, then there exists at most one morphism of formal R-models X → X. Example 3.5.11. — Assume that Y is affine, say Y = Spf(R{z1 , . . . , zr }[[w1 , . . . , ws ]]/(f1 , . . . , f )). The generic fiber Yη can be explicitly described as follows. Let B = M (K{z}) be the closed unit disk over K, and let E be the open unit disk over K. Then each power series fi ∈ O(Y) defines an analytic function on B r ×K E s , and Yη is the closed analytic subspace of B r ×K E s defined by the equations f1 = . . . = f = 0.

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(3.5.12) The Specialization Map. — For every formal R-scheme formally of finite type X, there exists a canonical specialization map spX : Xη → X. This is a morphism of locally ringed spaces with respect to the G-topology on Xη , but the map on underlying topological spaces is anti-continuous with respect to the Berkovich topology on Xη : the inverse image of a closed set in X is open in Xη . If R is a finite extension of R with quotient field K  and x is a K  -point on Xη , then spX (x) is the image of the unique extension of x to a morphism Spf(R ) → X. If Y is a locally closed subset of X0 , then sp−1 X (Y ) is an analytic domain in Xη that is canonically isomorphic to the generic fiber of the completion of X along Y . (3.5.13) Dimension and Irreducible Components. — The irreducible components of rigid K-varieties have been defined by Conrad (1999) via a process of normalization similar to the one we have used to define the rig-irreducible components of formal schemes in §3.4.7. A different and more general definition for analytic spaces was given by Ducros (2009), and he showed that his definition is equivalent to the one of Conrad for rigid varieties. In order to preserve the parallels with the definition for formal schemes, and since we only need to deal with Hausdorff strictly K-analytic varieties, we will follow Conrad’s definition, translated to the language of Berkovich spaces by means of the results in Ducros (2009). ( → X is defined Let X be a K-analytic space. The normalization h : X in Ducros (2009, 5.6, 5.10); it is a finite surjective morphism of K-analytic spaces (Ducros 2009, 5.11.1). We define the irreducible components of X to be the closed analytic subspaces defined by a coherent ideal sheaf of the form ker(OX → h∗ OC ) where ( This is equivalent to Ducros’ definition, C is a connected component of X. by (Ducros 2009, 5.15). Proposition 3.5.14. — Let X be a flat formal R-scheme formally of finite type. Then the generic fiber functor (·)η induces a bijection between the set of rig-irreducible components of X and the set of irreducible components of Xη . Proof. — This property was proven in Conrad (1999, 2.3.1) for rigid varieties, so that it suffices to show that the notion of irreducible component behaves well under the comparison functors between the categories of rigid varieties and analytic spaces in §3.5.5. For every K-analytic space X, we will denote the associated rigid variety by X rig . Then we must show that the functor (·)rig induces a bijection between the set of irreducible components of X and the set of irreducible components of X rig in the sense of Conrad (1999). This follows from the fact that (·)rig preserves connected components and commutes with normalization (since the constructions are the same for affinoid spaces).

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(3.5.15). — We refer to Ducros (2007, §1) and (Ducros 2009, 0.26) for the dimension theory of analytic spaces. If A is a K-affinoid algebra, then the dimension of M (A) is the Krull dimension of A. The dimension of a Kanalytic space X is the supremum of the dimensions of its affinoid domains. If X is irreducible, then all of its nonempty affinoid domains have the same dimension as X. We say that a K-analytic space X has pure dimension d if each of its irreducible components has dimension d; this is equivalent to saying that every nonempty affinoid domain in X has pure dimension d. Proposition 3.5.16. — Let X be a formal R-scheme formally of finite type. Then the dimension of Xη is at most the dimension of X0 . If X is flat over R, we have dim(X0 ) = dim(Xη ) = dim(X) − 1. In other words, the dimension of the generic fiber of the formal R-scheme X is at most the relative dimension of X, and both dimensions coincide when X is flat over R. We will say that X has pure relative dimension d if both Xη and X0 have pure dimension d. By Proposition 3.5.17 below, this happens, in particular, when X is flat over R and has pure dimension d + 1. Proof. — Replacing X by its maximal R-flat closed formal subscheme, it suffices to consider the case where X is flat. We may assume that X is affine, say X = Spf(A). Let d be the Krull dimension of A. Since X is catenary, the special fiber X0 has dimension d − 1 by flatness of X and the Krull Hauptidealsatz. We must show that the dimension of Xη also equals d − 1. Note that d − 1 is precisely the Krull dimension of A ⊗R K. By de Jong (1995, 7.1.9), there exists a bijective correspondence between the rigid points x of Xη (i.e., the points of (Xη )rig ) and the maximal ideals m of A ⊗R K. Moreover, the completion of the localization of A ⊗R K at m is isomorphic to the completion of the G-local ring of Xη at x. Since the completion of a noetherian local ring preserves its dimension, it now suffices to show that the dimension of Xη equals the supremum of the Krull dimensions of the G-local rings at the rigid points of Xη . This follows from Ducros (2007, 0.26.10). Proposition 3.5.17. — Let X be a flat formal R-scheme formally of finite type, and let d be a nonnegative integer. Then the following are equivalent: a) The formal scheme X has pure dimension d + 1; b) The generic fiber Xη has pure dimension d; c) The special fiber X0 has pure dimension d. Proof. — Since X is catenary, the equivalence of a) and c) is an easy consequence of Proposition 3.4.9, flatness of X and the Krull Hauptidealsatz. So let us prove the equivalence of a) and b). The rig-irreducible components of X are still flat over R, because the normalization of X is flat over R. Thus the result follows from Propositions 3.5.14 and 3.5.16.

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