Algebraic geometry 1

Contents

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Basics on commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Ideals and operations on ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 UFD’s and PID’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Polynomial rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Polynomials in D[x], where D a UFD . . . . . . . . . . . . . . . . 1.3.2 The case D = K a field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Resultant of two polynomials in D[x] . . . . . . . . . . . . . . . . . 1.3.4 Resultant of two polynomials in D[x1 , . . . , xn ] and elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Noetherian rings and the Hilbert basis theorem . . . . . . . . . . . . . . 1.5 R-modules, R-algebras and finiteness conditions . . . . . . . . . . . . . 1.6 Integrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Zariski’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Trascendence degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Tensor products of R-modules and of R-algebras . . . . . . . . . . . . . 1.9.1 Restriction and extension of scalars . . . . . . . . . . . . . . . . . . 1.9.2 Tensor product of algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Graded rings, homogeneous ideals . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.1 Homogeneous polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.2 Graded morphisms, graded modules and miscellanea . . . 1.11 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.1 Local rings and Localization . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebraic Affine Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Algebraic affine sets and ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Hilbert ”Nullstellensatz” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Some consequences of Hilbert Nullstellensatz and of Elimination theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Study’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Intersections of affine plane curves . . . . . . . . . . . . . . . . . . .

1 2 4 4 4 7 9 11 12 15 18 20 22 25 28 29 30 34 39 42 45 46 49 49 62 66 66 68

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2.4 Further remarks: K0 -rational points . . . . . . . . . . . . . . . . . . . . . . . . 69 2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3

Algebraic Projective Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.1 Algebraic Projective Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.2 Homogenoeus ”Hilbert Nullstellensatz” . . . . . . . . . . . . . . . . . . . . . 79 3.3 Fundamental examples and remarks . . . . . . . . . . . . . . . . . . . . . . . . 82 3.3.1 Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.3.2 Coordinate linear subspaces . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.3.3 Hyperplanes and the dual projective space . . . . . . . . . . . . 83 3.3.4 Fundamental affine open sets (or affine charts) of Pn . . . 84 3.3.5 Projective closure of affine sets . . . . . . . . . . . . . . . . . . . . . . 85 3.3.6 Projective subspaces and their ideals . . . . . . . . . . . . . . . . . 87 3.3.7 Projective and affine subspaces. Projective closure of affine subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.3.8 Homographies, projectivities, affinities and subspaces . . . 91 3.3.9 Projective cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.3.10 Projective hypersurfaces and projective closure of affine hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.3.11 Proper closed subsets of P2 . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.3.12 Affine and projective twisted cubics . . . . . . . . . . . . . . . . . . 97 3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

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Topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.1 Irreducible topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.1.1 Coordinate rings, ideals and irreducibility . . . . . . . . . . . . . 107 4.1.2 Algebraic varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.2 Noetherian spaces. Irreducible components . . . . . . . . . . . . . . . . . . 110 4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

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Regular and rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.1 Regular functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.2 Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.2.1 Consequences of Theorem ?? . . . . . . . . . . . . . . . . . . . . . . . . 129 5.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Appendix to Chapter 5: Basics on sheaves . . . . . . . . . . . . . . . . . . . . . . . 137 5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

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Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.1 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.2 Morphisms with (quasi) affine target . . . . . . . . . . . . . . . . . . . . . . . 143 6.3 Morphisms with (quasi) projective target . . . . . . . . . . . . . . . . . . . 151 6.3.1 The Veronese morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.3.2 Veronese morphism and consequences . . . . . . . . . . . . . . . . 157 6.3.3 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Contents

6.4 6.5

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Morphisms and local properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

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Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.1 Products of affine varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.1.1 Coordinate ring of V × W affine and tensor products . . . 166 7.2 Products of algebraic varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 7.2.1 Segre morphism and the product of projective spaces . . . 168 7.2.2 Products of projective varieties . . . . . . . . . . . . . . . . . . . . . . 170 7.2.3 Products of quasi-projective varieties . . . . . . . . . . . . . . . . . 172 7.3 Diagonals, graphs and affine open sets . . . . . . . . . . . . . . . . . . . . . . 173 7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

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Rational maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 8.1 Rational and birational maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 8.1.1 Some properties and examples of (bi)rational maps . . . . . 179 8.2 Unirational and rational varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 183 8.2.1 Stereographic projection of a rank-four quadric surface . 184 8.2.2 Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 8.2.3 Blow-up of Pn at a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 8.2.4 Blow-ups and resolution of singularities . . . . . . . . . . . . . . . 190 8.3 Birational transformations of an algebraic variety . . . . . . . . . . . . 194 8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

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Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9.1 Dimension of an algebraic variety . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

10 Completeness of projective varieties . . . . . . . . . . . . . . . . . . . . . . . . 203 10.1 Complete algebraic varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 10.2 The main theorem of elimination theory . . . . . . . . . . . . . . . . . . . . 205 10.2.1 Some consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 10.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 11 Tangent spaces and smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 11.1 Tangent space at a point of an affine variety. Smoothness . . . . . 209 11.2 Tangent space at a point of a projective variety. Smoothness . . 213 11.3 Zariski tangent space of an algebraic variety. Smoothness . . . . . 215 11.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

1 Basic notions on commutative rings with identity

All rings considered in this book will be commutative and with a multiplicative identity. If R is a ring, 0R and 1R (or simply 0 and 1, if no confusion arises) will respectively denote its additive and multiplicative identities. Given R and S two rings, any non-zero ring homomorphism ϕ : R → S will be such that ϕ(1R ) = 1S . An integral domain (or simply, domain) is a ring with no zero-divisors, i.e. where cancellation law holds. A field K is a domain in which every non-zero element is invertible, i.e. a unit. In symbols, U (K) = K \ {0}, where U (K) denotes the (multiplicative) group of units. For any field K the symbol K will denote its algebraic closure (K always exists and is uniquely determined up to field isomorphism, cf. M. Artin’s proof in e.g. [21, cf. V. § 2. Thm.2.5 and Cor. 2.6, 2.9]). Z will denote the domain of integers, whereas Q, R and C will denote the fields of rational, real and complex numbers, respectively. Zp will denote the finite field with p elements, p ∈ Z a prime. For the sake of completeness of this preliminary overview, we have to recall the classical result (first proved by Gauss in 1799), whose proof can be found e.g. in [21] (to which the interested reader is referred). Theorem 1.0.1 The field C is algebraically closed. Let R be a ring. An ideal I ⊆ R is a subset of R such that, for any a, b ∈ I and for any r ∈ R one has a + b ∈ I and ra ∈ I. If R is a domain, Q(R) will denote its quotient field, so that R ⊆ Q(R) where equality holds if and only if R is a field. For any ring R, R[x] denotes the ring of polynomials in one indeterminate x and with coefficients from R. More generally, if x := (x1 , . . . , xn ) are n indeterminates over R, R[x1 , . . . , xn ] denotes the ring of polynomials in the n indeterminates and with coefficients from R; for simplicity, sometimes we will use the symbol R[x] to denote this ring, if no confusion arises.

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1 Basics on commutative rings

1.1 Ideals and operations on ideals Let R be a ring. A proper ideal p ⊂ R is said to be prime if rs ∈ p implies that either r ∈ p or s ∈ p; in particular, p is prime if and only if R/p in an integral domain. Analogously, a proper ideal m ⊂ R is said to be maximal if no proper ideal I in R exists such that m ⊂ I ⊂ R, where all the inclusion are strict; in particular, m is maximal if and only if R/m is simple (i.e. with no proper ideals) which, by the assumptions on R, is equivalent to R/m being a field. It is therefore clear that a maximal ideal is also prime, but the converse is not true: e.g. consider the ideal (x) ⊂ Z[x], which is prime but not maximal. We recall some standard operations on ideals, which will be frequently used in the next chapters. Let I be an ideal of R. A set S of elements of R is called a set of generators for I if ( ) X I= ri si | si ∈ S, ri ∈ R , i

P

where in i ri si only finitely many ri ’s are non-zero. I is said to be a finitely generated ideal if I admits a finite set of generators S := {s1 , . . . , sn }; in such a case, we will write I := (s1 , . . . , sn ). In particular, when n = 1, I is called a principal ideal. If I1 , I2 are ideals, their sum is defined as I1 + I2 := {x1 + x2 | x1 ∈ I1 , x2 ∈ I2 } , which is the smallest ideal of R containing both I1 and I2 . More generally, if we have a (possibly infinte) family {Iα }α∈A of ideals of R, their sum is defined as ) ( X X xα | xα ∈ Iα Iα := α∈A

α∈A

where in the P sums almost all (i.e. up to a finite number) of the xα ’s equal 0. As above, α∈A Iα is the smallest ideal of R containing all the Iα ’s. T For a (possibly infinite) family {Iα }α∈A of ideals, their intersection α∈A Iα is an ideal. The product of two ideals I1 and I2 is denoted by I1 · I2 and defined to be the ideal generated by all the products x1 x2 , for any Px1 ∈ I1 and x2 ∈ I2 . The elements of this ideal are finite sums of the forms i xi yi , where xi ∈ I1 and yi ∈ I2 , for any i. Similarly, one can define the product of a (possibly infinte) family {Iα }α∈A of ideals in R. As a particular case of products of ideals, for any positive integer n, one has the notion of nth -power of an ideal I, which is denoted by I n (by convention, one poses I 0 := (1) = R). Thus, I n is the ideal generated by all the products of the form x1 x2 · · · xn , where xi ∈ I, for any 1 6 i 6 n. For any ideal I of R, one defines the radical of I to be the ideal

1.1 Ideals and operations on ideals

√

I := {x ∈ R | xn ∈ I, for some positive integer n}

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(1.1)

(some texts denote the radical with the √ symbol rad(I); see e.g. [1]). Notice that, for any ideal I, one clearly has I ⊆ I. √ Definition 1.1.1 I is said to be a radical ideal (or simply radical), if I = I, equivalently if R/I is a reduced ring, i.e. R/I has no nilpotents √ Lemma 1.1.2 (i) For any ideal I ⊆ R, I is radical. (ii) Any prime (respectively, maximal) ideal is radical. p√ p√ √ I. Now, f ∈ I implies Proof. (i) As for any ideal, one has I ⊆ √ there exists a positive integer n such that f n ∈ I. Thus, for some positive p√ √ √ integer m, (f n )m ∈ I, i.e. f ∈ I, which means that I ⊆ I. (ii) It directly follows from the definition of prime (resp., maximal) ideal. Let ϕ : R → S be any ring homomorphism. Definition 1.1.3 If J ⊆ S is any ideal, ϕ−1 (J) is an ideal in R, which is called the contracted ideal of J w.r.t. ϕ and denoted by J c (sometimes even by J ∩ R, even if ϕ not necessarily injective, cf. [1]). Conversely, if I ⊆ R is an ideal, the ideal of S generated by the subset ϕ(I) ⊆ S is called the extended ideal of I w.r.t. ϕ and denoted by I e . Remark 1.1.4 (i) Notice that ϕ(I) ⊆ S in general is not an ideal: consider e.g. ϕ : Z ֒→ Q to be the natural inclusion and I any proper ideal of Z. (ii) For any ideal J ⊆ S one has J ⊇ (J c )e and in general the inclusion is strict; ϕ consider e.g. Z2 ֒→ Z2 [x], where ϕ the natural inclusion, and J = (x2 + x + 1), so (J c )e = (0). Viceversa, for any ideal I ⊆ R one has I ⊆ (I e )c and in general the ϕ inclusion is strict; consider as in (i) Z ֒→ Q and I = (p), for some prime p ∈ Z, so (I e )c = (1). (iii) If p ⊂ S is a prime ideal, then pc ⊂ R is also prime; conversely, p ⊂ R prime ideal does not imply that pe ⊆ S is necessarily a prime ideal, cf. (ii) above. (iv) Notice that if m ⊂ S is a maximal ideal then mc ⊆ R in general is only ϕ

prime but not maximal. Indeed, consider e.g. Z[x] ։ Z defined by ϕ(q(x)) = q(0), for any q(x) ∈ Z[x], and J = (p), for any prime p ∈ Z; in such a case (p)c is prime but not maximal since Z[x]/(p)c ∼ = Zp [x] is an integral domain but not a field.

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1 Basics on commutative rings

1.2 UFD’s and PID’s Recall that a ring R is said to be a principal ring if every ideal I of R is principal. If R is an integral domain and a principal ring, then R is said to be a principal ideal domain (or simply a PID). Any field K and Z are easy examples of PID’s. Recall that, when K is a field, the ring K[x] is a Euclidean domain, roughly speaking it is an integral domain where Euclidean division algorithm holds. This implies that K[x] is a PID (cf. e.g. [21, IV § 1, Thm. 1.2]). From now on, let D denote an integral domain. Given a, b ∈ D we say that a divides b, or that b is divisible by a (in symbols a|b) if there exists c ∈ D s.t. b = ca; in particular, a is invertible if and only if a|1D . Recall that a, b ∈ D are called associate elements (simply associates) if a|b and b|a, i.e. b = ea for some e ∈ U (D). An element a ∈ D \ {0} is irreducible if it is not a unit in D and, whenever one can write a = bc, with b, c ∈ D, then either b or c is a unit in D; in other words a is irreducible iff it is divisible only by its associate elements and by the units in D. An element a ∈ D \ {0} is said to have a unique factorization into irreducible elements if there exist u ∈ U (D) and pi ∈ D irreducible elements, 1 6 i 6 r (not necessarily all distincts), such that a = up1 · · · pr and, if moreover we have two factorizations a = up1 · · · pr = u′ q1 · · · qs , we have r = s and, after a permutation of the indices, we have pi = ui qi for any 1 6 i 6 r, for some ui ∈ U (D). D is said to be a Unique Factorization Domain, UFD for short, if every non-zero element of D has a unique factorization into irreducible elements. In particular, any prime element (i.e. generating a prime ideal) of a UFD is an irreducible element (the converse being always true). At last, in a UFD one can define a greatest common divisor for finite set of elements in D (this is defined up to units of D). Recall that any PID is a UFD (cf. e.g. [21, II § 5, Thm. 5.2]); in particular, Z, any field K and K[x] are UFD’s.

1.3 Polynomial rings Here we collect some useful terminology and results on polynomials in one or more indeterminates, with coefficients from an integral domain, in particular from a UFD. 1.3.1 Polynomials in D[x], where D a UFD If D0 is an integral domain, definitions in § 1.2 apply to the integral domain D := D0 [x1 , . . . , xn ] giving rise to divisibility among polynomials, associate

1.3 Polynomial rings

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polynomials, irreducible polynomials, etc. Any polynomial of degree 1 with invertible coefficients is always irreducible. In general, the irreducibility of a polynomial depends on the domain D0 : x2 + 1 is irreducible as an element of R[x] but it factorizes into irreducible elements (x − i)(x + i) in C[x], where i2 = −1. Recall the following fundamental results, whose proofs can be found e.g. in [21, IV § 2], to which we refer the interested reader. Theorem 1.3.1 (Unique factorization theorem) Let D be a UFD and x an indeterminate over D. Then D[x] is a UFD. Remark 1.3.2 When K is a field, K[x1 , . . . , xn ] is a UFD, for any n > 1: indeed one recursively applies Theorem 1.3.1 to D′ := K[x1 , . . . , xn ] = (K[x1 , . . . , xn−1 ])[xn ] = D[xn ], with D := K[x1 , . . . , xn−1 ] a UFD. Theorem 1.3.3 (Gauss’ theorem) Let D be a UFD, K := Q(D) be its quotient field and x an indeterminate over D. Let f ∈ D[x] be a non-constant polynomial. Then: f factorizes in D[x] ⇔ f factorizes in K[x]. Remark 1.3.4 As a consequence of the previous result, prime (equivalently, irreducible) elements of D[x] are prime (equivalently, irreducible) elements of D and non-constant polynomials whose content is 1 and which are irreducible in K[x] (for details, cf. e.g. [21, IV § 2 Thm. 2.4]). Suppose to have f ∈ D[x] and consider its factorization into irreducible elements in D[x]; reaping the repeated irreducible factors, one gets f = g1e1 · · · gses ,

(1.2)

where the gi ’s are all the irreducible, distinct factors of f ∈ D[x] and ei ’s are non-negative integers, 1 6 i 6 s. The exponent ei is called the multiplicity of the factor gi in f , for 1 6 i 6 s. If ei > 1 then gi is said to be a multiple factor of f . One obviously has deg(f ) =

s X

ei deg(gi ).

i=1

Definition 1.3.5 If f ∈ D[x] is a non-constant polynomial then f = a0 + a1 x + a2 x2 + . . . + an xn ∈ D[x], for some aj ∈ D, 0 6 j 6 n, with an 6= 0. The integer n > 1 is the degree of f and an ∈ D is called the leading coefficient of f , which is also denoted by lc(f ). A non-costant polynomial is said to be monic if lc(f ) = 1.

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1 Basics on commutative rings

A non-constant polynomial f ∈ D[x], for which lc(f ) ∈ U (D), is always associated to a monic polynomial f ′ ∈ D[x]. In particular, for D = K a field, any non-constant polynomial is always associated to a monic one. Definition 1.3.6 Let f ∈ D[x] be a non-constant polynomial. An element α ∈ D is said to be a root of f , if f (α) = 0. The following is an easy straightforward result. Lemma 1.3.7 α ∈ D is a root of f ∈ D[x] if and only if (x − α) divides f in D[x]. Definition 1.3.8 α ∈ D is said to be a multiple root of f ∈ D[x], if (x − α) is a multiple factor of f , i.e. if one has f = (x − α)e g,

(1.3)

for some integer e > 2 (cf. decomposition (1.2)). The greatest integer e for which the factorization (1.3) exists is called the multiplicity of the root α for f . If e = 1, then α is said to be a simple root of f . If α is not a root of f , then one has e = 0. Notice that e is the multiplicity of the root α if and only if one has g(α) 6= 0 in (1.3); moreover, since deg(g) = deg(f ) − e, one has e 6 deg(f ), for any root of f . Corollary 1.3.9 A non-constant polynomial f ∈ D[x] of degree d has at most d roots in D (when the roots are counted with multiplicity). Proof. One can easily prove it by induction on d.

⊓ ⊔

It is obvious that the number of roots in D strictly depends on the domain D: e.g. x2 + 1 ∈ R[x] has no roots in R but it splits as (x + i)(x − i), i2 = −1, when viewed as element of C[x]. The notion of derivative of a polynomial can be given formally, with no use of infinitesimal calculus machinery. This allows one to consider derivatives of polynomials in D[x], for D any domain. Precisely, if f := ad xd + ad−1 xd−1 + · · · + a1 x + a0 ∈ D[x], one defines the (first) derivative of f to be df = f ′ := dad xd−1 + (d − 1)ad−1 xd−2 + · · · + a1 dx

(1.4)

and recursively, the k-th derivatives, for any k > 2, to be the polynomial given by
d f [k−1] dk f [k] = f := , dxk dx

where f [1] := f ′ .

1.3 Polynomial rings

7

Similarly, if f ∈ D[x1 , . . . , xn ], then one can define the partial derivative ∂f , to be the with respect to the indeterminate xj , 1 6 j 6 n, denoted by ∂x j (first) derivative of f considered as an element of (D[x1 , . . . , xj−1 , xj+1 , . . . xn ])[xj ]. It is therefore clear that one can also define higher order partial derivatives and that the Schwartz rule holds, as it can be easily verified on monomials. We will not dwell on this. We conclude this section with the following useful result. Lemma 1.3.10 α ∈ D is a multiple root of f ∈ D[x] if and only if it is a common root of f and f ′ . Proof. Since α ∈ D is a root of f , by Lemma 1.3.7, f = (x − α)e g, for some g ∈ D[x] s.t. g(α) 6= 0 and for some positive integer e. One has f ′ = e(x − α)e−1 g + (x − α)e g ′ . If f ′ 6= 0 (e.g. if D is of characteritic 0), then e = 1 if and only if f ′ (α) = g(α) 6= 0, proving the statement in this case. If otherwise f ′ = 0 (which can only occur in positive characteristic since f is non-constant by assumption), then f (x) = (h(x))p , for some non-constant polynomial h ∈ D[x] (cf. [21, Prop.1.12, pag. 179]), which implies that α is a root of multyplicity at least p. ⊓ ⊔ 1.3.2 The case D = K a field Here we focus on the case D = K a field. Proposition 1.3.11 Let K be any field. Then K[x] contains infinite irreducible monic polynomials. Proof. Obviously K[x] contains some irreducible polynomials, e.g. x − α ∈ K[x], for any α ∈ K. If K is an infinite field, we are done. If otherwise K is a finite field, assume by contradiction that f1 , . . . , fn ∈ K[x] are all the monic, irreducible polynomials, for some positive integer n. The polynomial f := 1 + (f1 · · · fn ) ∈ K[x] is monic and it must irreducible, indeed none of the fi ’s divides f , 1 6 i 6 n. This contradicts our assumption. ⊓ ⊔ As a direct consequence of Lemma 1.3.7, by induction on the degree of the polynomial one obtains the following Theorem 1.3.12 Let K be an algebraically closed field. Any non-constant polynomial f ∈ K[x] of degree d contains exactly d roots in K (when roots are counted with multiplicity). In particular, f factorizes in K[x] as

8

1 Basics on commutative rings

f = a(x − α1 ) · · · (x − αd ), where a ∈ K \ {0} and α1 , . . . , αd ∈ K are all the (not necessarily distinct) roots of f . Corollary 1.3.13 If K is an algebraically closed field, the K contains infinitely many elements. Proof. It directly follows from Proposition 1.3.11 and Theorem 1.3.12.

⊓ ⊔

For what concerns polynomial rings K[x1 , . . . , xn ], a fundamental result is the following. Theorem 1.3.14 Let K be any infinite field and let f ∈ K[x1 , . . . , xn ], n > 1, be a non-constant polynomial. If there exists a subset J ⊆ K with infinitely many elements s.t. f (a1 , . . . , an ) = 0 for any (a1 , . . . , an ) ∈ J n , where J n := J × . . . × J the n-tuple product of J, then f = 0 in K[x1 , . . . , xn ]. Proof. The proof is by induction on n. If n = 1, the statement directly follows from Corollary 1.3.9. Assume therefore n > 2 and that the statement holds for all polynomials in n − 1 indeterminates with coefficients from K. Take f ∈ K[x1 , . . . , xn ] which, by the inductive hypothesis, can be considered to be non-constant with respect to all the indeterminates (in particular with respect to xn ). Let d be the degree of f with respect to xn , i.e. when f is considered as an element of (K[x1 , . . . , xn−1 ])[xn ] ∼ = K[x1 , . . . , xn ]. Thus, one can write f = f0 + xn f1 + · · · + xdn fd ,

(1.5)

with fj ∈ K[x1 , . . . , xn−1 ], 0 6 j 6 d, and fd 6= 0 by the assumption on the degree. With this set-up, assume by contradiction there exists an infinte subset J ⊆ K for which f (a1 , . . . , an ) = 0, ∀ (a1 , . . . , an ) ∈ J n . Since fd 6= 0, by induction there exists (a1 , . . . , an−1 ) ∈ J n−1 s.t. fd (a1 , . . . , an−1 ) 6= 0. Thus, (1.5) gives φf (xn ) := f (a1 , . . . , an−1 , xn ) ∈ K[xn ], where φf (xn ) = f0 (a1 , . . . , an−1 )+xn f1 (a1 , . . . , an−1 )+· · ·+xdn fd (a1 , . . . , an−1 ) is a non-constant polynomial in K[xn ] since fd (a1 , . . . , an−1 ) 6= 0. This gives a contradiction; φf (xn ) ∈ K[xn ] and φf (a) = f (a1 , . . . , an−1 , a) = 0 for any a ∈ J which is infinite. ⊓ ⊔ Remark 1.3.15 The previous result implies in particular that, if K is infinite, the only way for a polynomial f ∈ K[x1 , , xn ] to be f (a1 , . . . , an ) = 0, for any (a1 , . . . , an ) ∈ Kn , is f = 0 in K[x1 , , xn ], n > 1. This is false in K is a finite field; indeed, consider f = xp − x ∈ Zp [x], where p ∈ Z a prime; one has f (α) = 0, for any α ∈ Zp (cf. e.g. Example 2.1.10 (iii) for details), even if f is not the zero-polynomial.

1.3 Polynomial rings

9

1.3.3 Resultant of two polynomials in D[x] For the whole section, D will denote a UFD. Here we shall briefly recall some basic results which allow one to enstablish if two polynomials f, g ∈ D[x] have a non-constant, common factor in D[x] (recall Theorem 1.3.1). Let f (x) = an xn + . . . + a0 ,

g(x) = bm xm + . . . + b0 ,

with an bm 6= 0, (1.6)

be non-constant polynomials in D[x]. Lemma 1.3.16 (Euler’s Lemma) In the above assumptions, f and g have a non-constant common factor in D[x] if and only if there exist non-zero polynomials p(x), q(x) ∈ D[x], with deg(p) < m, deg(q) < n, such that pf = qg.

(1.7)

Proof. If φ ∈ D[x] is a non-constant common factor of f and g, then f = φq,

g = φp,

for some p, q ∈ D[x], which verify the statement. Conversely, suppose (1.7) holds, with deg(p) < m and deg(q) < n. In particular, any irreducible factor of g in D[x] divides pf ; since deg(p) < m = deg(g), there exists at least one non-constant, irreducible factor φ of g which divides f , proving the statement. ⊓ ⊔ With notation and assumptions as in (1.6), one defines the Sylvester matrix of the two polynomials f, g ∈ D[x] to be the (m + n) × (m + n) matrix   a0 a1 . . . an 0 0 . . . 0  0 a0 a1 . . . an 0 . . . 0     ...     0 . . . 0 0 a0 a1 . . . an   S(f, g) :=  (1.8)  b0 b1 . . . bm 0 0 . . . 0  ,    0 b0 b1 . . . bm 0 . . . 0     ...  0 . . . 0 0 b0 b1 . . . bm where the block containing a0 , . . . , an consists of m = deg(g) rows whereas the block containing b0 , . . . , bm consists of n = deg(f ) rows. One denotes by R(f, g) := det(S(f, g)) ∈ D,

(1.9)

which is called the resultant of the two polynomials. One has Theorem 1.3.17 Let f and g be as in (1.6). Then f and g have a nonconstant common factor in D[x] if and only if R(f, g) = 0.

10

1 Basics on commutative rings

Proof. By Euler’s Lemma, the existence of polynomials p, q ∈ D[x] satisfying (1.7), with deg(p) < m and deg(q) < n, is equivalent to the existence of n + m elements in D ci , 0 6 i 6 m − 1, not all zero, dj , 0 6 j 6 n − 1, not all zero such that (cm−1 xm−1 + . . . + c0 ) f (x) = (−dn−1 xn−1 − . . . − d0 ) g(x),

(1.10)

holds. This is equivalent to the fact that the ci ’s and the dj ’s give rise to a non-zero solution of the homogeneous linear system of n + m equations a0 C0 = − b0 D0 a1 C0 + a0 C1 = − b1 D0 − b0 D1 ... ... an Cm−1 = − bm Dn−1

(1.11)

in the indeterminates Ci ’s and Dj ’s respectively, 0 6 i 6 m − 1, 0 6 j 6 n − 1, whose coefficient matrix is exactly the transpose of the Sylvester matrix S(f, g). One finally concludes by the well-known fact that the existence of a non-zero solution of (1.11) is equivalent to R(f, g) = 0 (by elementary tranformations, this condition is equivalent to the existence of a non-zero solution of (1.11) in the field K := Q(D); on the other hand, since (1.11) is homogeneous one has a non-zero solution in D). ⊓ ⊔ Corollary 1.3.18 If D = K is an algebraically closed field and f, g as in (1.6), then R(f, g) = 0 ⇔ f and g have a commmon root in K. Proof. Direct consequence of Theorem 1.3.17 and of the fact that, if K is algebraically closed, non-constant irreducible elements in K[x] are all of the form (x − α), α ∈ K. ⊓ ⊔ When f ∈ D[x] is such that n = deg(f ) > 2, one would like to find a condition similar to that given in Theorem 1.3.17 in order to enstablish if f has a multiple factor in D[x], i.e. f = g e h, for some integer e > 2 and some non-constant polynomial g ∈ D[x]. This question is obviously related to the existence of multiple roots of f , possibly in some field K containing D (cf. Def. 1.3.8 and Lemma 1.3.10). On the other hand, derivatives in positive characteristic not always behave well (recall e.g. the proof of Lemma 1.3.10); indeed when D is a UFD of positive characteritic, there actually exist non-contant polynomials f ∈ D[x] for which, either f ′ = 0 or f ′ constant even if deg(f ) > 1 (in both cases one cannot define R(f, f ′ )). For the scope of this book, we can limit ourselves to the characteristic-zero case and refer the interested reader to e.g. [21].

1.3 Polynomial rings

11

Thus, when D is a UFD of characteritic 0 and f ∈ D[x] is a polynomial of degree n > 2, then f ′ ∈ D[x] is a non-constant polynomial of degree n − 1. As above, one can define the (2n − 1) × (2n − 1)-Sylvester matrix S(f, f ′ ) and consequently R(f, f ′ ) ∈ D, which is simply denoted by ∆(f ) and called the discriminant of f . Proposition 1.3.19 Let D be a UFD of characteritic 0 and f ∈ D[x] a polynomial of degree n > 2. Then f has a multiple factor in D[x] if and only if ∆(f ) = 0. Proof. (⇒) By assumption there exists a non-constant g ∈ D[x] such that f = g 2 k, for some k ∈ D[x]. Then f ′ = 2gg ′ k + g 2 k ′ which shows that g|f and g|f ′ . Since deg(f ) > 2 and since the characteristic is zero, f ′ ∈ D[x] is a non-constant polynomial, so that ∆(f ) makes sense and one concludes by Theorem 1.3.17. (⇐) Assume ∆(f ) = 0. From Theorem 1.3.17, it follows that f and f ′ have a non-constant common factor g ∈ D[x] which we can assume to be irreducible. From f = gh, for some h ∈ D[x], we deduce f ′ = g ′ h+gh′ , where g ′ 6= 0. Since g|f ′ by assumption, then g|g ′ h. On the other hand, since deg(g) > deg(g ′ ) and since g is irreducible, we must have g|h, i.e. g 2 |f . ⊓ ⊔ As for Corollary 1.3.18, one has Corollary 1.3.20 If D = K is an algebraically closed field of characteristic 0 and f ∈ K[x] is a polynomial of degree n > 2, then ∆(f ) = 0 ⇔ f has a multiple root. 1.3.4 Resultant of two polynomials in D[x1 , . . . , xn ] and elimination In the previous section we introduced the notion of resultant for two polynomials in D0 [x], where D0 any UFD. In particular, if D is a UFD then D[x1 , . . . , xn ] = (D[x1 , . . . , xn−1 ])[xn ], with D0 := D[x1 , . . . , xn−1 ] a UFD (cf. Theorem 1.3.1 and Remark 1.3.2). Thus, we can apply the machinery developed in § 1.3.3 to any pair of nonconstant polynomials f, g ∈ D0 [xn ] to get Rxn (f, g) := R(f, g) ∈ D0 = D[x1 , . . . , xn−1 ], which is called the resultant polynomial of f and g with respect to xn . Notice that Rxn (f, g) is a polynomial where the indeterminate xn does not appear; this gives rise to the terminology elimination theory.

12

1 Basics on commutative rings

Proposition 1.3.1. Let D be a UFD and let f, g ∈ D[x1 , . . . , xn ] be nonconstant polynomials of the form f = ad xdn + . . . + a0 ,

g(x) = bm xm n + . . . + b0 ,

(1.12)

with ad , bm ∈ D[x1 , . . . , xn−1 ] \ {0}. Then the resultant polynomial R := Rxn (f, g) ∈ D[x1 , . . . , xn−1 ] belongs to the ideal generated by f and g in D[x1 , . . . , xn ]. More precisely there exist polynomials A, B ∈ D[x1 , . . . , xn ], with degrees w.r.t. the indeterminate xn at most m − 1 and d − 1, respectively, such that Af + Bg = R

(1.13)

Proof. One constructs the Sylvester matrix S(f, g) as in (1.8). Then one finds the following relations xm−1 f (x)

= ad xnd+m−1 + . . . + a0 xnm−1

xm−2 f (x) ...

=

f (x)

=

xd−1 g(x)

= bm xnd+m−1 + . . . + b0 xd−1 n

xd−2 g(x) ... g(x)

= =

ad xnd+m−2 + . . . + a0 xnm−2 ... ad xdn + . . . + a0 bm xnd+m−2 + . . . + b0 xd−2 n ... bm xm n + . . . + b0

If we first multiply each relation by the co-factor of the corresponding element in the last column of S(f, g) and then add-up all the relations obtained above, we get (1.13). ⊓ ⊔

1.4 Noetherian rings and the Hilbert basis theorem A ring R is said to be a Noetherian ring if any ideal I of R is finitely generated. Z and any field K are easy examples of Noetherian rings. Since K[x] is a Euclidean ring, it is a PID and so in particular it is a Noetherian ring. There actually exist rings which are not Noetherian: e.g. the ring K[x1 , x2 , x3 , . . .] of polynomials with coefficients from a field K and with infinite indeterminates is an integral domain which cannot be Noetherian as the maximal ideal (x1 , x2 , x3 , . . .) cannot be finitely generated. Notice that K[x1 , . . . , xn ], with n > 2, is not a PID: e.g. the maximal ideal (x1 , . . . , xn ) cannot be principal. On the other hand, when n > 2, K[x1 , . . . , xn ] is actually Noetherian. This is a consequence of the following more general result:

1.4 Noetherian rings and the Hilbert basis theorem

13

Theorem 1.4.1 (Hilbert’s basis theorem) Let R be a Noetherian ring and let x be an indeterminate. Then the ring R[x] is Noetherian. Remark 1.4.2 Let R be any ring and let x1 , . . . , xn be indeterminates over R. One clearly has
R[x1 , . . . , xn ] = R[x1 , . . . , xn−1 ] [xn ].

Thus, as a consequence of Theorem 1.4.1, if R is Noetherian then also R[x1 , . . . , xn ] is, for any positive integer n. In particular, this holds for R = K any field.

Proof (proof of Theorem 1.4.1). Let f (x) ∈ R[x] be any polynomial. If f (x) is not the zero-polynomial, then f (x) := a0 + a1 x + a2 x2 + . . . + ai xi ∈ R[x], where aj ∈ R, 0 6 j 6 i, with ai 6= 0 i.e the integer i > 0 is the degree of f (x) and ai := lc(f (x)) ∈ R (recall Def. 1.3.5). If otherwise f (x) is the zero-polynomial, its leading coefficient is 0 ∈ R by definition. Let I ⊂ R[x] be any non-trivial proper ideal (if I is either (0) or (1) there is nothing to prove). For any integer i > 0, denote by Ji the subset of R consisting of all leading coefficients of polynomials in I having degrees at most i. Similarly, denote by J the subset of R consisting of all leading coefficients of polynomials in I. It is immediate to observe that J0 ⊆ J1 ⊆ J2 ⊆ · · · ⊆ J and that 0 ∈ J0 . By straightforward computations, one also verifies that J and Ji , for any i > 0, are ideals of R. Being R Noetherian, all these ideals are finitely generated. Thus, for any i > 0, there exists a finite set {rih }h∈Hi of elements of R, with Hi a finite set of indices, such that Ji = (rih )h∈Hi . By the very definition of Ji , one can determine as many polynomials {fih (x)}h∈Hi s.t. fih (x) ∈ I, deg(fih (x)) 6 i and lc(fih (x)) = rih , ∀ h ∈ Hi . Similarly, there exists a finite set {rm }m∈M of elements of R, with M a finite set of indices, such that J = (rm )m∈M , and as many polynomials {fm (x)}m∈M s.t. fm (x) ∈ I and lc(fm (x)) = rm , ∀ m ∈ M. Let N := Maxm∈M {deg(fm (x))},

and consider the ideal I ′ generated by the polynomials {fm (x)}m∈M and {fih (x)}h∈Hi for i 6 N , i.e.

14

1 Basics on commutative rings

  I ′ := {fm (x)}m∈M , {fih (x)}h∈Hi ,i6N .

One has I ′ ⊆ I and I ′ is finitely generated. To conclude the proof, it suffices to show that I ′ = I. Assume by contradiction that the inclusion I ′ ⊂ I is strict. Thus, there exists an element g(x) ∈ I \ I ′ of minimal degree with respect to this property. If deg(g(x)) := d > N , then g(x) = b0 + b1 x + · · · +P bd xd , with 0 6= bd ∈ J, since g(x) ∈ I. By the assumption on J, we have bd = m∈M qm rm , for some qm ∈ R. Consider the polynomial X qm fm (x) xd−deg(fm (x)) ∈ I ′ . q(x) := m∈M

One has deg(q(x)) = deg(g(x)) and lc(q(x)) = lc(g(x)). Since g(x)−q(x) ∈ I and deg(g(x)−q(x)) < d = deg(g(x)), by the minimality condition on d = deg(g(x)), one must have g(x) − q(x) ∈ I ′ . This would imply g(x) = (g(x) − q(x)) + q(x) ∈ I ′ , a contradiction. If otherwise deg(g(x)) := i 6 N , g(x) = b0P + b1 x + · · · + bi xi , with 0 6= bi ∈ Ji . As above, by the assumption on Ji , bi = h∈Hi qih rih , for some qih ∈ R. Consider the polynomial X qih fih (x) xi−deg(fih (x)) ∈ I ′ , qi (x) := h∈Hi

so that deg(qi (x)) = deg(g(x)) and lc(qi (x)) = lc(g(x)). Since g(x) − qi (x) ∈ I and deg(g(x) − qi (x)) < i = deg(g(x)), one concludes as in the previous case. ⊓ ⊔ Remark 1.4.3 (1) If S is a subring of a Noetherian ring R, in general one cannot conclude that S is Noetherian too: consider e.g. the ring S := K[x1 , x2 , x3 , . . .] of polynomials in infinite indeterminates with coefficients from a field K; from Remark 1.4.2 S is not Noetherian on the other hand, since it is an integral domain, it admits a quotient field Q(S) := R so S is a subring of a Noetherian ring R. (2) On the contrary, any quotient R/I of a Noetherian ring R is Noetherian too, as it easily follows from the bijective correspondence between ideals of R/I and ideals of R containing I (cf. also [1, Prop. 7.1]). Proposition 1.4.4 A ring R is Noetherian if and only if every ascending chain I1 ⊆ I2 ⊆ I3 ⊆ . . . ⊆ In ⊆ In+1 ⊆ . . . ⊂ R (1.14)

of proper ideals is stationary, i.e. there exists an integer n0 among the indices s. t. In0 = In0 +h , for any integer h > 1.

1.5 R-modules, R-algebras and finiteness conditions

15

Proof. (⇒) Suppose that R is Noetherian and that (1.14) is an Pascending chain of proper ideals. Recalling what discussed in § 1.1, I := n In is a proper ideal of R; indeed if 1 ∈ I, there should exist an index n1 such that 1 ∈ In1 , which contradicts the assumptions on the chain (1.14). Since R is Noetherian, there exist r1 , . . . , rm ∈ I s.t. I = (r1 , . . . , rm ). Let ri ∈ In(i) and let n0 := Max16i6m {n(i)}. Then, by the ascending condition, In(i) ⊆ In0 so, in particular, ri ∈ In0 , for any 1 6 i 6 m. Thus I ⊆ In0 , i.e. In0 = In0 +h , for any integer h > 1. (⇐) Suppose that the second part of the statement holds true and let I be any proper ideal of R. If I were not finitely generated, one could find an infinite sequence a1 , a2 , a3 , . . . of distinct generators of I giving rise to an ascending sequence of proper ideals (a1 ) ⊂ (a1 , a2 ) ⊂ (a1 , a2 , a3 ) ⊂ . . . which could not be stationary, a contradiction.

⊓ ⊔

1.5 R-modules, R-algebras and finiteness conditions Let R be a ring. An abelian group (M, +) is called a R-module if there exists a R-multiplication map R × M → M, (a, m) → am such that for any a, b ∈ R and any m, n ∈ M the following hold: (a + b)m = am + bm, a(m + n) = am + an, (ab)m = a(bm), 1m = m. Easy examples of R-modules are e.g. M = R, M = I as well as M = R/I for any ideal I ⊂ R, M = Rn := R ⊕ . . . ⊕ R where the R-multiplication is considered componentwise. If R = K is a field, a K-module is nothing but a K-vector space. If N ⊂ M is a subgroup s.t. for any n ∈ N and any r ∈ R one has rn ∈ N , then N is said to be a R-submodule of M . If T ⊂ M is a subset of a R-module M , we denote by o nX ai ti | ti ∈ T, ai ∈ R hT i := P the subset of elements in M of the form ai ti , where in the sums only finitely many ai ’s are non-zero. The set hT i is a submodule of M which is called the R–module generated by T and T is said to be a set of generators of the Rmodule hT i. A R-module M is said to be finitely generated, if M = hT i for some finite subset T ⊂ M .

16

1 Basics on commutative rings

Let M and N be two R-modules. A map ϕ : M → N is a homomorphism of R-modules if it is a group homomorphism which is also R-linear, namely ϕ(x + y) = ϕ(x) + ϕ(y), ϕ(r x) = r ϕ(x), ∀ x, y ∈ M, f orall r ∈ R. A R-algebra is a ring S together with a ring-homomorphism ϕ : R → S (recall that by assumption ϕ(1R ) = 1S , so Ker(ϕ) ( R is always a proper ideal). S ′ ⊂ S is a R-subalgebra of S, if S ′ is a subring of S which is a Ralgebra. Remark 1.5.1 (i) Notice that a R-algebra is also a R-module by r s := ϕ(r) s, for any r ∈ R and any s ∈ S, where the two structure are compatible. (ii) Easy examples of R-algebras are e.g. the ring of polynomials R[x1 , . . . , xn ] for any n > 1, the quotient ring R/I for any ideal I ( R with ϕ = πI the canonical projection. (iii) If R = K is a field and S 6= 0, then ϕ necessarily is injective, i.e. K can be identified with its image in S. Thus a K-algebra is nothing but a ring containing the field K as a subring. If in particular also S = F is a field, K ֒→ F is said to be field extension. The degree of the field extension, denoted by [F : K], is dimK (F) as a K-vector space. (iv) Since R is commutative and with identity, there exits a unique ring homomorphism Z −→ R, n → n 1R ; in other words, any (commutative and with identity) ring is automatically a Z-algebra. (v) Let ϕ : R → S and ψ : R → T be ring homomorphism. A homomorphism of R-algebras, say η : S → T , is a ring homomorphism which is also a homomorphism of R-modules. This occurs if and only if η ◦ ϕ = ψ.

Definition 1.5.2 In the above notation, S is said to be a finite R-algebra if S is finitely generated as a R-module. When R = K is a field, easy examples of finite K-algebras are e.g. K[x]/(xn ) as well as any (algebraic) field extension of finite degree. ϕ Let S be a R-algebra, with R −→ S as above. Consider s1 , . . . , sn ∈ S and x1 , . . . , xn indeterminates over R. One can define a ring-homomorphism Φ := Φs1 ,...,sn : R[x1 , . . . , xn ] → S

(1.15)

by the rules Φ(xi ) = si , Φ(a) = ϕ(a), ∀ a ∈ R, 1 6 i 6 n. Then Im(Φ) ⊆ S is the smallest subalgebra of S containing all polynomial expressions in the elements s1 , . . . , sn with coefficient from the subring Im(ϕ) ⊆ S, which is simply denoted by R[s1 , . . . , sn ].

1.5 R-modules, R-algebras and finiteness conditions

17

Definition 1.5.3 A R-algebra S is said to be a R-algebra of finite type if S = R[s1 , . . . sn ], for some finitely many elements s1 , . . . , sn ∈ S. In particular, a ring R is said to be finitely generated if it is a Z-algebra of finite type.

Example 1.5.4 If S is a finite R-algebra, then it is also a R-algebra of finite type. The converse does not hold in general; indeed, R[x1 , . . . , xn ], where n > 1 any integer and xi indeterminates over R for 1 6 i 6 n, is a R-algebra of finite type which is not finite. Let K and F be fields such that F is a K-algebra. Let s1 , . . . , sn ∈ F \ K. One denotes by K(s1 , . . . , sn ) the quotient field Q (K[s1 , . . . , sn ]), which is the smallest subfield of F containing K and s1 , . . . , sn . Definition 1.5.5 A field extension K ⊂ F is called a finitely generated field extension if F = K(s1 , . . . , sn ), for some finitely many elements s1 , . . . , sn ∈ F. When n = 1, the field estension is called simple. Remark 1.5.6 Notice that if K and F are fields such that F is a K-algebra of finite type, then K ⊆ F is a finitely generated field extension. In particular, for a field F we have (∗) F is a finite K − algebra ⇓

(∗∗) F is a K − algebra of finite type ⇓

(∗ ∗ ∗) K ⊆ F is a finiteley generated field extension

The opposite implications do not hold in general: for (∗∗) does not imply (∗), recall Example 1.5.4 whereas, for (∗ ∗ ∗) does not imply (∗∗) we have the next result. Proposition 1.5.7 Let K be any field. Let x1 , . . . , xn be indeterminates over K, for any integer n > 1, and let K(x1 , . . . , xn ) := Q (K[x1 , . . . , xn ]) be the field of rational functions with coefficients from K, which is a finitely generated field extension of K. Then K(x1 , . . . , xn ) is not a K-algebra of finite type. Proof. For simplicity, we consider the case n = 1, the general one being similar. Therefore, in what follows we will simply put x1 = x. Assume by contradiction there exist finitely many s1 , . . . , st ∈ K(x) such that K(x) = K[s1 , . . . , st ], and let s ∈ K[x] be the product of all denominators of s1 , . . . , st ∈ K(x). In such a case, for any z ∈ K(x) there would exist a positive integer m (depending on z) such that sm z ∈ K[x]. This is a contradiction; indeed, it suffices to take any irreducible non-constant polynomial c ∈ K[x] ⊓ ⊔ which does not divide s in K[x], and consider z := 1c ∈ K(x).

18

1 Basics on commutative rings

1.6 Integrality Let R and S be rings such that R ⊆ S. Definition 1.6.1 An element v ∈ S is said to be integral over the subring R if there exists a monic polynomial f (t) ∈ R[t], t an indeterminate over R, s.t. f (v) = 0. Remark 1.6.2 (i) If v ∈ R, then v is integral over R. (ii) If R ⊆ S is a field extension, v ∈ S is integral over R if and only if it is algebraic over R. (iii) v ∈ Q is integral over Z if and only if v ∈ Z. Indeed, if v = rs ∈ Q, with r and s coprime, then an expression of the form v n + an−1 v n−1 + · · · + a0 = 0, ai ∈ Z, 0 6 i 6 n − 1, gives rn + an−1 rn−1 s + · · · + a0 sn = 0,

i.e. s divides rn . This implies that s divides r and so s = ±1. (iv) Let R be any UFD and let Q(R) be its quotient field. Then v ∈ Q(R) is integral over R if and only if v ∈ R. The proof is similar to that of (iii) above. Notice that the previous statement, in particular, applies to the polynomial ring R = K[t], where K any field and t an indeterminate over K. Proposition 1.6.3 Let R and S be rings s.t. R ⊆ S. The following conditions are equivalent: (i) v ∈ S is integral over R; (ii) R[v] is a finite R-algebra, i.e. it is a finitely generated R-module; (iii) there exists a subring B of S, containing R[v], which is a finitely generated R-module. Proof. (i) ⇒ (ii): from (i), there exists a positive integer n and elements ai ∈ R, 0 6 i 6 n − 1, such that for any non-negative integer r one has v n+r = −(an−1 v n+r−1 + · · · + a0 v r ). Applying recursively one gets that, for any integer m > 0, v m belongs to the R-submodule of S generated by the elements 1, v, v 2 , . . . , v n−1 . Thus, R[v] is a finitely generated R-module. (ii) ⇒ (iii): it suffices to consider B = R[v]. (iii) ⇒ (i): let R[v] ⊆ B ⊆ S be ring inclusions, such that B is a finitely generated R-module. Let v1 , . . . , vn ∈ B be generators of B as a R-module.

1.6 Integrality

19

Since v, vi ∈ B then vvi ∈ B, for any 1 6 i 6 n, since B is a ring. By the assumptions on B, there exist elements aij ∈ R, 1 6 i, j 6 n, such that vvi =

n X j=1

aij vi , ∀ 1 6 i 6 n.

Denoting by δij the Kronecher symbol, the previous equality gives det(δij v − aij ) vi = 0, ∀ 1 6 i 6 n. Since v1 , . . . , vn generates B, this implies det(δij v − aij ) = 0 so v is a root of the characteristic polynomial of the matrix (aij ), 1 6 i, j 6 n, which is a monic polynomial with coefficients from R. ⊓ ⊔ Corollary 1.6.4 Let R and S be rings s.t. R ⊆ S. The set C := {v ∈ S | v integral over R} is a subring S such that R ⊆ C ⊂ S. Proof. It is clear that as sets R ⊆ C. Take now v, w ∈ C any two elements. Since R ⊂ R[v] and since w is integral over R, then w is also integral over R[v]. From Proposition 1.6.3, R[v, w] is a finitely generated R[v]-module. By transitivity, R[v, w] is a finitely generated R-module. Therefore, from Proposition 1.6.3, R[v, w] ⊆ C; in particular, vw, v ± w ∈ C, which proves that C is a subring of S containing R as a subring. ⊓ ⊔ The subring C in Corollary 1.6.4 is called the integral closure of R in S. Definition 1.6.5 An integral domain R is said to be integrally closed if it coincides with its integral closure in Q(R), i.e. v ∈ Q(R) and v integral over R implies v ∈ R. From Remark 1.6.2 (iii) and (iv), Z and any UFD are integrally closed. Proposition 1.6.6 Let R be any integral domain and F := Q(R) be its quotient field. Let K be any field such that K ⊇ F. If α ∈ K is algebraic over F, then there exists d ∈ R such that dα ∈ K is integral over R. Proof. By assumptions, there exists a relation of the form αn + an−1 αn−1 + · · · +0 = 0, ai ∈ F, 0 6 i 6 n − 1. Let d ∈ R be the product of all denominators of the ai ’s. Then dai ∈ R, for any 0 6 i 6 n − 1. If we multiply the previous relation by dn , we get (dα)n + dan−1 (dα)n−1 + · · · + dn a0 = 0, i.e. dα is a root of a monic polynomial in R[x].

⊓ ⊔

20

1 Basics on commutative rings

Corollary 1.6.7 Let R be an integral domain and F := Q(R) be its quotient field. For any algebraic field extension F ⊆ K, consider C the integral closure of R in K. Then K = Q(C). Proof. The proof of the previous proposition shows that any element α ∈ K ⊓ ⊔ can be written as α = βd , where β integral over R and d ∈ R. In particular, one has: Proposition 1.6.8 Let R be any integrally closed domain and let F := Q(R) be its quotient field. Let K be a field, which is a finite extension of F. Then α ∈ K is integral over F if and only if the minimal polynomial fα (T ) ∈ F[T ] of α over F is with coefficients from R. Proof. If α ∈ F is integral over R, there exist ai ∈ R, 1 6 i 6 m, s.t. (∗) αm a1 αm−1 + . . . + am = 0. Let α′ ∈ K be any other conjiugate to α, i.e. any other root of the minimal polynomial fα (T ); then there exists a F-isomorphism σ : F[α] → F[α′ ], α → α′ . Applying σ to the above relation (∗) we get (∗∗) (α′ )m a1 (α′ )m−1 + . . . + am = 0, i.e. any conjiugate to α is integral over R. Since deg(fα )

fα (T ) =

Y

i=1

(T − αi ),

i.e. the coefficients of fα (T ) are given by elementary symmetric functions on the αi ’s, and since from Corollary 1.6.4 integral elements over R determine a subring of F, we get that coefficients of fα (T ) ∈ F[T ] are integral elements over R. On the other hand, since R is integrally closed, we conclude fα (T ) ∈ R[T ]. The converse is immediate since, by definition of minimal polynomial, fα (T ) is monic. ⊓ ⊔

1.7 Zariski’s Lemma In § 1.5 we reminded finiteness conditions for K-algebras, where K a field.

1.7 Zariski’s Lemma

21

Remark 1.7.1 Recall that for a field F which is a simple extension of K, i.e. F = K(v) for some v ∈ F \ K, two cases occur: ∼ K(x), with x an indeterminate, (i) if v ∈ F is trascendental over K, then F = and so Proposition 1.5.7 implies that K ⊂ F is a fininitely generated field extension which is neither a K-algebra of finite type nor a finite K-algebra; (ii) if v ∈ F is algebraic over K, then F = K[v] is a finite K-algebra (so in particular also a K-algebra of finite type) and the field extension K ⊂ F is finite (a fortiori finitely generated), i.e. [F : K] = deg(fv (x)) < +∞ where fv (x) ∈ K[x] is the minimal polynomial of v ∈ F. In other words, when v is algebraic the three conditions (∗), (∗∗) and (∗ ∗ ∗) in Remark 1.5.6 coincide. The next fundamental result shows that the same occurs for any finitely generated field extension. Lemma 1.7.2 (Zariski’s Lemma) Let K ⊂ F be a field extension. Then F is a K − algebra of finite type ⇔ [F : K] < +∞. Remark 1.7.3 (i) Notice that, in the assumptions of Zariski’s Lemma, K ⊂ F is in any case a finitely generated field extension (recall Remark 1.5.6). (ii) There are field extensions K ⊂ F which do not satisfy any of the finiteness conditions introduced in § 1.5. Take e.g. the field extension Q ⊂ Q (resp. Zp ⊂ Zp , where p ∈ Z a prime) given by algebraic closure. This cannot be a finitely generated field extension otherwise, being an algebraic extension, it would be of finite degree over Q (cf. [21, Prop. 1.6] for details), which is a contradiction since Q has infinite dimension as a Q-vector space. In particular, it is neither a finite Q-algebra nor a Q-algebra of finite type, as it easily follows from Zariski’s Lemma. The same discussion holds verbatim for Zp ⊂ Zp . Proof (Proof of Zariski’s Lemma). First of all, we can assume F with infinite elements, otherwise there is nothing else to prove. (⇐) This implication is obvious (cf. Example 1.5.4) (⇒) By assumption we have F = K[v1 , . . . , vn ], for some n. The proof proceeds by induction on n. The case n = 1 has already been discussed in Remark 1.7.1; therefore assume n > 2 and use inductive hypothesis. We have K ⊂ K1 := K(v1 ) ⊂ F. Since F = K[v1 , . . . , vn ], one also has F = (K(v1 )) [v2 , . . . , vn ] = K1 [v2 , . . . , vn ], i.e. F is a K1 -algebra of finite type. By induction, the extension K1 ⊂ F is algebraic of finite degree. By transitivity of degree extension, one has

22

1 Basics on commutative rings

[F : K] = [F : K1 ] [K1 : K] so, to deduce the finiteness of [F : K] it suffices to show that v1 is algebraic over K. Assume by contradiction that v1 is trascendental over K. Since F = K1 [v2 , . . . , vn ], with v2 , . . . , vn algebraic over K1 , from Proposition 1.6.6 there exists d ∈ K[v1 ] s.t. dvi is integral over K[v1 ], for any 2 6 i 6 n. Let f ∈ K1 be any element. Claim 1.7.4 For any integer N >> 0, dN f ∈ K[v1 , dv2 , . . . , dvn ]. Proof (Proof of Claim 1.7.4). Since f ∈ K1 ⊂ F, then X αj1 ...jn v1j1 v2j2 · · · vnjn . f= j1 ...jn

Thus, for any N sufficiently large, any monomial of dN f is of the form dN αj1 ...jn v1j1 v2j2 · · · vnjn = dN −(j2 +...+jn ) αj1 ...jn v1j1 (dv2 )j2 · · · (dvn )jn , where dN −(j2 +...+jn ) αj1 ...jn v1j1 ∈ K[v1 ].

⊓ ⊔

From Corollary 1.6.4, dN f is therefore integral over K[v1 ]. Since by assumption v1 is trascendental over K, then K[v1 ] ∼ = K[T ] is a PID (in particular, a UFD). From Remark 1.6.2 (iv), K[v1 ] is therefore integrally closed i.e. dN f ∈ K[v1 ]. If we take c ∈ K[v1 ] irreducible, not dividing d, and if we consider f := c−1 , from the previous discussion, for any N >> 0 we would get dN f ∈ K[v1 ], a contradiction. Thus v1 is algebraic and the theorem is proved. ⊓ ⊔

1.8 Trascendence degree Let K ⊂ F be a field extension and let s1 , . . . , sn ∈ F. Similarly to (1.15), consider the K-algebra homomorphism Φ : K[x1 , . . . , xn ] → F, Φ(xj ) = sj , 1 6 j 6 n,

(1.16)

where x1 , . . . , xn are indeterminates over K. Definition 1.8.1 The elements s1 , . . . , sn ∈ F are said to be algebraically independent over K if Ker(Φ) = (0). Otherwise, they are said to be algebraically dependent over K and any non-zero f ∈ Ker(Φ) is said to be a relation of algebraic dependence over K among the si ’s. If s1 , . . . , sn are algebraically independent over K, the homomorphism Φ above extends to an isomorphism Q(n) = K(x1 , . . . xn ) ∼ = K(s1 , . . . , sn ).

1.8 Trascendence degree

23

Definition 1.8.2 Let s1 , . . . , sn ∈ F be elements which are algebraically independent over K. The set {s1 , . . . , sn } is said to be a (finite) trascendence basis of F over K if F is an algebraic extension of K(s1 , . . . , sn ). Lemma 1.8.3 Let {s1 , . . . , sn } be a trascendence basis of F over K and let t1 , . . . , th ∈ F be algebraically independent over K. Then h 6 n Proof. Since F is an algebraic extension of K(s1 , . . . , sn ), there exists a nonzero polynomial f ∈ K[y1 , x1 , . . . , xn ] = A(n+1) s.t. f (t1 , s1 , . . . , sn ) = 0. Since s1 , . . . , sn are algebraically independent, the polynomial f is nonconstant with respect to the indeterminate y1 . Since moreover t1 ∈ F is trascendent over K, the polynomial f has to be non-constant with respect to at least one of the indeterminates xi , 1 6 i 6 n; up to a re-labeling of the indeterminates, we can assume this occurs for x1 . In such a case s1 is algebraic over K(t1 , s2 , . . . , sn ). By transitivity of algebraic extensions, F is an algebraic extension of K(t1 , s2 , . . . , sn ). Moreover t1 , s2 , . . . , sn are algebraically independent over K (otherwise s1 would be algebraically dependent to s2 , . . . , sn ). Thus {t1 , s2 , . . . , sn } is a trascendence basis of F over K. Consider the following statement, for some 1 6 i 6 h − 1: (Si ) Up to possibly permute the elements s2 , . . . , sn , the set {t1 , t2 , . . . , ti , si+1 , . . . sn } is a trascendence basis of F over K. (S1 ) has already been proved above. We proceed by induction on i; assume therefore we have proved (Si ), for some 1 6 i 6 h − 1, we want to prove (Si+1 ); by the inductive hypothesis, there exists a non-zero polynomial g ∈ K[yi+1 , y1 , . . . , yi , xi+1 , . . . , xn ] such that g(ti+1 , t1 , . . . , ti , si+1 , . . . , sn ) = 0. The polynomial g is non-constant with respect to the indeterminate yi+1 (since t1 , . . . , ti , si+1 , . . . , sn are algebraically independent) as well as with respect to at least one indeterminates xj , e.g. xi+1 (since t1 , . . . , ti , ti+1 are algebraically independent). Thus, si+1 is algebraic over K(t1 , . . . , ti+1 , si+2 , . . . , sn ). Moreover t1 , . . . , ti+1 , si+2 , . . . , sn are algebraically independent: indeed if one had h(t1 , . . . , ti+1 , si+2 , . . . , sn ) = 0 for some non-zero polynomial h ∈ K[y1 , . . . , yi+1 , xi+2 , . . . , xn ], by the inductive hypothesis the polynomial h could not be constant with respect to the indeterminate yi+1 , so ti+1 would be algebraic over K(t1 , . . . , ti , si+2 , . . . , sn ); in such a case si+1 would be algebraic over K(t1 , . . . , ti , si+2 , . . . , sn ], contradicting the inductive hypothesis. It follows that {t1 , . . . , ti+1 , si+2 , . . . , sn } is a trascendence basis of F over K, i.e. (Si+1 ) has been proved. By induction, (Sh ) holds true; in particular h 6 n. ⊓ ⊔

24

1 Basics on commutative rings

If F admits a finite trascendence basis over K, we will say that F has finite trascendence degree over K. In such a case, from Lemma 1.8.3, any two trascendence bases have the same cardinality. This non-negative integer is called trascendence degree of F over K and is denoted by trdegK (F).

(1.17)

Remark 1.8.4 (i) If K ⊆ F is an algebraic extension, one has trdegK (F) = 0. (ii) If F does not admit a finite trascendence basis over K (e.g. the quotient field of the polynomial ring K[x1 , x2 , . . .] with infinite indeterminates), one poses trdegK (F) = +∞. (iii) If x1 , . . . , xn are indeterminates over K, then for Q(n) := K(x1 , . . . , xn ) one has trdegK (Q(n) ) = n and {x1 , . . . , xn } is a trascendence basis of Q(n) over K. (iv) If F = K(s1 , . . . , sn ) and s1 , . . . , sn ∈ F are algebraically independent, then trdegK (F) = n and F ∼ = Q(n) . In such a case, F is said to be a purely trascendental extension of K. (v) For a finitely generated field extension K ⊂ F = K(s1 , . . . , sn ) with trdegK (F) = n, the field F is called a field of algebraic functions in the si ’s. Proposition 1.8.5 Let K ⊂ L ⊂ F be field extensions and assume trdegK (L) < +∞, trdegL (F) < +∞. Then trdegK (F) < +∞ and trdegK (F) = trdegK (L) + trdegL (F). More precisely, if s1 , . . . sn ∈ L is a trascendence basis of L over K and t1 , . . . tm ∈ F is a trascendence basis of F over L, then s1 , . . . sn , t1 , . . . tm is a trascendence basis of F over K. Proof. Assume by contradiction there exists a non-zero polynomial g ∈ K[x1 , . . . , xn , y1 , . . . , ym ], where the xi ’s and the yj ’s are indeterminates over K. Consider the polynomial hs (y1 , . . . , yn ) := g(s1 , . . . sn , y1 , . . . , ym ) ∈ L[y1 , . . . , ym ]. From the fact that s1 , . . . sn are algebraically independent over K, this polynomial is non-zero. In this case, hs (y1 , . . . , yn ) would give a non-zero algebraic relation over L among the elements t1 , . . . tm , contradicting the assumption. Thus s1 , . . . sn , t1 , . . . tm are algebraically independent over K. We are left to show that K(s1 , . . . sn , t1 , . . . tm ) ⊆ F

1.9 Tensor products of R-modules and of R-algebras

25

is an algebraic extension. Notice that L(t1 , . . . tm ) ⊆ F is an algebraic extension, since {t1 , . . . , tm } is a trascendence basis of F over L. The same occurs to the extension K(s1 , . . . , sn , t1 , . . . tm ) ⊆ L(t1 , . . . tm ), since {s1 , . . . , sn } is a trascendence basis of L over K. Therefore the composition K(s1 , . . . , sn , t1 , . . . tm ) ⊆ L(t1 , . . . tm ) ⊆ F ⊓ ⊔

is an algebraic extension and we are done.

Let K be a field and let R be an integral K-algebra with quotient field F := Q(R); by small abuse of terminology, one defines the trascendence degree of R over K, denoted by trdegK (R), to be the trascendence degree of F over K, i.e. trdegK (R) := trdegK (Q(R)). (1.18) In the sequel we will be concerned only on integral K-algebras having finite trascendence degree. Remark 1.8.6 (i) If x1 , . . . , xn are indeterminates over K, then trdegK (K[x1 , . . . , xn ]) = n. (ii) If Φ : R ։ R′ is a surjective homomorphism of integral K-algebras, then trdegK (R) > trdegK (R′ ).

(1.19)

Indeed, since R ⊂ Q(R), we can consider elements F1 , . . . , Fn ∈ R which are algebraically dependent over K. Let g ∈ K[x1 , . . . , xn ] be any non-zero polynomial s.t. g(F1 , . . . , Fn ) = 0. Posing fi := Φ(Fi ) ∈ R′ , for any 1 6 i 6 n, one has g(f1 , . . . , fn ) = Φ(g(F1 , . . . , Fn )) = Φ(0) = 0 the first equality following from the fact that Φ is a K-algebra homomorphism. (iii) If F = K(s1 , . . . , sn ) is a finitely generated field extension of K, then trdegK (F) 6 n as it follows from (i), (ii) and the fact that one has a surjective K-algebra homomorphism Φ : K[x1 , . . . , xn ] ։ K[s1 , . . . , sn ].

1.9 Tensor products of R-modules and of R-algebras Let M , N and P be R-modules. A map f : M × N −→ P

26

1 Basics on commutative rings

is said to be R-bilinear if, for any x ∈ M , the map f (x, −) : N −→ P, y −→ f (x, y) is R-linear and, for any y ∈ N , the map f (−, y) : M −→ P, x −→ f (x, y) is R-linear. In what follows we shall construct a R-module T , called the tensor product of the modules M and N , in such a way that, for any R-module P , R-bilinear maps M × N → P turn out to be in bijective correspondence with R-linear maps T → P . Indeed, one has Proposition 1.9.1 Let M and N be R-modules. There exists a pair (T, g), where T a R-module and g : M × N → T a bilinear map, satisfying the following property: (*) given any R-module P and any R-bilinear map f : M × N → P , there exists a unique linear map f ′ : T → P such that f = f ′ ◦ g. Moreover, if (T ′ , g ′ ) is another pair satisfying property (*), then there exists a unique isomorphism of R-modules j : T −→ T ′ such that j ◦ g = g ′ . Proof. Let C be the free R-module whose elements are given by (finite) linearP combinations of elements of M × N with coefficients from R, i.e. of the n form i=1 ri (xi , yi ), where n ∈ N, xi ∈ M , yi ∈ N . Let D be the R-submodule of C generated by elements of the form (x + x′ , y) − (x, y) − (x′ , y), (x, y + y ′ ) − (x, y) − (x, y ′ ), (r x, y) − r (x, y), (x, r y) − r (x, y) and put T := C/D. For any element (x, y) of the basis of C, we denote with the symbol x⊗y its image in T . Thus T is generated by elements of the form x ⊗ y and, from above, one has (x + x′ ) ⊗ y = x ⊗ y + x′ ⊗ y, x ⊗ (y + y ′ ) = x ⊗ y + x ⊗ y ′ , (r x) ⊗ y = x ⊗ r y = r x ⊗ y. Similarly, the quotient map g : M × N −→ T defined by g(x, y) := x ⊗ y

1.9 Tensor products of R-modules and of R-algebras

27

is R-bilinear. Let P be any R-module; any map f : M × N → P extends by linearity to a R-module homomorphism f : C → P . If in particular f is R-bilinear, from the above relations, f vanishes on all generators of D and so on D. Thus, it determines a well-defined homomorphism f ′ : T → P of R-modules such that f ′ (x ⊗ y) = f (x, y). The map f ′ is uniquely determined by this condition, so the pair (T, g) as above satisfies condition (*). Let (T ′ , g ′ ) be another pair satisfying (*); replacing (P, f ) with (T ′ , g ′ ) in the statement of (*), one obtains a unique R-linear map j : T → T ′ such that g ′ = j ◦ g. Chainging role between T and T ′ , one gets a R-linear map j ′ : T ′ → T such that g = j ′ ◦ g ′ . Since j ◦ j ′ and j ′ ◦ j are both the identities, it follows that j is an isomprphism. ⊓ ⊔ Definition 1.9.2 The R-module T in Proposition 1.9.1 is called the tensor product of M and N and it is denoted by M ⊗R N (or simply by M ⊗ N , if no confusion arises). By its definition, if (xi )i∈I and (yj )j∈J are families of generators of M and N , respectively, then M ⊗ N is generated by the elements xi ⊗ yj , i ∈ I, j ∈ J; in particular, if M and N are finitely generated R-modules, the same occurs for M ⊗ N . Example 1.9.3 One word of warning; the expression x ⊗ y strictly depends on the modules M and N . Take e.g. R = Z. Let M = Z, N = Z2 and let M ′ = 2Z be the submodule of M generated by 2. Let 1 ∈ Z2 and consider z := 2 ⊗ 1. Viewed as an element of M ⊗ N = Z ⊗ Z2 , z is the zero element indeed 2 ⊗ 1 = 2 · 1 ⊗ 1 = 1 ⊗ 2 · 1 = 1 ⊗ 0 = 0M ⊗N . On the other hand, as an element of M ′ ⊗ N = 2Z ⊗ Z2 , z is non zero.

The proof of Proposition 1.9.1 can be extended to multilinear maps f : M1 × · · · × Mr → P of R-modules, giving rise to multi-tensor product M1 ⊗ · · · ⊗ M r

(1.20)

of modules. We left to the reader to check straigtforward computations (cf. Exercise 1.12.10). Proposition 1.9.4 Let M , N and P be R-modules. The following are uniquely determined isomorphisms: (i) M ⊗ N ∼ = N ⊗ M , defined by x ⊗ y → y ⊗ x, (ii) (M ⊗ N ) ⊗ P ∼ = M ⊗ N ⊗ P , defined by (x ⊗ y) ⊗ z → = M ⊗ (N ⊗ P ) ∼ x ⊗ (y ⊗ z) → x ⊗ y ⊗ z, (iii)(M ⊕ N ) ⊗ P ∼ = (M ⊗ P ) ⊕ (N ⊗ P ), defined by (x, y) ⊗ z → (x ⊗ z, y ⊗ z),

28

1 Basics on commutative rings

(iv) R ⊗ M ∼ = N , defined by r ⊗ x → r x. Proof. Left to the reader as Exercise 1.12.9 (cf. also [1, Proposition 2.14]. ⊓ ⊔ 1.9.1 Restriction and extension of scalars f

Let R −→ S be a ring homomorphism and let N be a S-module. Then N has a natural struture of R-module defined by: r x := f (r) x, for any r ∈ R and any x ∈ N. This structure of R-module on N is said to be obtained by restriction of scalars. In particular, the homomorphism f naturally defines a structure of R-module on the ring S (cf. § 1.5). Proposition 1.9.5 With notation as above, assume that N is finitely generated as a S-module and that S is finitely generated as a R-module. Then N is finitely generated as a R-module. Proof. Let y1 , . . . , yn be a system of generators of N as a S-module and x1 , . . . , xm a system of generators of S as a R-module. The set {xi yj }, 1 6 i 6 n, 1 6 j 6 m, is a system of mn generators of N as a R-module. ⊓ ⊔ Let M be a R-module; from above one can consider the R-module MS := S ⊗R M ; this R-module has also a structure of S-module given by: s(s′ ⊗ x) = (ss′ ) ⊗ x, for any s, s′ ∈ S and any x ∈ M. The S-module MS is said to be obtained from M by extension of scalars. Proposition 1.9.6 Assume that M is a finitely generated R-module. Then MS is a finitely generated S-module. Proof. If x1 , . . . , xm is a system of generators of M as a R-module, the elements 1 ⊗ xi , 1 6 i 6 m, generate MS as a S-module. ⊓ ⊔ Let now M , N and P be R-modules and f :M ×N →P be a R-bilinear map. Since, for any x ∈ M , the induced map f (x, −) : N → P, y → f (x, y) is R-linear, then f induces a map

1.9 Tensor products of R-modules and of R-algebras

29

M −→ Hom(N, P ) which is R-linear, since f is R-linear with respect to the variable x ∈ M . Conversely, any R-homomorphism φ

M −→ HomR (N, P ) defines a R-bilinear map fφ : M × N → P, (x, y) −→ φ(x)(y). This shows there exists a bijective correspondence between the set of Rbilinear maps M × N → P and Hom(M, Hom(N, P )). At the same time, by definition of tensor product, the set of R-bilinear maps M ×N → P bijectively corresponds to Hom(M ⊗R N, P ). Thus, one has a canonical isomorphism of R-modules Hom(M ⊗R N, P ) ∼ = Hom(M, Hom(N, P )). 1.9.2 Tensor product of algebras ϕ

ψ

Let S and T be two R-algebras, with R −→ S and R −→ T the corresponding structural morphisms. From Remark 1.5.1 (i), we can consider the R-module U := S ⊗R T. In what follows, we will endow U with a structure of R-algebra. Consider the map S × T × S × T −→ U, (s, t, s′ , t′ ) → ss′ ⊗ tt′ ; it is R-linear in each factor. By definition of multi-tensor product (1.20) (cf. also Exercise 1.12.10), the previous map induces a homomorphism of R-modules S ⊗ T ⊗ S ⊗ T → U.

From Proposition 1.9.4 this homomorphism corresponds to a homomorphism of R-modules U ⊗U →U which, by Proposition 1.9.1, corresponds to a R-bilinear map:

µ : U × U −→ U, µ(s ⊗ t, s′ ⊗ t′ ) := s s′ ⊗ t t′ ; by linearity one has   X X X (si s′j ⊗ ti t′j ). (s′j ⊗ t′j ) = µ  (si ⊗ ti ), i

j

i,j

In other words, the map µ defines a product on U .

(1.21)

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1 Basics on commutative rings

Proposition 1.9.7 The product µ endows the R-module U with a structure of commutative ring, with identity 1 ⊗ 1. Furthermore, U is a R-algebra. Proof. The first part of the statement is a straightforward verification. For the last part, notice that the map R −→ U, r → ϕ(r) ⊗ 1 = 1 ⊗ ψ(r) is a ring homomorphism; indeed one has a commutative diagram of ring homomorphisms S ր ϕ ցf R U ψ ց րg T where f and g are respectively defined by f (s) = s ⊗ 1 and g(t) = 1 ⊗ t.

1.10 Graded rings, homogeneous ideals Let S be a ring (commutative and with identity, as always) and let moreover G(+) be an abelian group. Definition 1.10.1 S is said to be a G–graded ring (equivalently, endowed with a G–graduation), if S has a decomposition M Sg , (1.22) S= g∈G

where each Sg is a sub-group of the abelian group S(+) s.t.: (i) 1 ∈ S0 , and (ii) for any (g, h) ∈ G × G one has Sg · Sh ⊆ Sg+h , where for any subsets A, B ⊆ S one denotes A · B := {ab | a ∈ A, b ∈ B} and A + B := {a + b| a ∈ A, b ∈ B}. The group Sg is called the degree-g graded part of S whereas its elements are called homogeneous elements of degree g of S. When in particular G = Z, then S is simply called a graded ring. For any non-empty subset F ⊆ S of a G–graded ring, one poses Fg := F ∩ Sg , ∀ g ∈ G and H(F) :=

[

Fg ,

g∈G

where the latter is called the set of homogeneous elements of F.

(1.23) (1.24)

1.10 Graded rings, homogeneous ideals

31

Notation 1 From now on in this book, to distinguish homogeneous elements from non-homogeneous ones, homogeneous elements of a graded ring S will be denoted by capital letters A, B, etcetera. When, moreover, we want more precisely specify the degree of a given homogeneous element, we will sometimes use the symbol Ag to stress that it is an element in Sg . On the contrary, elements in S which are not homogeneous will be simply denoted by small letters like a, b, etcetera. Remark 1.10.2 From the previous definitions, one has that: (a) any f ∈ S can be uniquely written as a finite sum f = Fg 1 + . . . + F g n ,

(1.25)

where Fgi ∈ Sgi , for distinct integers 1 6 i 6 n and distinct elements g1 , . . . , gn ∈ G; Fg1 , . . . , Fgn are called the homogeneous components of f and (1.25) is called the decomposition of f into its homogeneous components; (b) S0 is a subring of S, whereas Sg is a S0 -submodule of S, for any g ∈ G. In particular, if S0 ∼ = K is a field, then Sg is a K–vector space; (c) if G = Z, for any integer n we set M Sd . S>n := d>n

If Sd = {0} for any d < 0, then S is more precisely a non-negatively graded ring and S>n is an ideal of S, for any integer n. In particular, one has: \ S = S>−1 and S>n = {0}. n∈N

Definition 1.10.3 The ideal S>0 is simply denoted by S+ and is called the irrelevant ideal of S. (cf. Chapter 3 for geometric motivations of this terminology). Definition 1.10.4 Let I be an ideal of a graded ring S. Then I is said to be a homogeneous ideal if M Ig , I= g∈G

i.e. f ∈ I if and only if all the homogeneous components of f belong to I.

Example 1.10.5 It is easy to see that e.g. the polynomial ring S := K[x1 , x2 ] is a (non-negatively) graded ring, where the graduation is induced by the degree (cf. § 1.10.1 for a more general treatement). Then one easily verifies that S+ = (x1 , x2 ) is a homogeneous ideal. On the other hand I := (x1 + x22 , x1 x2 ) is not homogeneous. To see this, consider e.g. I ∋ f = (x1 + x22 )(1 + 2x2 ) + (x1 x2 )x21 = (x1 ) + (2x1 x2 + x22 ) + (2x32 ) + (x31 x2 ),

32

1 Basics on commutative rings

where the right-hand-side of the previous equality is the decomposition of f into its homogeneous parts; then F1 := x1 ∈ S1 and F2 := (2x1 x2 + x22 ) ∈ S2 do not belong to I, whereas F3 = 2x32 = 2x2 (x1 + x22 ) − 2(x1 x2 ) ∈ I and F4 = x31 x2 ∈ I. Proposition 1.10.6 Let S be a G–graded ring and let I be an ideal. Then I is a homogeneous ideal if and only if I can be generated by a family of homogeneous elements. Proof. (⇒) Let I := (fℓ )ℓ∈L . By Remark 1.10.2 (a), any generator fℓ of I can be uniquely decomposed by means of its homogeneous components: fℓ = Fℓ,gtℓ + Fℓ,gtℓ +1 + · · · + Fℓ,gtℓ +kℓ , for suitable integers tℓ , kℓ depending on the choice of ℓ ∈ L. Since I is homogeneous, for any ℓ ∈ L these components belong to I so that
I := Fℓ,gj ℓ∈L,t 6j6t +k , ℓ

ℓ

ℓ

as desired. (⇐) Let I := (Fℓ )ℓ∈L , where {Fℓ }ℓ∈L is a family of homogeneous generators of I. Then, any a ∈ I is of the form X aℓ Fℓ , a= ℓ∈L

for some aℓ ∈ S. Decomposing any aℓ as in (1.25), one easily sees that any homogeneous component of a is in I, i.e. I is homogeneous. ⊓ ⊔ Proposition 1.10.7 (a) Let S be a G–graded ring and let I1 , I2 be homogeneous ideals. Then I1 · I2 , I1 ∩ I2 , I1 + I2 are homogeneous ideals. (b) If S is a non-negatively graded ring and I is a homogeneous ideal, then: √ I is homogeneous; (i) (ii) I is prime if and only if, for any pair (F, G) ∈ H(S) × H(S) s.t. F G ∈ I, then either F ∈ I or G ∈ I. Proof. (a) is a straightforward consequence of the definitions. √ To prove (b) − (i), for any f ∈ I consider its decomposition into its homogeneous components f = Fℓ + Fℓ+1 + · · · + Fℓ+m , (1.26) √ for some integers m, ℓ > 0. By definition of I, there exists a positive integer r > 1 s.t. f r ∈ I. From (1.26), the decomposition of f r into homogeneous components is of the form f r = Fℓr + o(rℓ),

1.10 Graded rings, homogeneous ideals

33

√ where o(rℓ) ∈√ (S+ )rℓ . Since I is homogeneous then Fℓr ∈ I, i.e. Fℓ ∈ I. Thus, also f − Fℓ ∈ I. Recursively applying the same reasoning, one can conclude. For (b) − (ii), the implication (⇒) directly follows from the definition of prime and homogeneous ideal. Let us prove the other implication. Take elements a, b ∈ S such that ab ∈ I. Consider the decompositions of a and b in S into their homogeneous components a := As + As+1 + · · · + As+t and b := Bn + Bn+1 + · · · + Bn+m , for some integers t, s > 0, m, n > 0. Then X Hi ∈ I, ab = i

where for any index i s.t. n + s 6 i 6 nP+ m + s + t, Hi is the degree-i homogeneous component of ab, with Hi = As+j Bn+k , where summation is on indices 0 6 j 6 t and 0 6 k 6 m such that s + j + m + k = i. Since I is homogeneous, then Hi ∈ I for any n + s 6 i 6 n + m + s + t. In particular, this holds for Hs+t+n+m = As+t Bn+m ∈ I. By the assumption on I, either As+t ∈ I or Bn+m ∈ I. We can reduce to the case that e.g. As+t ∈ /I and Bn+m ∈ I (if indeed both of them are in I, one replaces a and b with a − As+t and b − Bn+m respectively and proceeds). By induction assume one has proved Bn+m , Bn+m−1 , . . . , Bn+m−r ∈ I, for some r > 0. Consider Hm+n+s+t−r−1 ∈ I, where Hm+n+s+t−r−1 = As+t Bn+m−r−1 + As+t−1 Bn+m−r + · · · + As+t−r−1 Bn+m . By inductive hypothesis and the assumption As+t ∈ / I, one has Bn+m−r−1 ∈ I. By recursive application of the same strategy, one proves that all homogeneous components of b are in I, i.e. I is prime. ⊓ ⊔ By using decomposition in homogeneous components, one can easily prove the following easy result. Proposition 1.10.8 Let S be a G–graded ring and let I ⊆ S be an ideal. Consider the canonical projection π : S ։ S/I. (i) For any g ∈ G one has π(Sg ) ∼ = Sg /Ig ; in particular X Sg /Ig . (1.27) S/I ∼ = g∈G

(ii) Moreover, (1.27) is a direct sum if and only if I is homogeneous.

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1 Basics on commutative rings

From the previous result, when I is a homogeneous ideal, if one poses (S/I)g := Sg /Ig , for any g ∈ G, then the ring S/I is a endowed with a G–graduation induced by the G– graduation of S. We conclude this section by giving the following important definition. Definition 1.10.9 If S is a graded, integral domain, the field Q(S) contains G1 such that G1 , G2 ∈ H(S) as a sub-field the set consisting of all fractions G 2 are of the same degree, with G2 6= 0. This sub-field will be denoted by Q0 (S) and called the sub-field of degree-zero, homogeneous fractions of S.

1.10.1 Homogeneous polynomials We now discuss in more details a concrete example of non-negatively graded ring. Let K be any field and let X0 , . . . , Xn be indeterminates over K. A non-zero polynomial F ∈ K[X0 , . . . , Xn ] is said to be homogeneous if all its monomials have the same degree. Definition 1.10.10 For any integer d > 0, we will denote by K[X0 , . . . , Xn ]d the set consisting of the zero-polynomial together with all degree-d, homogeneus polynomials in K[X0 , . . . , Xn ]. This is a K-vector space, whose canonical basis is given by monic monomials of degree d in the indeterminates X0 , . . . , Xn . Lemma 1.10.11 For any integer d > 0, one has p(n, d) := dimK (K[X0 , . . . , Xn ]d ) = with the convention

n 0






 n+d , d

(1.28)

= 1.

Proof. If d = 0, for any n one has K[X0 , . . . , Xn ]d = K and we are done in this case. Therefore, we may assume d > 1. If n = 1, for any integer d, (1.28) holds as the canonical basis of K[X0 , X1 ]d is given by X0d , X0d−1 X1 , X0d−2 X12 , . . . , X1d ,
monomials. We can therefore assume n > 2 and consisting of d + 1 = d+1 d proceed by double induction on n and d. Let Mdn denote the canonical basis of K[X0 , . . . , Xn ]d ; its cardinality |Mdn | equals dimK (K[X0 , . . . , Xn ]d ). The set Mdn is a disjoint union of the two nonempty subsets

1.10 Graded rings, homogeneous ideals

35

Mdn = M′ ∪ M′′ ,

where M′ is the set of monomials in Mdn containing Xn whereas M′′ = Mdn \ M′ . On the one hand, M′′ is the canonical
basis of K[X0 , . . . , Xn−1 ]d ; by the in; on the other, if we divide each monoductive hypothesis on n, |M′′ | = n−1+d d mial appearing in M′ by Xn , we get the canonical basis of K[X0 , . . . , Xn ]d−1 which by construction bijectively corresponds to M′ and which, by the induc
elements. tive hypothesis on d, consists of n+d−1 d−1 One concludes by       n+d−1 n−1+d n+d |Mdn | = |M′ | + |M′ | = + = , d−1 d d ⊓ ⊔

as desired. Remark 1.10.12 (i) Previous results show that the polynomial ring (n)

S(n) := K[X0 , . . . , Xn ] = ⊕d>0 Sd ,

(1.29)

(n)

is a non-negatively graded ring. In particular, S0 = K and each graded (n) (n) piece Sd , for any d > 1, is a (free) S0 -module i.e. K-vector space of finite dimension given by (1.28). Graduation of S(n) is given by the degree of the homogeneous components of a polynomial. (ii) One comment for the reader; we use here S(n) to denote K[X0 , . . . , Xn ] (and not other notation like A(n+1) which instead will be used later on, cf. e.g. Chapter 2) since we want to shed light onto the graded structure of this ring. Moreover, the index (n) is used instead of (n + 1), the correct number of indeterminates, since as customary S(n) will be identified with the ring of homogeneous coordinates of the projective space Pn in Chapter 3. Notation 2 Following Notation 1, homogeneous polynomials in S(n) will be denoted by capital letters like F , G, etcetera. When, in particular, we con(n) sider a polynomial in Sd , we will sometimes more precisely denote it by Fd , to explicitly remind its degree. On the contrary, polynomials which are not homogeneous will be denoted by small letters f , g, etc., as done in the previous sections. If f ∈ S(n) is of degree m, then f can be uniquely decomposed as a linear combination of its degree-d homogeneous parts, for 0 6 d 6 m, which we write m X Fd , (1.30) f= d=0

where not necessarily all Fd 6= 0.

Proposition 1.10.13 (i) F is homogeneous of degree d if and only if the following identity between polynomials in K[X0 , . . . , Xn , t] holds:

36

1 Basics on commutative rings

F (tX0 , . . . , tXn ) = td F (X0 , . . . , Xn ).

(1.31)

(ii) If F is homogeneous of degree d, then the Euler’s identity n X i=0

Xi

∂F =dF ∂Xi

holds. (iii) Let F, g ∈ S(n) be non-constant polynomials, where F is homogeneous and g divides F . Then g = G is also homogeneous. In particular, irreducible factors of a homogenoeus polynomial are all homogeneous. Proof. (i) Condition (1.31) is obviously necessary. The fact that it is also sufficent directly follows from the decomposition as in (1.30). Indeed, for f ∈ S(n) , condition f (tX0 , . . . , tXn ) = td f (X0 , . . . , Xn ) forces f to be an element (n) of Sd . (ii) Taking the partial derivative with respect to the indeterminate t of (1.31) and then evaluating for t = 1 gives Euler’s identity. (iii) One has F = gh, for some h ∈ S(n) . Assume by contradiction that g is not homogeneous. From (1.30), there exist integers k > 0, j > 0 such that g = Gj + Gj+1 + . . . + Gj+k , with Gj , Gj+k 6= 0. Since F is homogeneous, then h has to be non-homogeneous, i.e. there exist integers r > 0, i > 0 such that h = Hi + Hi+1 + . . . + Hi+r , with Hi , Hi+r 6= 0. Now F = gh would give F = Gj Hi + (Gj Hi+1 + Gj+1 Hi ) + . . . + Gj+r Hi+k , where Gj Hi 6= 0, Gj+r Hi+k 6= 0, since S(n) is an integral domain. Thus i + j = deg(Gj Hi ) < deg(Gj+r Hi+k ) = i + j + r + k would contradict the hypothesis on F . ⊓ ⊔ We now define important operators on homogeneous and non-homogeneous polynomials, which will be frequently used in the next chapters (cf. e.g. § 3). In what follows, x1 , . . . , xn and X0 , . . . , Xn will denote indeterminates over K. Definition 1.10.14 For any integer i ∈ {0, . . . , n}, let δi : K[X0 , . . . , Xn ] → K[X0 , . . . , Xi−1 , Xi+1 , . . . , Xn ] be the map defined by

(1.32)

1.10 Graded rings, homogeneous ideals

37

δi (f (X0 , . . . , Xi−1 , Xi , Xi+1 , . . . , Xn )) := f (X0 , . . . , Xi−1 , 1, Xi+1 , . . . , Xn ). Viceversa, let hi : K[x1 , . . . , xn ] → K[X0 , . . . , Xn ]

(1.33)

be the map defined in such a way that, if f ∈ K[x1 , . . . , xn ] is of degree d, then   X0 X1 Xn d hi (f (x1 , . . . , xn )) := Xi f , ,..., , Xi Xi Xi where in the above definition one has replaced xh with

Xh Xh−1 , for 1 6 h 6 i, and with for i + 1 6 h 6 n. Xi Xi

Lemma 1.10.15 For any integer i ∈ {0, . . . , n}: (i) the map δi is a K-algebra homomorphism; (ii) the map hi is not a K-algebra homomorphism; on the other hand it is multiplicative, i.e. hi (f g) = hi (f )hi (g) and, when deg(f ) = deg(g) = deg(f + g), it is also additive, i.e. hi (f + g) = hi (f ) + hi (g); (iii) for any f ∈ K[x1 , . . . , xn ] of degree d, hi (f ) ∈ K[X0 , . . . , Xn ]d ; (iv) δi ◦ hi = idK[x1 ,...,xn ] ; (v) Xi does not divide f if and only if δi (f ) has the same degree of f ; (vi) let F ∈ S(n) be any homogeneous polynomial and let m > 0 be the multiplicity of Xi as a factor of F (recall (1.2)), then hi (δi (F )) = XFm . i

Proof. Properties (i), (ii), (iv) and (v) are straightforward, whereas (iii) follows by applying Proposition 1.10.13 (i). To prove (vi), it suffices to consider the case where Xi does not divide F and verifying in this case that hi (δi (F )) = F . To prove this, taking into account (v), it suffices to prove it on monomials of degree d and then use linearity on each monomial appearing in F . ⊓ ⊔ Notice that, if F ∈ S(n) is homogeneous, in general δi (F ) does not remain homogeneous (it remains homogeneous only if F is constant w.r.t. Xi ). For this reason δi is called the dehomogenizing operator w.r.t. Xi and, correspondingly, δi (F ) the dehomogenized polynomial of F w.r.t. Xi . On the contrary, by Lemma 1.10.15 (ii), hi (f ) is called the homogenized polynomial of f w.r.t. Xi and, consequently, hi is called the homogenizing operator w.r.t. Xi . As in § 1.3.4, one can consider resultant of two homogeneous polynomials with respect to a given indeterminate. One has:

38

1 Basics on commutative rings

Theorem 1.10.16 Let F, G ∈ S(n) be non-constant, homogeneous polynomials, with deg(F ) = d and deg(G) = m. Write F = Ad + Ad−1 Xn + . . . + A0 Xnd and G = Bm + Am−1 Xn + . . . + B0 Xnm , where Aj ∈ K[x0 , . . . , Xn−1 ]j , Bk ∈ K[x0 , . . . , Xn−1 ]k , with 0 6 j 6 d, 0 6 k 6 m and A0 B0 6= 0. Then the resultant polynomial RXn (F, G) is either 0 or it is homogeneous of degree dm. Proof. If F, G ∈ (K[X0 , . . . , Xn−1 ])[Xn ] =: D[Xn ] have a non-constant common factor then, by Theorem 1.3.17, one concludes RXn (F, G) = 0. Assume therefore that F and G have no non-constant common factor, i.e. RXn (F, G) ∈ K[X0 , . . . , Xn−1 ] \ {0}. For simplicity, let R(X0 , . . . , Xn−1 ) := RXn (F, G). By (1.8) and the fact that the polynomials Aj and Bi are homogeneous, for any t ∈ K one has that R(tX0 , . . . , tXn−1 ) is given by the determinant of the (m + d) × (m + d) matrix:  d  t Ad td−1 Ad−1 ... . . . A0 0 0 ... 0  0 td Ad td−1 Ad−1 . . . . . . A0 0 ... 0     ... ... ... ... ... ... . . . . . . . . .    ... ... ... . . . td Ad td−1 Ad−1 . . . . . . A0  . C :=  tm Bm tm−1 Bm−1  ... .. B0 0 ... 0   m m−1  0 t Bm t Bm−1 . . . . . . .. B0 . . . 0     ... ... ... ... ... ... ... ...  ... ... ... tm Bm . . . ... . . . . . . B0

If we multiply the ith -row of C by tm−i+1 , 1 6 i 6 m, and the (m + j)th -row of C by td−j+1 , 1 6 j 6 d, we get the following matrix  m+d  t Ad tm+d−1 Ad−1 . . . . . . ... ... tm A0 0 ... 0  0 tm+d−1 Ad . . . . . . ... ... tm A1 tm−1 A0 . . . 0     ... ... ... ... ... ... ... ... .. ..     0 0 . . . td+1 Ad . . . ... ... ... .. tA0  e  . C =  m+d Bm tm+d−1 Bm−1 . . . td+1 B1 td B0 0 0 ... .. 0  t   0 tm+d−1 Bm . . . td+1 B2 td B1 .. 0 ... .. 0     ... ... ... ... ... ... ... ... .. ..  0 0 ... ... 0 tm+1 Bm tm Bm−1 . . . .. tB0

e once with respect to rows and then with respect Using linearity for the det(C), to columns, we get the following relation

e = tq R(X0 , . . . , Xn ), tp R(tX0 , . . . , tXn ) = det(C)

whereas + d+1 where p := m + (m − 1) + · · · + 1 + d + (d − 1) + · · · 1 = m+1 2 2
. Thus, R(tX , . . . , tXn ) = q := (m + d) + (m + d − 1) + · · · + 1 = m+d+1 0 2 tq−p R(X0 , . . . , Xn ) and one then concludes by Proposition 1.10.13 (i) and by q − p = md. ⊓ ⊔

1.10 Graded rings, homogeneous ideals

39

To conclude this section, we notice that for homogeneous polynomials in two variables, we have the following useful result. Proposition 1.10.17 Let K be algebraically closed, d a positive integer and F ∈ K[X0 , X1 ]d . Then there exist λ ∈ K \ {0} and d pairs (ai , bi ) ∈ K2 , 1 6 i 6 d (not necessarily all distinct pairs but each of them different from (0, 0)) such that F (X0 , X1 ) = λ(a1 X1 − b1 X0 ) · · · (ad X1 − bd X0 ). The pairs (ai , bi ) are uniquely determined up to order whereas each pair is determined up to proportionality. These pairs are called roots of the homogeneous polynomial F . Proof. Assume that r is the multiplicity of X0 as a factor of F , with 0 6 r 6 d. Thus, F = X0r G, where G ∈ K[X0 , X1 ]d−r is not divisible by X0 . If r = d, the G = λ is a constant and the pairs in the statement all coincide with (0, 1), which is therefore counted with multiplicity d. Let us assume therefore that r < d. Then G ∈ K[X0 , X1 ]d−r is a nonconstant, homogeneous polynomial (cf. Proposition 1.10.13 (iii)), which is not divisible by X0 ; thus δ0 (G) ∈ K[X1 ]. One concludes by applying Theorem 1.3.12 to the polynomial δ0 (G) and then by homogenizing via h0 each linear factor. ⊓ ⊔ 1.10.2 Graded morphisms, graded modules and miscellanea If S is a G–graded ring and S′ is H–graded ring, for G(+) and H(+) abelian groups, a ring homomorphism f : S → S ′ is said to be homogeneous if there exists a group homomorphism φ : G → H s.t., for any g ∈ G one has f (Sg ) ⊆ S′φ(g) . If f and φ are both isomorphisms, then f is an isomorphism of φ-graded rings. If moreover G = H, a homomorphism f : S → S′ is said to be homogeneous of degree 0 if φ = idG . An isomorphism of degree 0 is simply called an isomorphism. If G = H = Z and if f : S → S′ is homogeneous, then φ : Z → Z is given by the multiplication with an integer d, so f (Sg ) ⊆ S′dg , for any g ∈ Z. In this case, f is said to be homogeneous of degree d. If S is a G–graded ring, an S–modulo M is called a G–graded module if M admits a decomposition M Mg , M= g∈G

where each Mg is an abelian sub-group of the group M(+), s.t., for any (g, h) ∈ G × G, Sg · Mh ⊂ Mg+h , where we used notation as above.

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1 Basics on commutative rings

As an easy example, any homogeneous ideal of a G–graded ring S is a G–graded S-module. Same terminology and definitions introduced for graded rings can be extended to graded modules, in particular one has the notion of homogeneous homomorphisms of graded modules. Given a G–graded S–module M, one can change its graduation by using the following procedure: fix h ∈ G and define M M(h)g where M(h)g := Mh+g . M(h) := g∈G

M (h) is a G-graded S–module which is isomorphic to M as an S-module but not as a G–graded S-module (cfr. Exercise 1.12.13). Example 1.10.18 Let K be any field and let V be a K-vector space of dimension n + 1 > 1. The symmetric algebra over V is M Sym(V ) = Symd (V ), d∈N

which is a non-negatively graded ring, simply denoted by S(V ); its degree d homogeneous part will be therefore denoted by S(V )d . A similar proof as for (1.28) shows that   n+d . dim(S(V )d ) = p(n, d) = d ∼ K and S(V ) is generated as a K–algebra by S(V )1 . As Moreover, S(V )0 = usual, for F ∈ S(V )d , we write d = deg(F ). The choice of an (ordered) basis (e0 , . . . , en ) for V (also called a frame for V ) induces the dual frame (e0 , . . . , en ) on V ∗ ∼ = Hom(V, K), which is defined by ei (ej ) = ej (ei ) = δij where δij the Kronecker’s symbol. One usually poses ei := Xi and ei := ∂i , 0 6 i 6 n. In this way, S(V ∗ ) is naturally identified with S(n) = K[X0 , . . . , Xn ] as in § 1.10.1. Dually, the ring S(V ) is identified with the ring of differential operators, which is denoted by Dn := K[∂0 , . . . , ∂n ] and which is (abstractly) isomorphic to S(n) as a graded ring. The degree–d homogeneous part Dn,d of the ring Dn is called K-vector space of degree–d homogeneous differential operators, i.e. differential operators where only degree–d monomials in ∂0 , . . . , ∂n appear (the zero-differential operator is considered to be homogeneous of any degree). Let X := (X0 , . . . , Xn ) be indeterminates and, for any multi–index i := (i0 , . . . , in ) ∈ Nn+1 , let |i| := i0 + . . . + in be the length of the multi–index. We will pose

1.10 Graded rings, homogeneous ideals

41

X i := X0i0 . . . Xnin . (n)

Thus, any degee–d homogeneous polynomial in Sd X F = Fi X i

can be written also as

|i|=d

(similar notation can be used for differential operators). In the above notation, from (1.31) F is homogeneous of degree d if and only if F (tX) = td F (X), ∀t ∈ K and Euler’s identity (cf. Prop. 1.10.13 (ii)) reads dF (X) =

n X

∂i F (X).

i=0

Example 1.10.19 Let Vi , 1 ≤ i ≤ h, be K-vector spaces of dimensions ni +1, respectively. Put V := ⊗hi=1 Vi . From Example 1.10.18, S(V ) has a natural structure of graded ring induced by V . On the other hand, S(V ) has also a Nh –graduation in which the degree– d = (d1 , . . . , dh ) part is given by ⊗hi=1 Sym Vdi . When S(V ) is intended to be endowed with the latter graduation, then S(V ) will be more precisley denoted by S(V1 , . . . , Vh ) and, correspondingly, its homogeneous part of degree d will be denoted by S(V1 , . . . , Vh )d . As observed in Example 1.10.18, when one introduces frames for the vector spaces Vi , 1 6 i 6 h, the ring S(V1∗ , . . . , Vh∗ ) can be identified with the polynomial ring K[X1 , . . . , Xh ], where Xi := (Xi0 , . . . , Xin ). To underline its Nh –graduation, this ring will be denoted by S(n1 ,...,nh ) , or simply by Sn , with n := (n1 , . . . , nh ) a multi–index. Its degree–d = (n) (d1 , . . . , dh ) part, i.e. Sd , is the K-vector space of pluri–homogeneous polynomials of degree d in the indeterminates X1 , . . . , Xh . These are the homogeneous polynomials having, respectively, degrees di in the indeterminates Xi = (xi0 , . . . , xin ), for any i ∈ {1, . . . , h}. Similarly to Lemma 1.10.11, one has  h  Y ni + d i . p(n, d) := dim(Sn ) = d di i=1

Dually, the ring S(V1 , . . . , Vh ) is identified with the ring of differential operators K[∂ 1 , . . . , ∂ h ] (where ∂ i := (∂i0 , . . . , ∂in )) and, similarly to Example 1.10.18, this is also denoted by either Dn1 ,...,nh or Dn . Its degree–d part,

42

1 Basics on commutative rings

which is denoted by Dn,d , is the K-vector space of pluri–homogeneous differential operators of degree d = (d1 , . . . , dh ) operating on the indeterminate X1 , . . . , Xh . (n) Similarly as in Example 1.10.18, a pluri–homogeneous polynomial in Sd will be written as h h Y X X F = X j ij . Fi1 ,...,ih j=1 |ij |=di

j=1

Analogous notation can be used for pluri–homogeneous differential operators.

1.11 Localization Let R be a ring and let S ⊂ R be a subset. S is said to be a multiplicative system if (i) 0 ∈ / S, (ii) 1 ∈ S, (iii) for any s, t ∈ S, one has st ∈ S.

In the cartesian product R × S one poses the following relation: (a, s) ≡ (b, t) ⇔ ∃ u ∈ S s.t. (at − bs)u = 0.

(1.34)

It is easy to see that ≡ is an equivalence relation. Remark 1.11.1 When in particular R is an integral domain and S = R∗ , then ≡ coincides with the equivalence relation used to construct the field of fraction Q(R), i.e. (a, s) ≡ (b, t) ⇔ (at − bs) = 0. Definition 1.11.2 The quotient set R × S/ ≡ will be denoted by RS and called localization of R with respect to S. The ≡-equivalence class of a pair (a, s) ∈ R×S will be denoted by as . Operations on R endow RS with a structure of a (commutative and with identity) ring by the following rules: a b at + bs a b ab + = and = . s t st s t st One has a natural ring homomorphism j : R → RS , defined by a →

a , 1

which is called the localization homomorphism. Proposition 1.11.3 (i) With notation as above, one has Ker(j) = {a ∈ R | sa = 0, for some s ∈ S}. (ii) For any s ∈ S, j(s) =

s 1

∈ RS is invertible.

1.11 Localization

43

Proof. (i) Take a ∈ R and assume that as = 0 in R, for some s ∈ S. Then j(a) = a1 is such that (a, 1) ≡ (as, s) = (0, s), i.e. j(a) = 0 ∈ RS . Conversely, for any a ∈ Ker(j), one has (a, 1) ≡ (0, s), for some s ∈ S. This means there exists u ∈ S s.t. 0 = u(as − 0) = a(us) in R, i.e. a is a zero-divisor in R. (ii) j(s)−1 = 1s . Remark 1.11.4 (i) From Proposition 1.11.3, if R is an integral domain then j is injective, in which case R can be identified with its image j(R) and so considered as a subring of RS . Moreover, for any choice of multiplicative system S ⊂ R, RS is identified with a subring of Q(R), the field of fractions of R. (ii) When otherwise R is an arbitrary ring and S coincides with the set of all non-zero divisors in R, RS is called the ring of total fractions of R Definition 1.11.5 If R is a graded ring and S is a multiplicative system of R, the elements of RS are endowed with a graduation induced by that of R in the following way: a := deg(a) − deg(s). deg s The subset o na ∈ RS | a, s ∈ H(R), s ∈ S, deg(a) = deg(s) R(S) := s

is a subring of RS which is called the homogeneous localization of R with respect to S. In other words, R(S) is the subring of homogeneous, degree-zero elements in RS . Recalling Definition 1.1.3, one has:

Proposition 1.11.6 Let R be a ring, S ⊂ R a multiplicative system and j the associated localization homomorphism. (i) Any ideal of RS is an extended ideal. (ii) Let J ⊂ R be a proper ideal. J e ⊆ RS is a proper ideal of RS if and only if J ∩ S = ∅. (iii) Let I1 and I2 be ideals in RS . Then I1 = I2 if and only if I1c = I2c . (iv) If R is Noteherian, then RS is Noetherian. Proof. (i) Let I ⊆ RS be any ideal. For any as ∈ I, one has j(s) as = 1s as = a ∈ I and a ∈ I c . Thus j(a) 1s = as ∈ (I c )e , i.e. I ⊆ (I c )e . By Remark 1.1.4 (ii) one concludes that I = (I c )e . (ii) 1 ∈ J e if and only if there exist a ∈ R and s ∈ S s.t. 1 = as ∈ J e , i.e. if and only if (a, s) ≡ (d, d), for some d ∈ S. By (1.34), this is equivalent to the existence of u ∈ S s.t. uad = uds ∈ S, since S is a multiplicative system, i.e. if and only if a(ud) ∈ J ∩ S.

44

1 Basics on commutative rings

(iii) The statement follows from (i) and the fact that for any ideal I ⊆ RS one has I = (I c )e . (iv) Let I1 ⊆ I2 ⊆ · · · ⊆ RS be any ascending chain of ideals of RS . Since R is noetherian, the ascending chain I1c ⊆ I2c ⊆ · · · ⊆ R is stationary (cf. Prop. 1.4.4). This mean there exists a positive integer n0 s.t. Inc 0 = Inc 0 +k for any integer k > 1. From (i) and (iii) we get In0 = (Inc 0 )e = (Inc 0 +k )e = In0 +k , for any k > 1, i.e. the original ascending chain of ideals of RS is stationary too. ⊓ ⊔ Remark 1.11.7 From Proposition 1.11.6 (ii) and (iii), the correspondence {Ideals of RS } → {Ideals of R}, defined by I → I c , is injective and it restricts to a bijective correspondence {Proper ideals of RS } → {Ideals of R not intersecting S}. by

If J ⊂ R is an ideal s.t. J ∩ S = ∅, the ideal J e of RS is usually denoted JRS

(1.35)

(cf. e.g. [1]). Corollary 1.11.8 (i) If J ⊂ S is an ideal s.t. J ∩ S = ∅, then (RS /JRS ) ∼ = (R/J)S . (ii) If p ⊂ R is a prime ideal s.t. p ∩ S = ∅, then pRS is prime, such that c (pRS ) = p. (iii) Any prime ideal of RS is of the form pRS , for some prime ideal p of R s.t. p ∩ S = ∅. Proof. (i) By assumption, the sequence ι

π

0 → J → R → R/J → 0 is exact, i.e. ι is injective, Im(ι) = Ker(π) and π is surjective. From the fact that localization is an exact operation (cf. for details [1, Prop. 3.3]), this means that the previous exact sequence gives rise to the exact sequence 0 → JRS → RS → (R/J)S → 0, which proves the statement. (ii) Assume first that p is a prime ideal in R. If as bt ∈ pRS , there exists u ∈ S s.t. uab = stu ∈ p. Since u ∈ S and S ∩ p = ∅, then ab ∈ p, which implies either a ∈ p or b ∈ p. Thus, either as ∈ pRS or pt ∈ pRS . c Let now b ∈ (pRS ) be any element. By definition of pRS , this means there exist a ∈ p and s, t ∈ S s.t. bt = as ∈ pRS . By (1.34), there exists u ∈ S

1.11 Localization

45

s.t. u(at − bs) = 0, i.e. ubs = uat ∈ p. On the other hand, since us ∈ S and p ∩ S = ∅, then b ∈ p because p is prime. Thus (pRS )c ⊆ p. By Remark 1.1.4 c (ii), one deduces (pRS ) = p. (iii) Let q ⊂ RS be any prime ideal. From Remark 1.1.4 (iii) and Prop.1.11.6 (ii), p := qc is a prime ideal of R s.t. p ∩ S = ∅. From Prop.1.11.6 (i) and (iii), q = (p)RS . ⊓ ⊔ Remark 1.11.9 Previous results show, in particular, that there exists a bijective correspondence between prime ideals of RS and prime ideals of R not intersecting S. 1.11.1 Local rings and Localization A ring R is said to be a local ring if it contains a unique maximal ideal m. The field R/m is called the residue field of the local ring (R, m). Any field F is a local ring, of maximal ideal (0) and residue field F; for any prime p ∈ Z and any positive integer n, Zpn is a local ring, of maximal ideal (p) and residue field Zp . Proposition 1.11.10 Let R be a ring and m be a maximal ideal of R. Then (R, m) is local if and only if U (R) = R \ m. Proof. If (R, m) is local, then every non-invertible element is contained in m. Conversely, assume that U (R) = R \ m; for any proper ideal I ⊂ R one necessarily has I ⊆ m, i.e. all proper ideals of R are contained in m which is therefore the only maximal ideal of R. ⊓ ⊔ We will now briefly discuss how localization in some cases can produce local rings from a given ring R. Remark 1.11.11 (i) If p is any prime ideal of a ring R, the set S := R \ p is a multiplicative system. In this case, one usually denotes the localization RS by Rp (1.36) which, by abuse of terminology, is called the localization of R w.r.t. p (cf. e.g. [1]). From Remark 1.11.9, there is therefore a bijective correspondence between prime ideals of Rp and prime ideals q of R contained in p. If moreover R is a graded ring, and p is a homogeneous prime ideal, the homogeneous localization R(S) (cf. Def 1.11.5) with S := R \ p is denoted by R(p) .

(1.37)

(ii) If f ∈ R is any non-nilpotent element, the set S := {1, f, f 2 , f 3 , · · · } is a multiplicative system and the localization RS will be denoted by Rf ;

(1.38)

46

1 Basics on commutative rings

when moreover R is a graded ring and f is a homogeneous element in R, the homogeneous localization R(S) will be denoted by R([f ]) .

(1.39)

(iii) If R is an integral domain, the ideal (0) is prime and one has R(0) = Q(R). If moreover R is a graded ring, then R((0)) = Q0 (R), where Q0 (R) as in Definition 1.10.9. In the above notation, one has the following: Proposition 1.11.12 (i) For any prime ideal p in R, (Rp , pRp ) is a local ring with residue field isomorphic to Q(R/p). (ii) For any graded ring R and any homogeneous, prime ideal p, (R(p) , pR(p) ) is a local ring with residue field isomorphic to Q0 (R/p). Proof. (i) From Corollary 1.11.8 (ii), pRp is a prime ideal of Rp ; in particular all its elements are not invertible in Rp . Consider any as ∈ Rp \ pRp ; then a ∈ / p. Thus, j(a) = a1 is invertible in Rp . One concludes that (Rp , pRp ) is local by Proposition 1.11.10. At last, by definition of localization, it is clear that Rp /pRp ∼ = Q(R/p). (ii) It follows from (i) and the fact that R(p) ⊂ Rp as a subring. ⊓ ⊔

1.12 Exercises Exercise 1.12.1. Let R be a ring and I ⊆ R be a proper ideal. Prove that I is prime if and only if, for any expression I = I ′ ∩ I ′′ , with I ′ , I ′′ ideals in R, then either I ′ = I or I = I ′′ . Exercise 1.12.2. Let R be a ring and let I, J ⊂ R be ideals. Show that the set (I : J) := {r ∈ R | r J ⊆ I} is an ideal, which is called the quotient ideal of I by J. Show that, for any ideals I and J, one has: (i) I ⊆ (I : J), (ii) (I ⊆ I, √ √ √ : J) J √ ∩ J = I ∩ J, (iii) IJ = Ip √ √ √ I + J. (iv) I + J = Exercise 1.12.3. Let R be ring and x be an indeterminate over R. Let f (x) := a0 + a1 x + a2 x2 + · · · + an xn ∈ R[x]. Show that f (x) is invertible if and only if a0 is invertible in R and a1 , . . . , an are nilpotent elements.

1.12 Exercises

47

Exercise 1.12.4. Notation as in Exercise 1.12.3. Show that f (x) ∈ R[x] is nilpotent if and only if a0 , a1 , . . . , an are nilpotent elements. Show that f (x) is a zero-divisor if and only if there exists r ∈ R∗ = R \ {0} such that r f (x) = 0. Exercise 1.12.5. Let K be any field, d any positive integer and x an indetereminate over K. Let f, g ∈ K[x] be polynomials of degree at most d. Prove that if there exists d + 1 elements of K over which f and g assume the same values, then f = g. Exercise 1.12.6. Show that if K is a finitely generated ring (i.e. a Z-algebra of finite type) which is also a field, then K is a finite field (cf. [1, Ex. 6, pag. 128]). Exercise 1.12.7. Let K be a field and let R ⊂ K be a subring such that K is a finite R-algebra. Show that R is also a field (cf. [17, Lemma 1.22]). Exercise 1.12.8. Let R be a Noetherian ring and let S ⊂ R be an arbitrary multiplicative part. Show that S −1 R is Noetherian (cf. [1, Proposition 7.3]). Exercise 1.12.9. Prove Proposition 1.9.4. Exercise 1.12.10. Let R be a ring and M1 , . . . , Mn be R-modules. Following the proof of Proposition 1.9.1, construct the pair (M1 ⊗· · ·⊗Mn , g), where M1 ⊗· · ·⊗Mn the multi-tensor product module and g : M1 ⊗ · · · ⊗ Mn → M1 ⊗ · · · ⊗ Mn the canonical multi-linear map (x1 , ldots, xn ) → x1 ⊗ · · · ⊗ xn , such that for any Rmodule P and any multi-linear map f : M1 × · · · × Mn → P , there exists a unique homomorphism f ′ : M1 ⊗ · · · ⊗ Mn → P such that f ′ ◦ g = f . Show moreover that the pair (M1 ⊗ · · · ⊗ Mn , g) is uniquely determined up to isomorphism. Exercise 1.12.11. Let A and B be rings, M be an A-module, P a B-module and N an (A, B)-bimodule (i.e. N is both an A-module and a B-module and the two structures are compatible which means a(xb) = (ax)b, for any a ∈ A, b ∈ B, x ∈ N ). Show that: (i) M ⊗A N has a natural structure of B-module, (ii) N ⊗B P has a natural structure of A-module, (iii) one has (M ⊗A N ) ⊗B P ∼ = M ⊗A (N ⊗B P ) (cf. [1, Excercise 2.15 p.51]. Exercise 1.12.12. Extend (i) and (ii) in Proposition 1.10.7 to the case where G is ordered, i.e. there exists a total order 6 on G which is compatible with the group structure, i.e. g 6 g ′ and h 6 h′ implies g + h 6 g ′ + h′ . Exercise 1.12.13. Let M be a S–graded module. Consider M(h), for h ∈ G, defined as above. Show that M(h) is still a S–graded module which is isomorphic to M as a S–module. Given an example where M(h) is not isomorphic to M as a S–graded module. Exercise 1.12.14. Let M and N be Z–graded modules and let f : M → N a homogeneous homomorphism of degree d, i.e. f (M n ) ⊆ N n+d , for any n ∈ Z. Prove that f : M(−d) → N and f : M → N(d) are homogeneous homomorphisms of degree 0.

48

1 Basics on commutative rings

Exercise 1.12.15. Let X := (X0 , . . . Xn ) and Y := (Y0 , . . . Ym ) be indeterminates. (n) Let Gi (X) ∈ Sd , 0 6 i 6 m, be homogeneous polynomials of degree d. Denote by G(X) := (G0 (X), . . . , Gm (X)). Show that the map

F (Y ) ∈ S(m) −→ F (G(X)) ∈ S(n)

is a homogeneous homomorphism of degree d. This homomorphism is said to be obtained by homogeneous substitution of degree d of the indeterminates. Exercise 1.12.16. Show that a homogeneous substitution of indeterminate of degree d is an isomorphism if and only if n = m, d = 1 and the polynomials (n) Gi (X) ∈ S1 , 0 ≤ i ≤ n, are linearly independent. Prove moreover that any homogeneous isomorphism is of this form. (n)

Exercise 1.12.17. Let F ∈ Sd and let (F ) be the principal, homogeneous ideal generated by F . Prove that g ∈ S(n) → gF ∈ (F ) is an isomorphism of degree d of S(n) –modules and that Sn (−d) → (F ) is an isomorphism (of degree 0). (n)

(n)

Exercise 1.12.18. Let d > 1 be an integer and let F ∈ Sd , G ∈ S1

be such that

F := Ad + Ad−1 Xn + · · · + A0 Xnd , Ai ∈ K[X0 , . . . , Xn−1 ]i , 0 6 i 6 d and G := B1 + B0 Xn , Bi ∈ K[X0 , . . . , Xn−1 ]i , 0 6 i 6 1.

Show that the resultant polynomial RXn (F, G) ∈ K[X0 , . . . , Xn−1 ]d equals A0 B1d − A1 B0 B1d−1 + A2 B02 B1d−2 + · · · + (−1)d Ad B0d .

2 Algebraic affine sets

From now on K will denote a field. To start with, we discuss basic facts with no further assumptions on K. On the other hand, we will present some examples showing where difficulties come out in this general setting; finally, we will focus once and for all on the case of K algebraically closed (unless otherwise explicitely stated), in particular containing infinitely many elements (cf. Corollary 1.3.13).

2.1 Algebraic affine sets and ideals For any field K, let AnK (or simply An , when the field K is understood), the n–dimensional numerical (standard) affine space over K. This is the set Kn of ordered n-tuples (p1 , . . . , pn ), where pi ∈ K, 1 6 i 6 n. The use of the symbol An (instead of Kn ) resides on the fact that we want to take into account the geometric nature of Kn , i.e. considering its elements as points instead as vectors. We will indeed endowed An K with a structure of topological space (cf. Definition 2.1.13), and this structure we want to be kept in mind to be completely different from that of Kn as n-dimensional vector space over K. An element P = (p1 , . . . , pn ) in An will be called a point of the affine space and p1 , . . . , pn will be its coordinates in An . Correspondingly, the (numerical) vector (p1 , . . . , pn ) ∈ Kn , i.e. the vector having those components with respect to the canonical basis of Kn , will be simply denoted by P and called coordinate vector of the point P . The point O ∈ An , whose coordinate vector is O = (0, 0, . . . , 0) ∈ Kn , is the origin of An . Let x := (x1 , . . . , xn ) be a n–tuple of indeterminates over K. In this (n) chapter, we will denote by AK , or simply by A(n) , the polynomial ring K[x] = K[x1 , . . . , xn ]. An element f (x) = f (x1 , . . . , xn ) ∈ A(n) will be simply denoted by f , if no confusion arises. It is then clear that any f ∈ A(n) can be regarded as a map

50

2 Algebraic Affine Sets

f : An → K,

P → f (P ) = evP (f ),

(2.1)

where evP (f ) denotes the evaluation of f ∈ A(n) at P ∈ An . Definition 2.1.1 The subset of An Za (f ) := f −1 (0) = {P ∈ An | f (P ) = 0}, will be called the set of zeroes (or simply the zero–set) of f . The subscrpit a in Za (−) stands for the word affine, to distinguish from the projective case which will be introduced in Chapter 3. Remark 2.1.2 From Definition 2.1.1 it is already clear the reason why, for geometric problems, one is mostly concerned with algebraically closed fields. Indeed, consider the polynomials 1, x21 + 1 ∈ A(1) . Whatever K is, Za (1) = ∅. On the other hand, if e.g. K = R then Za (1) = ∅ = Za (x21 + 1) since x21 + 1 is irreducible in A(1) . If otherwise K = C, in this case Za (1) = ∅ ⊂ Za (x21 + 1) = {i, −i} and the cardinality of the zero–set of the polinomial x21 + 1 ∈ A(1) equals its degree. More generally, one poses Definition 2.1.3 For any F ⊂ A(n) finite subset of polynomials, \ Za (f ) Za (F ) := f ∈F

is called the zero–set of F . More precisely, if F := {f1 , . . . , ft }, then Za (F ) = Za (f1 , . . . , ft ) =

t \

i=1

Za (fi ) = {P ∈ An | fi (P ) = 0, ∀ i = 1, . . . , n} .

Definition 2.1.4 A subset Y ⊆ An is called an Algebraic Affine Set (AAS, for short), if Y = Za (F ), for some finite subset F ⊂ A(n) . When F is explicitely given as F = {f1 , . . . , ft }, the polynomials fi ∈ A(n) , 1 6 i 6 t, are said to be a system of equations defining Y . Given F = {f1 , . . . , ft } as above, consider the ideal these polynomials generate inside A(n) . Thus I := (f1 , . . . , ft ) is a finitely generated ideal. Proposition 2.1.5 For any field K, one has Za (F ) = Za (I) := {P ∈ An | f (P ) = 0, ∀ f ∈ I} . Proof. It is clear Pt that Za (I) ⊆ Za (f1 , . . . , ft ). On the other hand, since any f ∈ I is f = i=1 gi fi , for suitable gi ∈ A(n) , 1 6 i 6 t, it is clear that P ∈ Za (f1 , . . . , ft ) implies P ∈ Za (I). ⊓ ⊔

2.1 Algebraic affine sets and ideals

51

Remark 2.1.6 From Proposition 2.1.5, Y does not depend on the system of equations defining it, in particular any finite set of generators of the same ideal I defines the same AAS Y in An . Example 2.1.7 As above, let K be any field. (i) In A1 , one has {0} = Za (x1 ) = Za ((x1 )) = Za ({xn1 }n∈F ), for F ⊂ N any (possibly infinite) subset. (ii) In A(2) consider the three ideals I = (x1 , x2 ), J = (x1 , x2 − x21 ), K = (x2 , x2 − x21 ). One has I = J; indeed J is a proper indeal in A(1) , I ⊆ J (since x1 , x2 = 1(x2 − x21 ) + x1 (x1 ) ∈ J) and I is maximal (since A(2) /I ∼ = K is a field). On the other hand, K $ I, as it easily follows from the fact that K = (x2 , x21 ). However Za (I) = Za (J) = Za (K) = O = (0, 0) ∈ A2 . To have a look to the geometric meaning of the three ideals, take for simplicity K = R. The origin O is the AAS cut out by the generators of the 3 ideals I, J and K above. The main difference among the 3 ideals is the following. Generators of I and J transversally intersect along O: in I the intersection is between the coordinate axes, whereas in J this intersecion is between the vertical axis x1 = 0 and the parabola x2 − x21 = 0 with vertex at O. For what concerns K, the two generators are the parabola x2 − x21 = 0 and its tangent line x2 = 0 at O, i.e. x2 = 0 has intersection multiplicity 2 with the parabola at O (cf. Figure 2.1 below). In all the three cases, passing to Za (−) discards multiplicities from the picture. We will give more precise algebraic motivations for this phenomenon in Remark 2.1.14 (iv).

Fig. 2.1. Geometric representation of Za (I), Za (J) and Za (K)

A direct consequence of Theorem 1.4.1 is the following Corollary 2.1.8 Let K be any field. Any AAS in An is of the form Za (I), for some ideal I ⊆ A(n) . Remark 2.1.9 In particular, Definitions 2.1.3, 2.1.4 make sense for any subset F ⊆ A(n) ; no matter F is finite or not, the ideal I := (F ) is always finitely generated. We discuss easy examples, some of which show basic differences between the cases with K algebraically closed or not and infinite or not.

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Example 2.1.10 (i) In A(1) , consider the subset F := {xn1 }n∈N . Then, the ideal I generated by F is I = (F ) = (x1 ), since K[x1 ] is a PID, and Za (F ) = Za ((F )) = Za ((x1 )) = {0}. (ii) Consider f ∈ A(1) a non-constant polynomial. If K is not algebraically closed, there exist polynomials f with no root in K (e.g. when f is irreducible of deg(f ) > 2). In such a case, Za ((f )) = ∅ = Za ((1)) even if the inclusion (f ) ⊂ (1) = A(1) is strict (being (f ) a proper maximal ideal in A(1) ). When otherwise K is algebraically closed, for any f ∈ A(1) , one has ∅ 6= Za ((f )); more precisely, Za ((f )) = {α1 , . . . , αk }, where k 6 deg(f ) and αi are all the distinct roots of f , the case k = n occurring if and only if f has distinct (i.e. simple) roots. (iii) Conversely to (ii), taking α1 , . . . , αk ∈ K distinct elements and posing Qk f := i=1 (x1 − αi ) ∈ K[x1 ], one has {α1 , . . . , αk } = Za (f ) = Za ((f )) ⊂ A1K . (iv) If K is a finite field, there do exist non-zero polynomials f ∈ A(1) for which Za (f ) = A1 . Take e.g. K = Zp , p ∈ Z a prime, and consider f := xp1 − x1 = x − 1p + (p − 1)x − 1 ∈ Zp [x]. One easily finds that Za (f ) = A1 = Za (0); indeed, as a consequence of Lagrange’s theorem andQthe fact that U (Zp ) is cyclic of order p − 1, f totally splits in Zp [x] as f = i∈Zp (x − i). Recalling operations on ideals as in § 1.1, one can discuss first properties of AAS’s in An .

Proposition 2.1.11 Let K be any field. Then: (i) for any subsets F and G of A(n) one has F ⊆ G =⇒ Za (F ) ⊇ Za (G).

(2.2)

In particular, for any ideals I1 ⊆ I2 in A(n) , one has Za (I1 ) ⊇ Za (I2 ); (ii) for any I1 , I2 ⊆ A(n) ideals, Za (I1 ∩ I2 ) = Za (I1 ) ∪ Za (I2 ); (iii) for any family {Iα }α∈A of ideals in A(n) , ! \ X Za (Iα ) ; Iα = Za α∈A

α∈A

(iv) for any P ∈ An , whose coordinates are (p1 , . . . , pn ), the ideal mP := (x1 − p1 , . . . , xn − pn )

(2.3)

is maximal in A(n) and Za (mP ) = {P }. √ (v) for any ideal I ⊆ A(n) , Za (I) = Za ( I). Proof. (i) For any P ∈ Za (I2 ) and for any f ∈ I2 , one has f (P ) = 0. Since I1 ⊆ I2 , then P ∈ Za (I1 ) also. Finally, (2.2) follows from Remark 2.1.9 and what proved for ideals.

2.1 Algebraic affine sets and ideals

53

(ii) Since I1 ∩ I2 ⊆ I1 , I2 , by (i) we have Za (I1 ) ∪ Za (I2 ) ⊆ Za (I1 ∩ I2 ). On the other hand, if P ∈ / Za (I1 )∪Za (I2 ), there exist f1 ∈ I1 and f2 ∈ I2 s.t. f1 (P ) 6= 0 and f2 (P ) 6= 0; f1 f2 ∈ I1 ∩ I2 is such that (f1 f2 )(P ) := f1 (P )f2 (P ) 6= 0, since K is a field, which shows that P ∈ / Za (I1 ∩ I2 ). P P by (i), Za ( α∈A Iα ) ⊆ (iii) One has Iα ⊆ α∈A Iα , for any α ∈ A. PTherefore, T Za (Iα ), for any α ∈ A, which implies T Za ( α∈A Iα ) ⊆ α∈A Za (Iα ). On the other hand, for any P ∈ α∈A Za (Iα ), one P has that fα (P ) = 0, for any fα ∈PIα and for any α ∈ A. By definition of α∈A Iα , this implies that P ∈ Za ( α∈A Iα ). (iv) For any P ∈ An , we have the K-algebra homomorphism ΦP : A(n) → K defined by the rules ΦP (k) = k, ∀ k ∈ K, ΦP (xi ) = pi , 1 6 i 6 n; in particular, for any f ∈ A(n) , ΦP (f ) = evP (f ), where evP (−) is the evaluation at the point P as above. Clearly, the morphism ΦP is surjective. Thus Ker(ΦP ) is a maximal ideal of A(n) and, by the definition of mP as in (2.3), one has mP ⊆ Ker(ΦP ). To prove the maximality of mP it therefore suffices to show that equality holds. To do this, denote by y := (y1 , . . . , yn ) a set of new indeterminates in such a way that y := x − P , where P is the coordinate vector associated to the point P ∈ An . For any f = f (x) ∈ Ker(ΦP ), one has f (x) = f (y + P ) =: g(y) ∈ K[y] ∼ = A(n) . By the assumptions on f (x), one has g(y) ∈ (y1 , . . . , yn ) ⊂ K[y], i.e. g(y) = y1 h1 (y) + · · · + yn hn (y), for some hi (y) ∈ K[y], 1 6 i 6 n. This implies that f (x) = g(x − P ) = (x1 − p1 )h1 (x − P ) + · · · + (xn − pn )h1 (x − P ) ∈ mP , which shows that Ker(ΦP ) ⊆ mP . The last part of the assertion is trivial. √ (v) The inclusion Za ( I) ⊆√Za (I) directly follows from (i) and the fact that, for any ideal I one has I ⊆ I (recall §√1.1). On the other hand, for any g ∈ I, there exists a positive integer n such that g n ∈ I so that, for any P ∈ Za (I), one has (g n )(P ) = 0. Since √ (g n )(P ) = (g(P ))n , as a power in K, this forces g(P ) = 0, i.e. P ∈ Za ( I), which proves the other inclusion. ⊓ ⊔ Notice that, for any field K, one has ∅ = Za ((1)) and An = Za ((0)), i.e. they are AAS’s. Therefore, using also Proposition 2.1.11 (ii) and (iii), one easily observes:

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Proposition 2.1.12 The set Cna := {AAS’s in An }

(2.4)

is the set of closed subsets of a topology on An . Definition 2.1.13 The topology on An having Cna as the set of closed subsets is called the Zariski topologyof An and will be denoted by ZarAn , or simply by Zarna . Remark 2.1.14 (i) From now on, unless otherwise specified, An will be always considered endowed with the topology Zarna . The AAS’s will be called also (Zariski) closed subsets of An (for brevity, the term Zariski will be omitted in the sequel). An open subset in Zarna will be of the form Za (I)c , for some ideal I in A(n) , i.e. the complement of a closed subset of An . Any subset Y ⊆ An will be endowed from now on with the topology induced by Zarna on Y . This topology will be called the Zariski topology of Y and denoted by Zarna,Y (or simply ZarY , when the inclusion Y ⊆ An is clearly fixed). (ii) From Proposition 2.1.11 (iv), it follows that Zarna is T1 and that Cna contains all finite subsets of An . It is clear that for n = 1 these, together with the empty–set, are the only proper (i.e. strictly contained) closed subsets of A1 ; if moreover K is finite, these are the only closed subsets at all. If K is infinite (e.g. when K is algebraically closed, cf. Corollary 1.3.13), any non-empty open subset of A1 contains infinite elements and any two nonempty open subsets intersect, i.e. Zara1 is not T2 . We will show that this more generally holds for any n > 1 (cf. Remark 4.1.1). This is false when K is finite: take e.g. K = Z2 , U1 = Za (x)c and U2 = Za (x − 1)c . (iii) It is then clear that, when K is either R or C, the euclidean topology of AnK is finer than Zarna . (iv) Proposition 2.1.11 (v) shows that we do not have a bijective correspondence between ideals in A(n) and elements in Cna , since any ideal √ I which is not radical, defines the same closed subset of its radical ideal I. On the other hand, passing to radical ideals allows one to discard non-reduced pieces from the scene. To understand why, consider the following examples. Example 2.1.15 (i) Let K be any field. Consider e.g. x2 , x22 ∈ A(2) ; the 2 associated closed set in A2 is the same Za (x2 ) = Za (x p2 ), given by the x1 -axis. Algebraically speaking, this is due to the fact that (x22 ) = (x2 ). The point set given by Za (x2 ) = Za (x22 ) does not take into account that the x1 -axis in the second case should have to be considered deemed to have multiplicity 2 at any point. In terms of ideals, the situation is completely different: (x2 ) is prime, being A(2) /(x2 ) an integral domain, whereas (x22 ) is not prime, since A(2) /(x22 ) is not a reduced ring (recall Definition 1.1.1). This kind of phenomenon has already appeared in Example 2.1.7 (ii). The ideal I = J is radical, since maximal (cf. Lemma 1.1.2 (ii)). On the contrary,

2.1 Algebraic affine sets and ideals

55

K is not radical since K = (x2 , y) but x ∈ / K. On the other hand, one easily √ finds that K = I, which motivates O = Za (I) = Za (K). (ii) Let K = R and let f := x21 + x22 − 1, g := x1 − 1 ∈ A(2) . One has Za (f, g) = {(1, 0) = P } as a point of A2R . The p same point is defined also by Za ((x1 − 1, x2 )) and indeed one easily finds (f, g) = (x1 − 1, x2 ). Geometric interpretation of the two systems of polynomial equations are given in picture below (see Figure 2.2)

Fig. 2.2. Geometric representation of Za ((f, g)) and Za ((x − 1, y))

(iii) Similarly, consider f := x21 + x22 + 2x1 , g := x21 + x22 − 2x1 ∈ R[x1 , x2 ] and let J = (f, g) be the ideal they generate. One has Za (J) = Za (mO ) = Za ((x1 , x2 )) = O, where mO is a maximal ideal (cf. Proposition 2.1.11 (iv)). Notice that J ⊂ mO , where the inclusion is strict, since x2 ∈ / J. On the other √ hand, x1 = 41 (f − g) ∈ J and x22 = f − x1 (x1 + 2) ∈ J, i.e. mO ⊆ J √ √ / J. The and so mO = J, by the maximality of mO and the fact that 1 ∈ geometric counterpart of the previous analysis is that Za (f ) and Za (g) are two circles passing through O and sharing the same tangent line at O, which is the x2 -axis. The intersection at O between the two circles is not transverse; the radical ideal replaces the system of equations of O with the two coordinate axes (see Figure 2.3)

Fig. 2.3. Geometric representation of the two circles and of the two axes

When K is not algebraically closed, the failure of the bijective correspondence between ideals of A(n) and elements in Cna is even worse: indeed it fails even if we restrict to radical ideals of A(n) . Remark 2.1.16 (i) Let K be a non-algebraically closed field. Let f ∈ A(1) be a non-constant polynomial with no roots in K (e.g. when f is irreducible). Then Za ((f )) = ∅ = Za ((1)), even if (f ) is a proper ideal. If we moreover assume that f ∈ A(1) is irreducible, then (f ) is a maximal ideal (since A(1) is a PID), and so radical (as p it has been proved in Lemma 1.1.2 (ii)). Thus Za ((f )) = ∅, even if (f ) = (f ) ⊂ (1) = A(1) . (ii) Consider once again Example 2.1.10 (iv), where we found Za (xp − x) = Za (0) = Zp , even if (0) ⊂ (xp − x). Notice moreover that the ideal (xp − x) is

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T radical: indeed (xp − x) = a∈Zp (x − a), where each (x − a) is a maximal (so prime ideal) in Zp [x] (cf. the primary decomposition of an ideal, [1, Ch. 4]).

(iii) Let f := x21 + x22 + 1 ∈ K[x1 , x2 ], where we consider K to be either R or C. In both cases, f is irreducible. Since K[x1 , x2 ] is a UFD (but not a PID), the ideal (f ) is prime (so radical) but not maximal (since K[x1 , x2 ]/(f ) is not a field). Thus, from the algebraic point of view, properties of the ideal (f ) remain unchanged, no matter if K is either R or C. On the contrary the geometric situation is completely different. When p K = R, one has Za ((f )) = ∅ = Za ((1)) even if (f ) = (f ); this is a nondegenerate empty conic in the real affine plane. When otherwise K = C, Za ((f )) 6= ∅ and its support is a non-degenerate ellipse in the complex affine plane. Once one identifies A2C with A4R , by posing x1 := a + ib, x2 := c + id, where a, b, c, d, ∈ R and i2 = −1, one can easily show that the locus Za ((f )) (considered as endowed with the topology induced by the euclidean topology of A4R ) is homemorphic to a 2sphere in A3R minus 2 points. This is the open Riemann surface associated to Za (f ). Below we suggest how to proceed. Hints: (1) First, consider the pencil of lines in A2C , given by Lt : x2 − ix1 − t = 0, for any t = t1 + it2 ∈ C, with i2 = −1. With the use of the pencil Lt , define a bijective map   t2 − 1 t2 + 1 , x2 (t) = ψ : C \ {0} → Za ((f )) ⊂ A2C , ψ(t) := x1 (t) = − 2it 2t which is a (complex) rational parametrization of Za (f ). (2) Identify A2C with A4R via x1 (t) := a(t1 , t2 ) + ib(t1 , t2 ) and x2 (t) := c(t1 , t2 ) + id(t1 , t2 ), where a(t1 , t2 ), b(t1 , t2 ), c(t1 , t2 ), d(t1 , t2 ) ∈ R. Show that equations for Za ((f )) in A4R are given by  2 a + c2 − b2 − d2 + 1 = 0 . ab + cd = 0 The map ψ can be considered as a map ψ ∗ : A2R \ {(0, 0)} → Za ((f )) ⊂ A4R . Show that ψ ∗ is a homeomorphism (in euclidean topologies). (3) Consider A3R , with coordinates (t1 , t2 , t3 ), in such a way that A2R above is identified with the affine plane t3 = 0. In A3R consider the 2-sphere S, centered at C = (0, 0, 21 ) with radius r = 12 . Its North-pole is N = (0, 0, 1) whereas

2.1 Algebraic affine sets and ideals

57

its South-pole is O = (0, 0, 0). Let π : S \ {N } → A2R be the stereographic projection from the North-pole. Deduce that ψ ∗ ◦ π : S \ {N } → Za ((f )) ⊂ A4R is a homemorphism (in the euclidean topologies). If on the other hand we consider g := x21 + x22 − 1 ∈ K[x1 , x2 ], a similar discussion holds. The main difference is that Za ((g)) in A4R is given by  2 a + c2 − b2 − d2 − 1 = 0 , ab + cd = 0 so the totally real plane b = d = 0 in A4R cuts Za ((g)) along the plane conic  2  a + c2 − 1 = 0 b=0 .  d=0

This is the real trace of Za ((g)) ⊂ A2C , which coincides with Za ((g)) ⊂ A2R , when g is considered with coefficients in K = R. Example 2.1.17 (Coordinate subspaces) Let K be any field, which we will assume infinite for simplicity, and let 0 < m 6 n be integers. For any sequence of integers 0 < i1 < i2 < i3 < · · · < im 6 n, one has an injective map ϕ := ϕi1 ,i2 ,i3 ,··· ,im : Am ֒→ An defined as follows: (b1 , b2 , . . . , bm ) −→ (0, . . . , 0, b1 , 0, . . . , 0, b2 , 0, . . . , 0, bm , 0, . . . , 0) i1 i2 im . ϕ is a homeomorphism between (Am , Zarm a ) and Im(ϕ) endowed with the topology induced by Zarna . This image is the (Zariski) closed subset of An defined by the homogeneous linear system xj = 0, ∀j 6= i1 , . . . , im ; Im(ϕ) is called a m-dimensional coordinate affine subspace of An . The case m = 1 gives rise to the ith -coordinate axis of An , for 1 6 i 6 n. Example 2.1.18 (Affine subspaces) Let K be any field, which we will assume infinite for simplicity, and let ℓ1 , . . . , ℓk ∈ A(n) be linearly independent linear forms, i.e. ℓi = ai1 x1 + · · · + ain xn , aij ∈ K, 1 6 i 6 k, 1 6 j 6 n, and b1 , . . . , bk ∈ K. Consider

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Y := Za (ℓ1 − b1 , . . . , ℓk − bk ), i.e. the set of solutions of the linear system Ax = b, where A is the k×n matrix whose ith -row is filled-up by the coefficients of the linear form ℓi , 1 6 i 6 k, and where x and b are the n × 1 and k × 1 matrices of indeterminates and constant terms, respectively. By assumptions on the linear forms, rank(A) = k, so Ax = b is compatible. Compatibility condition is independent from the field K and Rouch´e-Capelli’s theorem ensures that the set of solutions Y is a linear variety in An of dimension n − k, i.e. Y has a linear parametric representation of the form x1 = c1 + λ1 (t1 , . . . , tn−k ), . . . , xn = cn + λn (t1 , . . . , tn−k ), where ci ∈ K are constant, λi are linear homogeneous polynomials in the parameters tj and with coefficients from K, 1 6 i 6 n, 1 6 j 6 k. Y is called a (n − k)-dimensional affine subspace of AnK . If I = (ℓ1 − b1 , . . . , ℓk − bk ) ⊂ A(n) is the ideal generated by the linear equations defining Y , then I is a radical ideal: consider indeed the K-algebra homomorphism π : K[x1 , . . . , xn ] −→ K[t1 , . . . , tn−k ], defined by xi −→ ci + λi (t1 , . . . , tn−k ), 1 6 i 6 n; π is surjective as it follows from Gauss-Jordan elimination theory, and its kernel is given by I, as its generators are representatives of the equivalence class of linear systems whose zero–set is exactly Y . Thus, I is prime, since K[t1 , . . . , tn−k ] is an integral domain, so it is radical (cf. Lemma 1.1.2). Example 2.1.19 (Affine hypersurfaces) Let K be any field and let f ∈ A(n) be a non-constant polynomial. The AAS Y := Za (f ) = Za ((f )) is called the affine hypersurface of equation f = 0 in An . Since A(n) is a UFD then, up to units, f factorizes as f = f1r1 f2r2 · · · fℓrℓ ,

(2.5)

where f1 , . . . , fℓ ∈ A(n) are non-proportional, irreducible polynomials and r1 , . . . , rℓ are positive integers. From Proposition 2.1.11, it is clear that Za (f ) = Za (f1 · · · fℓ ) =

ℓ [

i=1

Za (fi );

(2.6)

2.1 Algebraic affine sets and ideals

59

p indeed, by the primary decomposition of ideals in a UFD, one has (f ) = (f1 · · · fℓ ) (cf. [1, Ch. 4], for more details). Y is called the affine hypersurface determined by f ∈ A(n) and the polynomial fred = f1 f2 · · · fℓ

(2.7)

is called the reduced equation of Y , which is indeed another equation for Y . The integer deg(fred ) is called the degree of Y . When n = 1, d = 1 gives a point, d = 2 gives either the emptyset or two distinct points (it dependes on the field K), etcetera; if n = 2, d = 1 gives a line, d = 2 gives a conic, d = 3 gives a plane cubic curve, etcetera; for n = 3, d = 1 is a plane, d = 2 is a quadric, d = 3 is a cubic surface, and so on. Definition 2.1.20 Let K be an infinite field and let f ∈ A(n) be a non constant polynomial. The open set Ua (f ) := Za (f )c = An \ Za (f ) is called a principal open (affine) set of Zarna . Lemma 2.1.21 Principal open (affine) sets form a basis for Zarna . Proof. Any open set of Zarna is of the form U = Za (I)c , for some ideal I ⊂ A(n) . Since A(n) is T Noetherian, I = (f1 , . . . , fm ), forSsome fi ∈ A(n) , m m 1 6 i 6 m, and Za (I) = i=1 Za (fi ). This shows that U = i=1 Ua (fi ). ⊓ ⊔

Example 2.1.22 (Products of AAS’s. Cylinders) (i) Let Z1 ⊆ Ar and (r) Z2 ⊆ As be (Zariski) closed sets. Denote by Ax := K[x1 , . . . , xr ] and (s) Ay := K[y1 , . . . , ys ] the ring of polynomials which are (evaluating) functions operating on the affine spaces Ar and As , respectively. As a set, it is clear that Ar × As = Ar+s , and the ring of functions operating on this affine (r+s) space will be denoted by Ax,y := K[x1 , . . . , xr , y1 , . . . , ys ]. (r)

Now, Z1 = Za (I1 ), for some ideal I1 = (f1,1 , . . . , f1,n ) ⊆ Ax , and (s) Z2 = Za (I2 ), for some ideal I2 = (f2,1 , . . . , f2,m ) ⊆ Ay , as it follows from Noetherianity. Since we have natural inclusions (r+s) A(r) ←֓ A(s) x ֒→ Ax,y y ,

the polynomials generating the ideals I1 and I2 can be regarded as elements (r+s) in Ax,y . One poses Z1 × Z2 := Za ((f1,1 , . . . , f1,n , f2,1 , . . . , f2,m )) ⊂ Ar+s ,

(2.8) (r+s)

where the ideal (f1,1 , . . . , f1,n , f2,1 , . . . , f2,m ) is intended as an ideal in Ax,y . Z1 × Z2 is called the product of the two AAS’s; it is a closed subset of Ar+s .

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For example, if p1 ∈ A1x1 and p2 ∈ A1y1 , their product is given by the point P = Za ((x1 − p1 , y1 − p2 )) = (p1 , p2 ) ∈ A2 . (ii) Particular cases of products are given by cylinders. If Z1 ⊂ Ar is a (Zariski) closed proper subset as above, then Z1 × A1 ⊂ Ar+1 is called the cylinder over Z1 in Ar+1 whereas Z1 is called a directrix of the cylinder. Since A1 in itself is determined by the ideal (0) (moreover this is the only possibility, when K is algebraically closed), the cylinder is simply given by Z × A1 := Za ((f1,1 , . . . , f1,n , 0)) = Za ((f1,1 , . . . , f1,n )) ⊂ Ar+1 ;

(2.9)

in other words, equations defining Z1 in Ar coincide with equations defining the cylinder over Z1 in Ar+1 . For example, if p1 ∈ A1x , the cylinder in A2 over p1 is {p1 } × A1y , whose defining equation is Za (x1 − p1 ); this is nothing but the line x1 = p1 in the plane. The closed subsets Za (x21 + x22 − 1), Za (x21 − x22 − 1) and Za (x21 − x2 ) regarded in A3 are the elliptic, hyperbolic and parabolic quadric cylinders in A3 (see Figure 2.4).

Fig. 2.4. Elliptic, hyperbolic and parabolic cylinders in the affine 3-space

(iii) When if K is infinite notice that the topology Zarr+s is finer than the a topology Zra × Zsa . To see this, consider e.g. A2 = A1 × A1 as a set. From Remark 2.1.14 (ii), closed subsets in Zar1a are only ∅, finite number of points and A1 . Therefore, if we endow A2 with the topology Zar1a × Zar1a , the only closed subsets are ∅, A2 , finite unions of points, finite unions of parallel lines to the x1 -axis and finite unions of parallel lines to the x2 -axis. On the contrary, since K is infinite, the set Za (x1 − x2 ) ⊂ A2 is a closed subset for Zara2 , which is not closed in Zar1a × Zar1a . However, given Z1 × Z2 ⊂ Ar+s , endowed with the (natural) Zariski topology Zarr+s a,Z1 ×Z2 (recall notation as in Remark 2.1.14 (i)), the topology induced by Zarr+s a,Z1 ×Z2 on {p1 } × Z2 (Z1 × {p2 }, respectively), with p1 ∈ Z1 (p2 ∈ Z2 , resp.), coincides with Zarsa,Z2 (Zarra,Z1 , resp.). Up to now, the ”operation” Za (−) allows one to bridge from ideals in An to (Zariski closed) subsets of A(n) . We now want to pass from subsets of An to ideals in An . To do this, we introduce the following: Definition 2.1.23 Let K be any field. For any subset Y ⊆ An , one denotes by

2.1 Algebraic affine sets and ideals

61

n o Ia (Y ) := f ∈ A(n) | f (P ) = 0, ∀ P ∈ Y .

It is an ideal, which is called the ideal of Y in A(n) .

Remark 2.1.24 Notice that for any f ∈ Ia (Y ), one has Y ⊆ Za ((f )); geometrically speaking, if Y 6= ∅, An , elements in Ia (Y ) give rise to hypersurfaces in An containing Y . From Definition 2.1.23 it follows that Y1 ⊆ Y2 ⇒ Ia (Y1 ) ⊇ Ia (Y2 ),

(2.10)

for any subsets Y1 and Y2 of An . Proposition 2.1.25 Let K be any field and let Y be any subset of An . Then Za (Ia (Y )) = Y , where Y is the closure (in the topology Zarna ) of Y in An . In particular, for any subset Y of An one has Y ⊆ Za (Ia (Y )), where equality holds if and only if Y is already (Zariski) closed in An . Proof. By Definition 2.1.23, Za (Ia (Y )) is a closed subset in An such that Y ⊆ Za (Ia (Y )); then Y ⊆ Za (Ia (Y )). Conversely, let W := Za (J) be any closed subset containing Y , J some ideal in A(n) . Since Y ⊆ W , from (2.10) one has J ⊆ Ia (W ) ⊆ Ia (Y ). By Proposition 2.1.11 (i), one has therefore W = Za (J) = Za (Ia (W )) ⊇ Za (Ia (Y )), i.e. any closed subset containing Y contains also Za (Ia (Y )). In particular, Y ⊇ Za (Ia (Y )). ⊓ ⊔ We consider some easy examples where the inclusion Y ⊂ Y is strict. In all the examples below, we will consider K an infinite field. Example 2.1.26 (i) Let Y := A1 \ {0}. Then Y = Za (x1 )c is an open set of A1 strictly contained in it. On the other hand Ia (Y ) = Ia (A1 ) = (0), since non-zero, non-constant polynomials f ∈ A(1) have at most finitely many roots in K. In particular Y = A1 . The same conclusion holds for e.g. any principal open set Ua (f ) ⊂ A1 , for any f ∈ A(1) \ K. (ii) Let T ⊂ K be an infinite subset of K and, for any integer n > 2, let T n denote the n-tuple product T × · · · × T ⊂ An . If U ( An is a subset containing T n , one has Ia (U ) = (0), as it follows from Theorem 1.3.14, and so U = An as in (i). The same conclusion holds when, e.g. K is either R or C, and U contains an open polydisk of the euclidean topology of An .

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By Proposition 2.1.11 (i) and (2.10), one is therefore left with inclusion– reversing maps: Ia (−)  {Subsets of An } −→ Ideals of A(n) Y → Ia (Y )

(2.11)

Za (−)  {Subsets of An } ←− Ideals of A(n) Za (J) ← J.

(2.12)

and

Remark 2.1.27 (i) By definition of Za (−), the map (2.12) is neither surjective nor injective. The non-surjectivity follows from Im(Za (−)) = Cna ( {Subsets of An } ; similarly, for any field K, the non–injectivity of the map (2.12) is a direct consequence of Proposition 2.1.11 (v), as it follows for any choice of ideal I which is not radical. (ii) Notice that, when K is not algebraically closed, injectivity of (2.12) fails also on radical ideals. Consider e.g. K = R and m = (x2 + 1) ( A(1) ; m is maximal so radical (cf. Lemma 1.1.2), on the other hand ∅ = Za (m) = Za ((1)). Similarly, for any prime p ∈ Z, the ideal I = (xp − x) ⊂ Zp [x] is a proper, radical ideal (cf. Remark 2.1.16 (ii)) on the other hand Za (I) = Za ((0)) = A1Zp . n (iii) Examples2.1.26 show that (2.11) is not injective; on the other hand, Ca (n) surjects onto Ideals of A .

An important task is to understand how domain and target of the previous maps Ia (−) and Za (−) have to be restricted in order to get bijectivity for both of them. A second task is to understand wether Ia (−) is the inverse map of Za (−). The results of the next sections will give answers to these questions, when K is algebraically closed (cf. Corollary 2.2.5).

2.2 Hilbert ”Nullstellensatz” In Proposition 2.1.11 (iv) we proved that, for any field K and for any P ∈ An , the ideal mP = (x1 −p1 , . . . , xn −pn ) ⊂ A(n) is maximal. The next fundamental result will show that, when K is algebraically closed, all maximal ideals of A(n) are of this form. Theorem 2.2.1 (Hilbert ”Nullstellenstaz”-weak form) Let K be an algebraically closed field. Then m ⊂ A(n) is a maximal ideal ⇔ m = mP , for some P = (p1 , . . . , pn ) ∈ An .

2.2 Hilbert ”Nullstellensatz”

63

Remark 2.2.2 (i) The assumption K algebraically closed is essential for the (1) (⇒)-part. Indeed, taking e.g. m1 = (x2 + 1) ⊂ AR and m2 = (x2 + x + 1) ⊂ (1) AZ2 , one gets maximal ideals in both cases but none of them is of the form prescribed by Theorem 2.2.1. (ii) Theorem 2.2.1 can be viewed as the anologue in A(n) , for any integer n > 1 and for any algebraically closed field K, of the Fundamental theorem of (1) Algebra for AC (recall Theorem 1.0.1). Indeed, if I ⊂ A(n) is a proper ideal then I ⊆ m, for some maximal ideal m ⊂ A(n) ; on the other hand, m = mP for some P ∈ An , since K is algebraically closed therefore ∅ = 6 Za (I) ⊇ Za (mP ) = {P }, as it follows from Proposition 2.1.11 (i) and (iv). In other words, if K is algebraically closed, Za (I) = ∅ ⇔ I = (1). In both examples in (i), one has instead ∅ = Za (m1 ) ⊂ A1R and ∅ = Za (m2 ) ⊂ A1Z2 even if the ideals are maximal. Proof (Proof of Hilbert ”Nullstellensatz”-weak form). (⇐) This is Proposition 2.1.11 (iv), which holds for any field K. (⇒) Let m ⊂ A(n) be any maximal ideal; then F := A(n) /m is a field. The compositon of K-algebra homomorphisms ι

π

K ֒→ A(n) ։ F, where ι the natural inclusion and π the canonical projection, gives rise to a K-algebra homomorphism ϕ:K→F which is necessarily injective, since K is a field. ϕ Then K ֒→ F is a field extension. Since F is, by construction, also a Kalgebra of finite type then, by Zariski’s Lemma (cf. Lemma 1.7.2), the field extension K ⊂ F is algebraic of finite degree. Thus F = K, since K is algebraically closed. Posing pi := π(xi ) ∈ F = K, for any 1 6 i 6 n, then mP := (x1 − p1 , . . . , xn − pn ) ⊆ Ker(π) = m. On the other hand, mP is maximal in A(n) , therefore equality must hold.

⊓ ⊔

Theorem 2.2.3 (Hilbert ”Nullstellenstaz”-strong form) Let K be an algebraically closed field and let n > 1 be any integer. For any ideal I ⊆ A(n) one has √ Ia (Za (I)) = I. Remark 2.2.4 The assumption K algebraically closed is essential for the √ equality Ia (Za (I)) = I. Indeed, taking e.g. A1R one has ∅ = Za ((1)) = (1) Za ((x2 + 1)) even if (x2 + 1) is radical since maximal in AR . Similarly, in (1) A1Zp , p ∈ Z a prime, Za ((0)) = Za ((xp − x)) even if (xp − x) is radical in AZp (recall Remark 2.1.16 (ii)).

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Proof (Proof of Hilbert ”Nullstellensatz”-strong form). For simplicity denote I is a proper ideal. √Y := Za (I). We can assume that√ Notice that I ⊆ Ia (Y ). Indeed, for any g ∈ I there exists a positive integer s = s(g) such that g s ∈ I so, for any P ∈ Y , 0 = g s (P ) = (g(P ))s which implies g ∈ Ia (Y ), since K is a field. √ We now prove the other inclusion Ia (Y ) ⊆ I. Let g ∈ Ia (Y ) be any element; by Noetherianity of A(n) , let I = (f1 , . . . , fm ), where fi ∈ A(n) suitable non-constant polynomials. Consider y another indeterminate over K; in the polynomial ring A(n+1) := K[x1 , . . . , xn , y] ∼ = A(n+1) x,y consider the ideal J := (f1 , . . . , fm , yg − 1).

Since g ∈ Ia (Y ), then Za (J) = ∅ ⊂ An+1 . The fact that K is algebraically closed implies that J = (1), as it follows from the Hilbert ”Nullstellensatz”weak form. (n+1) Thus, there exist polynomials q1 , . . . , qm , p ∈ Ax,y such that 1 = q1 f1 + · · · + qm fm + p(yg − 1),

(2.13)

(n+1)

as a polynomial identity in Ax,y . If one poses y = g −1 in (2.13) (this is the so called Rabinowithch’s trick), one gets 1 = q˜1 f1 + · · · + q˜m fm

(2.14)

where q˜1 , . . . , q˜m ∈ K[x1 , . . . , xn , g −1 ]. Taking any sufficiently large positive integer N and multiplying both members of (2.14) by g N yields ∗ g N = q1∗ f1 + · · · + qm fm ,

with qi∗ ∈ A(n) , for 1 6 i 6 m. This implies g N ∈ I so g ∈ √ I.

√

I, i.e. Ia (Y ) ⊆ ⊓ ⊔

Important consequences of the previous results are the following. Corollary 2.2.5 Let K be any algebraically closed field. Then: (i) for any subset Y ⊆ An , one has Ia (Y ) = Ia (Y ); (ii) the maps (2.11) and (2.12) induce bijections: 1−1

n {(Zariski) closed S subsets of A } ←→ n

{Points of A }

1−1

←→





(n) Radical ideals S of A

Maximal ideals of A



(n)



(2.15)

2.2 Hilbert ”Nullstellensatz”

65

Proof. (i) By Proposition 2.1.25, for any subset Y ⊆ An , Y = Za (Ia (Y )). On the other hand, one also has Y = Za (Ia (Y )), i.e. Za (Ia (Y )) = Za (Ia (Y )). Since K is algebraically closed, by the Hilbert Nullstellensatz-strong form we q p have Ia (Y ) = Ia (Y ) which implies Ia (Y ) = Ia (Y ) since, by definitions, they are both radical ideals. (ii) The correspondence between points in An and maximal ideals is the content of Hilbert ”Nulstellensatz”-weak form. The bijective correspondence between radical ideals of A(n) and (Zariski) closed subsets of An follows from the fact that, if I is a radical ideal, by the Hilbert ”Nulstellensatz”-strong form one has Ia (Za (I)) = I whereas, if Y is (Zariski) closed then Za (Ia (Y )) = Y , as it follows from Proposition 2.1.25. ⊓ ⊔ Remark 2.2.6 (a) In Chapter 4, we will see that part (i) of Corollary 2.2.5 more generally holds for any field K. Indeed, this will be a direct consequence of Theorem 1.3.14 and the fact that Y is dense in its closure Y . (b) The algebraic counter-part of the Hilbert ”Nullstellensatz”-strong form is the following: Claim 2.2.7 If K is any algebraically closed field and I ⊆ A(n) is any ideal, then \ \ √ m = I = mP . m maximal, m⊇I

P ∈Za (I)

Proof (Proof of Claim 2.2.7). The second equality follows from the Hilbert Nullstellensatz-weak form. For the first let MI := {m ⊂ A(n) maximal √ √ equality, T ideal |m ⊇ I}. If √ m ∈ MI then I ⊆ m = m (cf. Lemma 1.1.2), so I ⊆ MI m. To prove the other inclusion, observe that since K is algebraically closed n any maximal ideal of A(n) is of the form mT P , for P ∈ A , as the Hilbert Nullstellenstaz-weak form states. Thus, f ∈ MI m if and only if f (P ) = 0 √ for any P ∈ Za (I), i.e. if and only if f ∈ Ia (Za (I)) = I, the last equality following from Hilbert Nullstellensatz-strong form. ⊓ ⊔ Proposition 2.2.8 Let K be any algebraically closed field. Then (2.15) is an inclusion-reversing bijective correspondence for which the following further properties hold: (i) for any closed subsets Y1 , Y2 ∈ Cna , one has Ia (Y1 ∪ Y2 ) = Ia (Y1 ) ∩ Ia (Y2 ). The same holds for any finite union of closed subsets of An ; (ii) for any collection {Yα }α∈A of closed subsets, one has

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Ia

\

α∈A

Yα

!

=

sX

Ia (Yα ).

α∈A

Proof. Statement about inclusion-reversing property directly follows from Proposition 2.1.11 (i) and formula (2.10).  (i) By definition Ia (Y1 ) ∩ Ia (Y2 ) := f ∈ A(n) | Y1 , Y2 ⊆ Za (f ) , which coincides with Ia (Y1 ∪ Y2 ). (n) (ii) Let Yα := Z Ta (Jα ), for Jα ⊆ A P radical ideal for any α ∈ A. By Proposition 2.1.11, Ia ( α∈A Yα ) = Ia (Za ( α∈A Jα )). Since K is algebraically closed, pP by the Hilbert Nullstellensatz-strong form, the latter equals α∈A Jα , as desiderd. ⊓ ⊔ Remark 2.2.9 (a) The inclusion-reversing property stated in Proposition 2.2.8 is true for any field K, for any pair of ideals I ⊆ J in A(n) and for any pair of subsets Y1 ⊆ Y2 in An (cf. Proposition 2.1.11 (i) and formula (2.10)). (b) Property (i) more generally holds for any field K and for any pairs Y1 , Y2 of subsets of An , as the proof of Proposition 2.2.8 shows. pP (c) The use of α∈A Ia (Yα ) in part (ii) is unavoidable, even if Ia (Yα ) is a radical ideal for any α ∈ A. For example, for any field K, I1 = (x21 − x2 ) and I2 = (x21 + x2 ) are both radical ideals in A(2) , since they are prime ideals; on the other hand I1 + I2 is not radical as x21 = 21 (x21 − x2 ) + 21 (x21 + x2 ) ∈ I1 + I2 but x1 ∈ / I1 + I2 . (d) From Example 2.1.19, when K is algebraically closed (2.15) implies that hypersurfaces in An are in 1 − 1 correspondence with principle radical ideals in A(n) (which are not maximal when n > 2). Recalling the decomposition (2.7), the closed subsets Za (f1 ), . . . , Za (fℓ ) are called the irreducible components of the hypersurface Za (f ) = Za (fred ). These are the hypersurfaces corresponding to the irreducible factors of fred in A(n) ; notice that, since A(n) is a UFD, each such irreducible factor is also a prime element in A(n) , therefore it generates a prime ideal in A(n) . The closed subset Za (fi ) ⊂ An is also called an irreducible hypersurface in n A , for any 1 6 i 6 ℓ. Thus, (2.15) implies that irreducible hypersurfaces in An are in 1 − 1 correspondence with principle prime ideals in A(n) .

2.3 Some consequences of Hilbert Nullstellensatz and of Elimination theory We will dicuss some nice consequences of the machinery developed up to this point.

2.3 Some consequences of Hilbert Nullstellensatz and of Elimination theory

67

2.3.1 Study’s principle In this section, we will focus on the case of K an algebraically closed field. Theorem 2.3.1 (Study’s principle) Let K be algebraically closed and let n > 1 be an integer. Let f, g ∈ A(n) be non-constant polynomials, with f irreducible. If g(p1 , . . . , pn ) = 0 for any P = (p1 , . . . , pn ) ∈ Za (f ), then f divides g in A(n) . Proof. First proof (using Hilbert Nullstellensatz-strong form). By the assumptions on f and g, one has g ∈ Ip a (Za (f )). Since K is algebraically closed, from Theorem 2.2.3, Ia (Za ((f ))) = (f ). On the other hand, since f ∈ A(n) is irreducible and A(n) is a UFD, then f is prime. Thus, (f ) is prime so radical (cf. Lemma 1.1.2). Thus g ∈ (f ), i.e. f |g. Second proof (using resultants and elimination). By assumption on f , up to a permutation of the indeterminates we can assume f to be of the form as in (1.12), with ai ∈ K[x1 , . . . , xn−1 ], 0 6 i 6 d, d > 0 and ad 6= 0. We claim that g cannot be constant with respect to xn . Indeed if, by contradiction, g ∈ K[x1 , . . . , xn−1 ], from the facts that K is infinite, ad , g 6= 0, and from Theorem 1.3.14, it would follow there exist (p1 , . . . , pn−1 ) ∈ An−1 s.t. g(p1 , . . . , pn−1 ), ad (p1 , . . . , pn−1 ) 6= 0. Thus,

φf (xn ) := f (p1 , . . . , pn−1 , xn ) = a0 (p1 , . . . , pn−1 ) + · · · + ad (p1 , . . . , pn−1 )xdn is a polynomial in K[xn ] of degree d, since ad (p1 , . . . , pn−1 ) 6= 0. From the fact that K is algebraically closed, it follows there exists α ∈ K such that φf (α) = f (p1 , . . . , pn−1 , α) = 0. On the other hand, by assumption g(p1 , . . . , pn−1 ) 6= 0, g would not vanish at (p1 , . . . , pn−1 , α), contradicting the hypotheses in the statement and proving what claimed. Since f and g are both non-constant with respect to xn , we can consider their resultant polynomial R := Rxn (f, g) ∈ K[x1 , . . . , xn−1 ] with respect to xn (recall § 1.3.4). From Proposition 1.3.1, we get R = Af + Bg, for some polynomials A, B ∈ K[x1 , . . . , xn ] with degxn (A) < degxn (g) and degxn (B) < degxn (f ), where degxn (−) denotes the degree w.r.t. the indeterminate xn . By the assumptions on f and g, for any P = (p1 , . . . , pn ) ∈ Za (f ), one has R(p1 , . . . , pn−1 ) = 0. Reasoning as done above for the polynomial g, from Theorem 1.3.14 it follows that R = 0. Thus applying Theorem 1.3.17 to the polynomial R, with D = K[x1 , . . . , xn−1 ], we conclude that f and g have a non-constant, common factor in K[x1 , . . . , xn ]. Since f is irreducible by assumption, then f |g. ⊓ ⊔

Remark 2.3.2 If K is not algebraically closed, Study’s principle does not hold. Take e.g. f, g ∈ R[x1 ] any two distinct, non-constant, irreducible polynomials; then Za (f ) = Za (g) = ∅ even if neither f |g or g|f . Similar considerations can be done e.g. in the ring Z2 [x1 ].

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2.3.2 Intersections of affine plane curves Another interesting application of elimination theory, is related to intersection of affine plane curves (i.e. hypersurfaces in A2 ) and classification of proper (Zarisky) closed subsets of A2 . In what follows, the term curve is always meant to be an affine (possibly reducible) plane curve. Theorem 2.3.3 Let K be any infinite field. The intersection of two curves, if not empty, consists of either finitely many points, or of a curve, or of a union of a curve and of finitely many points. From the previous result, it immediately follows Corollary 2.3.4 For any infinite field K, non-empty, proper closed subsets in Zar2a consist only of finitely many points, curves and unions of a curve and finitely many points. Proof (Proof of Theorem 2.3.3). Let Za (f ), Za (g) be two curves, with f, g ∈ A(2) non-constant polynomials. If f and g have a common, non-constant factor h ∈ A(2) , the curve Za (h) is contained in the intersection Za (f, g) of the two originary curves. Therefore, from now on we will assume that f and g have no common, non-constant factor. If both f and g are constant with respect to the same indeterminate, e.g. x1 , then Za (f, g) = ∅: if f and g have some powers of linear factors in their own factorizations, these factors represent lines which are parallel to the x1 axis and, by the assumptions on f and g, one concludes. If otherwise f is constant with respect to x1 whereas g is constant with respect to x2 , then Za (f, g) consists of at most finitely many points (Za (f, g) = ∅ if no powers of linear factors appear in the factorization of either f or g), and we are done also in this case. We can therefore assume that f and g are both non-constant with respect to the same indeterminate, e.g. x2 . By the assumption on f and g, the resultant R := Rx2 (f, g) ∈ K[x1 ] is non-zero (cf. Theorem 1.3.17 and § 1.3.4). Thus if (p1 , p2 ) ∈ Za (f, g), one must have R(p1 ) = 0 as it follows from Theorem 1.3.17 applied to f ′ (x2 ) := f (p1 , x2 ), g ′ (x2 ) = g(p1 , x2 ) ∈ K[x2 ], which must have (x2 − p2 ) ∈ K[x2 ] as a common, non-constant factor. Since R ∈ K[x1 ] has at most finitely many roots in K, one can have at most finitely many choices for p1 ∈ K. If we make same considerations with respect to the other indeterminate, we arrive at the same conclusion for p2 ∈ K. Thus, Za (f, g) consists of at most finitely many points. ⊓ ⊔ As a consequence of the previous results, we have Theorem 2.3.5 (Bezout’s theorem (weak-form)) Let Z1 = Za (f ) and Z2 = Za (g) be curves in A2 , with f, g ∈ A(2) \ K. If φ := g.c.d.(f, g) in A(2) then

2.4 Further remarks: K0 -rational points

69

Z1 ∩ Z2 = Za (φ) ∪ Z3 , where Z3 is either empty or a finite set of points.

2.4 Further remarks: K0 -rational points Even if in the next chapters we will focus on K algebraically closed (and, in the contrary case, we will always say so), this does not mean that algebraic geometry does not apply to studying questions concerned with algebraically non-closed fields K0 . However, applications of this kind most frequently involve passing to an algebraically closed field containing K0 . In the case of R, one usually passes to C, which often allows one to guess or to prove purely real relations. Some other situations in which questions arise involving algebraic geometry over a non-algebraically closed field K0 , and whose study usually requires passing to a bigger field (or even algebraically closed) K ⊃ K0 are e.g. the following: (a) K0 = Q: the study of e.g. points of an algebraic plane curve Za (f ), where (2) (2) f ∈ AQ ⊂ AR , in such a way that the coordinates of points we want to study are in A2Q instead those in A2R or in A2C . This is one of the fundamental problems in number theory (the so called Diophantine problems). For example, Fermat’s last theorem requires to describe points (p1 , p2 ) ∈ A2Q of the affine plane curve xn1 + xn2 = 1, ∀ n > 3. (b) K0 = Zp , for some p ∈ Z prime: studying the points with coordinates in K0 on the algebraic curve given by e.g. a polynomial f ∈ Z[x1 , x2 ], is another problem of number theory, concerning the solution of the congruence f ≡0

(mod p).

(c) K0 = K(x3 ): if one consider a (hyper)surface Z ⊂ A3 , this is given by (3) the vanishing locus of a polynomial f ∈ AK . By putting e.g. the indeterminates x3 into the coefficients, i.e. f ∈ K[x1 , x2 , x3 ] = (K[x3 ])[x1 , x2 ] ⊂ (K(x3 ))[x1 , x2 ], one can consider Z = Za (f ) as a curve defined over the field K(x3 ). As already observed in this book, this sometimes is a technical step which allows one to deduce several important properties on the surface itself; moreover, this approach is an extremely fertile method in studying algebraic surfaces. Examples (a)-(c) above justify the following Definition 2.4.1 Let K0 ⊂ K be fields and let Y be an AAS given by Za (f1 , . . . , fm ) ⊂ AnK ,

(2.16)

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2 Algebraic Affine Sets (n)

where each fi ∈ AK0 . (2.16) is called a system of algebraic equations defining Y , the field K0 is called the field of definition of Y whereas K is called the coordinate field of Y . The set Y (K0 ) := Y ∩ AnK0 is called the set of K0 -rational points of Y . For example, Fermat’s last theorem deals with the description of Q-rational points of the affine curve xn1 + xn2 = 1, for any n > 3. Notice that in general, as K0 varies in all sub-fileds of a given algebraically closed field K, the sets Y (K0 ) can have extremely different properties. For example, in Remark 2.1.16 (iii) we gave the full description of Y = Za (x21 + x22 + 1) ⊂ A2C ; on the other hand Y (R) = Y (Q) = ∅. In Example 2.4.2 we will describe extremely simple complex plane curves Y as conics such that, either Y (R) is a real plane algebraic curve and Y (Q) consists of infinitely many points or Y (R) is still a plane algebraic curve but Y (Q) = ∅. Example 2.4.2 [Some parametrized curves] (i) Q-rational points of the unit circle and the Pythagorean triples: integral triples satisfying the equation X12 + X22 = X02 are the so called Pythagorean triples. How do you find all integral solutions? The equation is quadratic and homogeneous so X0 = 0 implies also X1 = X2 = 0, therefore all non-trivial integral solutions gives rise to Q-rational points of the circle Y := Za (x21 + x22 − 1) ⊂ A2R ,

X2 X1 , x2 = X with X0 6= 0. Notice that Y contains the integral where x1 = X 0 0 point P = (1, 0) ∈ Y , i.e. a point with integral coordinates. The pencil of lines of A2 through P is given by

ℓm : x2 − 1 = mx1 , m ∈ R. Considering the movable part of the intersection Y ∩ ℓm , as m varies in R, one gets x1 (m) = −

1 − m2 x1 (m) 2m x2 (m) = , where m = . 2 2 1+m 1+m 1 − x2 (m)

This gives all integral solutions of the original problem: X1 = −2λµ, X2 = λ2 − µ2 , X0 = λ2 + µ2 , with λ, µ ∈ Z coprime, or each divided by 2 when λ and µ are both odd (cf. e.g. [27, § 1.1]). The previous construction is nothing but a linear projection of the real unit circle from the point P ∈ Y onto an affine line A1R , whose points are in

2.5 Exercises

71

1 − 1 correspondence with the slope m ∈ R of the lines ℓm of the pencil. Using this projection, we notice that the parametric equation x1 (m) and x2 (m) are rational functions defined over Q; for m varying, the line ℓm and Y (both defined over Q) meets in the fixed point P and in a further point Qm = (x1 (m), x2 (m)) ∈ Y . So when m ∈ Q, this gives all Q-rational points of the unit circle (cf. [27, § 1.1]). To sum-up, points Qm in Y (Q) are in bijective correspondence with rational points in A1 ∋ m; in particular, Y (Q) is infinite. (ii) On the contrary, consider the real circle Z = Za (x21 + x22 − 3) ⊂ A2R , whose defining equation is with coefficient in Z, therefore its field of definition is Q. If one performs the same construction as in (i) above, where the projection is √ taken from the point P ′ = (0, 3) ∈ Z, one obtains real parametrizations √ √ 1 − m2 2 3m x2 (m) = 3 , m ∈ R. x1 (m) = − 2 1+m 1 + m2 Claim 2.4.3 Y (Q) = ∅. Proof. One is reduced to prove that (a, b, c) ∈ Z3 s.t. a2 + b2 = 3c2 ⇒ (a, b, c) = (0, 0, 0). By the fact that the equation in a quadratic form, we can assume that c 6= 0 and that 3 does not divide all a, b and c. Now, if 3|a and 3|b, then 9|3c2 , i.e. 3|c2 so 3|c a contradiction to our assumption. We are therefore reduced to understanding when a, b ∈ Z3 , with (a, b) 6= (0, 0) the reduction modulo 3 of a and b respectively, are such that 2 a2 + b = 0 in Z3 . Since 2 is not a square in Z3 , the previous equation in Z3 has only the trivial solution, which concludes the proof. ⊓ ⊔ For further readings about K0 -parametrizations of plane algebraic curves and connections with their K0 -rational points, the reader is referred e.g. to [29, § 1.2] and to [27, Appendix to Chapter 1].

2.5 Exercises Exercise 2.5.1. If K is a finite field show that, for any n > 1, every subset of An K is an AAS. Exercise 2.5.2. Show that the following sets are AAS’s: (a) {(cos(t), sin(t)) ∈ A2R | t ∈ R}; (b) the set of points in A2R whose polar coordinates (r, θ) satisfy the equation r = sin(θ) (cf. [12, Ex. 1.11, p.5]). Exercise 2.5.3. Show that the following sets are not AAS’s: (a) {(x1 , x2 ) ∈ A2R | x2 = sin(x1 )}; (b) {x1 , x2 ) ∈ A2C | |x1 | + |x2 | = 1}, where |z|2 := a2 + b2 for z = a + ib ∈ C; (b) {cos(t), sin(t), t) ∈ A3R | t ∈ R} (cf. [12, Ex. 1.11, p.5]).

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2 Algebraic Affine Sets

Exercise 2.5.4. Let K be an infinite field and let f1 , . . . , fc ∈ A(n) be polynomials of degree 1 which are linearly independent over K. Assume that either Za (f1 , . . . , fc ) 6= ∅ or c = n+1 (in which case Za (f1 , . . . , fc ) = ∅). Prove there exists an automorphism of the K-algebra A(n) which maps the ideal (f1 , . . . , fc ) to (xn , . . . , xn−c+1 ). Exercise 2.5.5. Notation and assumptions as in Exercise 2.5.4. Deduce that (f1 , . . . , fc ) is a prime ideal in A(n) . Exercise 2.5.6. Let K be an infinite field. Let Y := {(t, t2 ) ∈ A2 | t ∈ K}. Show that Y is an AAS and find explicit generators of Ia (Y ). Show moreover that Ia (Y ) is a prime ideal. What happens to Ia (Y ) if K = Zp , for some prime p ∈ Z? Exercise 2.5.7. If K is a finite field show that, for any n > 1, every subset of An K is an AAS. Exercise 2.5.8. Let K be an algebraically closed field and let Y = {(0, 0), (1, 0)}. Determine Ia (Y ). Show that A(2) /Ia (Y ) is isomorphic to K[t]/(t2 − t). Deduce that Ia (Y ) cannot be prime. Exercise 2.5.9. Let K be an infinite field and let Y := Za (x21 − x2 x3 , x1 x3 − x1 ) ⊂ A3K . Prove that Ia (Y ) is not prime. Exercise 2.5.10. Let K = R. Show that Ia (Za (x21 + x22 + 1)) = (1). Exercise 2.5.11. Let K = R. Show that every algebraic subset of A2 (R) is equal to Za (f ), for some f ∈ R[x1 , x2 ] (cf. [12, Ex. 1.30, p.10]). 2

Exercise 2.5.12. Let K be an infinite field. Identify Am with the set of all m × m K matrices with entries in K. Show that, for any non negative integer r 6 m, the 2 subset of matrices with rank at most r is an AAS of Am K . Exercise 2.5.13. Let K be algebraically closed. Let Y ⊂ A3 be the union of the three coordinate axes. Prove that Ia (Y ) cannot be generated by less that 3 elements although Y has codimension 2 in A3 . Exercise 2.5.14. Let K be algebraically closed. Let Y ⊂ A4 be the union of the two planes Y ′ = Za (x1 , x2 ) and Y ′′ = Za (x3 , x1 − x4 ). Determine I := Ia (Y ) ⊂ A(4) .

Exercise 2.5.15. Notation as in Exercise 2.5.14. For any a ∈ K, let Ia ⊂ K[x1 , x2 , x3 ] be the ideal obtained by substituting t = a in I above and let Ya := Za (Ia ). Show that the family of AAS’s Ya , a ∈ K, describes two skew lines in A3 approaching each other, until they finally intersect transversally for a = 0. Show that √ the ideals Ia are radicals for a 6= 0 but that I0 is not. Find moreover elements in I0 \ I0 .

2.5 Exercises

73

Exercise 2.5.16. Let K be algebraically closed and let f1 , . . . , fk ∈ A(n) . Show that the following are equivalent: (a) (f1 , . . . , fk ) = (1); (b) the principal open sets Ua (f1 ), . . . , Ua (fk ) determine an open covering of An ; (c) for any point P ∈ An , fi (P ) 6= 0 for some i ∈ {1, . . . , k}. Exercise 2.5.17. Let K = C. Consider in A(3) the three ideals I1 = (x1 x2 + x22 x1 x3 + x2 x3 ), I2 = (x1 x2 + x22 x1 x3 + x2 x3 + x1 x2 x3 + x22 x3 ) and

I3 = (x1 x22 + x32 , x1 x3 + x2 x3 ).

(a) Does Ik = Il , for some k 6= l ∈ {1, 2, 3}? (b) Does Za (Ik ) = Za (Il ), for some k 6= l ∈ {1, 2, 3}?(cf. [17, Ex. 3, p.58]). Exercise 2.5.18. Find a parametrization of the conic C := Za (2x21 +x22 −5) ⊂ A2 (R) and show that C(Q) = ∅ (cf. [27, § 1.2, p.20]).

3 Algebraic Projective Sets

From now on, unless otherwise stated, K will be always considered algebraically closed. Let V be a (n+1)-dimensional K–vector space. One can define the following relation on V \ {0}: v ∼ w ⇐⇒ ∃ t ∈ K∗ := K \ {0} s.t. w = tv. It is easy to see that this is an equivalence relation, which is called proportionality. We will denote by [v] the equivalence class of v ∈ V \ {0}. The quotient set (V \ {0}) P(V ) := ∼ is called the projective space associated to V . We will denote by πV (or simply π if no confusion arises) the canonical projection (3.1) π : V \ {0} ։ P(V ). The integer n will be called the dimension of the projective space, which is denoted by dim(P(V )) (the empty–set is considered as the projective space of dimension −1, which is associated to the vector space V = {0}). Elements of P(V ) will be called points. When in particular V = Kn+1 is the (n + 1)–dimensional numerical (standard) K-vector space, the associated projective space will be simply denoted by Pn and called the n–dimensional numerical projective space over K. If v = (v0 , . . . , vn ) ∈ Kn+1 is a non–zero numerical vector, its proportionality equivalence class [v] will be also denoted by [v0 , . . . , vn ]. (v0 , . . . , vn ) is called a vector of homogeneous coordinates for v. Thus, homogeneous coordinates of a point P ∈ Pn are such that: (a) not all of them equal zero, and (b) they are defined up to proportionality by a factor t ∈ K∗ . For any 0 6 i 6 n, we will denote by Pi ∈ Pn the point whose homogeneous coordinates are proportional to the components of the vector ei of

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3 Algebraic Projective Sets

the canonical basis of Kn+1 . The points Pi are called the vertices of the fundamental (n + 1)-hedron (or pyramid) of Pn . The point Pn+1 = [1, . . . , 1], is called the unit point of Pn ; the points Pi , with 0 6 i 6 n + 1, are called the fundamental points of Pn .

3.1 Algebraic Projective Sets Consider the projective space P(V ) of dimension n and fix a homogeneous element F ∈ S(V ∗ )d (recall Example 1.10.18 and § 1.10.1). Remark 3.1.1 Once we fix a (ordered) basis on V , we have a natural identification of Kn+1 with V . This induces a projectivity ϕ : Pn → P(V ), which gives homogeneous coordinates on P(V ). Recalling notation (1.29), for any d > 0, in this case S(V ∗ ) naturally identifies with S(n) by means of the isomorphism ϕ e : S(n) −→ S(V ∗ )

induced by the choice of the (ordered) basis in V . If F ∈ S(V ∗ )d , then ϕ e−1 (F ) is a homogeneous polynomial of degree d. Therefore F vanishes at P ∈ P(V ) (where P the point corresponding to the equivalence class [P ] = [p0 , . . . , pn ] of the vector P ∈ V and the homogeneous coordinates [p0 , . . . , pn ] are w.r.t. the given basis of V ) means that ϕ e−1 (F )(p0 , . . . , pn ) = 0 which holds if and −1 only if ϕ e (F )(tp0 , . . . , tpn ) = 0, for any t ∈ K∗ (recall (1.31)). One says −1 that ϕ e (F )(X) = 0 is a homogeneous equation of Z(F ) in Pn , where X := (X0 , . . . , Xn ) indeterminates. If one introduces two different sets of homogeneous coordinates on P(V ), ϕ : Pn → P(V ),

ψ : Pn → P(V ),

S(V ∗ ) is identified to S(n) in two different ways. In other words, there are two isomorphisms ϕ e : S(n) → S(V ∗ ), ψe : S(n) → S(V ∗ )

determined by the two distinct bases for V . If X = (X0 , . . . , Xn ) and Y := (Y0 , . . . , Yn ) denote the induced indeterminates, respectively, and if A denotes the matrix associated to the base change in V , the map ωA := ψe−1 ◦ ϕ e : S(n) → S(n)

is an isomorphism of graded rings, which sends ϕ e−1 (F )(X) to the polynomial

ωA ϕ e−1 (F )(X) := ψe−1 (F ) Y At

where X t = A Y t . This isomorphism depends on A, so it is determined up to a non-zero proportionality factor.

3.1 Algebraic Projective Sets

77

Any such isomorphism of S(n) is said to be a homogeneous linear substitution of the indeterminates (cfr. Exercise 1.12.15). In particular the homogeneous equation ψe−1 (F )(Y ) = 0 can be deduced from ϕ e−1 (F )(X) = 0 by means of a homogeneous linear substitution in the indeterminates, which preserves the degrees. From what discussed above and from (1.31), differently from the affine case an element F ∈ S(V ∗ )d does not determine a function on P(V ). On the other hand, it makes sense to ask for F ∈ S(V ∗ )d to vanish at P : if P = [p] ∈ P(V ), one has F (p) = 0 if and only if F (tp) = 0, for any t ∈ K∗ . In such a case, the point P is called a zero of F . Given F ∈ S(V ∗ )d , therefore it makes sense to consider Zp (F ) ⊆ P(V ), which is called the zero–set of F in P(V ).

Remark 3.1.2 (i) Given any identification of P(V ) with Pn as above, any polynomial f ∈ S(n) of degree d can be uniquely decomposed into its homogeneous components f = F0 + . . . + Fd , (n)

where Fi ∈ Si , 0 6 i 6 d. Since K has infinitely many elements (Corollary 1.3.13), by Theorem 1.3.14 f vanishes at P ∈ Pn (in the above sense) if and only if all of its homogeneous components Fi do, 0 6 i 6 d. (ii) When K is finite, what stated in (i) does not hold in general. Take e.g. K = Zp , x := x0 and f = xp − x ∈ Zp [x]: f vanishes at all a ∈ Zp (cf. Example 2.1.10) but its homogeneous components vanish only at 0 ∈ Zp . Recalling notation (1.24), from Remark 3.1.2 (i) one poses: Definition 3.1.3 For any subset T ⊆ S(n) , the subset of Pn \ Zp (T ) := Zp (F ), F ∈H(T )

is called the zero-set of T in Pn . The subscript p in the above definition stands for projective, to distinguish it from the affine case considered in Definition 2.1.1. Notice that Zp (T ) = Zp ((H(T ))), (3.2) where (H(T )) ⊆ S(n) is a homogeneous ideal (cf. Proposition 1.10.6). Since S(n) is Noetherian, there exist F1 , . . . , Fm ∈ H(T ) s.t. (H(T )) = (F1 , . . . , Fm ); thus, as in Definition 2.1.4, the set {F1 , . . . , Fm } is called a system of homogeneous equations defininig Zp (T ). In particular, one has

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3 Algebraic Projective Sets

Definition 3.1.4 A subset Z ⊆ Pn is called an Algebraic Projective Set (APS, for short), if there exist polynomials F1 , . . . , Fm ∈ H(S(n) ) s.t. Z = Zp (F1 , . . . , Fm ); equivalently, Z = Zp (I) for a homogeneous ideal I ⊆ S(n) . More generally, Z ⊆ P(V ) is a APS if there exists a subset T of S(V ∗ ) such that Z = Zp (T ). Recalling Propositions 2.1.11 and 1.10.7, one has Proposition 3.1.5 (i) Let T1 ⊆ T2 be any subsets of S(n) ; then one has Zp (T2 ) ⊇ Zp (T1 ). In particular for any pair of homogeneous ideals I1 ⊆ I2 in S(n) , then Zp (I1 ) ⊇ Zp (I2 ). (ii) For any homogeneous ideals I1 , I2 ⊆ S(n) , Zp (I1 ∩ I2 ) = Zp (I1 ) ∪ Zp (I2 ). (iii) For any family {Iα }α∈A of homogeneous ideals in S(n) , ! \ X Zp (Iα ) . Iα = Zp α∈A

α∈A

Proof. The proof easily follows by using similar strategies as in the proofs of Proposition 2.1.11 (i), (ii) and (iii). ⊓ ⊔ From the previous proposition, Cnp := {APS’s in Pn }

(3.3)

is the set of closed subsets of a topology on Pn . This topology will be called the Zariski topology on Pn , which will be denoted by ZarPn (or simply Zarnp ). Proposition 3.1.5 can be stated more generally for APS’s in P(V ), using S(V ∗ ) instead of S(n) . Therefore, CVp will denote the set of APS’s of P(V ), which is the set of closed subsets of the Zariski topology on P(V ). This topology will be denoted by ZarP(V ) . Open sets in the Zariski topology will be complementary sets of APS’s. In particular, for any F ∈ S(V ∗ )d , Up (F ) = Zp (F )c

(3.4)

is called a principal open set of P(V ). As in Lemma 2.1.21, principal open sets form a basis for ZarP(V ) . Any non-empty subset Y ⊆ P(V ) will be endowed, from now on, with the topology induced on Y by ZarP(V ) . This will be called the Zariski topology

3.2 Homogenoeus ”Hilbert Nullstellensatz”

79

of Y and will be denoted by ZarP(V ),Y (respectively, Zarnp,Y or simply ZarY , when the inclusion Y ⊆ Pn is clearly fixed). From Remark 3.1.1, in what follows we will focus for simplicity on the case of Pn ; statements can be easily extended to P(V ), replacing S(n) with S(V ∗ ). As in the affine case, from Remark 3.1.2 (i) and the Noetherianity of S(n) , one has: Definition 3.1.6 For any subset Y ⊆ Pn , n o Ip (Y ) := f ∈ S(n) | f (P ) = 0, ∀ P ∈ Y

is a homogeneous, finitely generated ideal, which is called the homogeneous ideal of Y in S(n) . Same strategies used in the affine case show the following

Proposition 3.1.7 (i) For any subsets Y1 ⊆ Y2 ⊆ Pn , Ip (Y1 ) ⊇ Ip (Y2 ); (ii) For any subsets Y1 , Y2 ⊆ Pn , Ip (Y1 ∪ Y2 ) = Ip (Y1 ) ∩ Ip (Y2 ).

3.2 Homogenoeus ”Hilbert Nullstellensatz” From the results in the previous section, as in the affine case, one has natural inclusion–reversing maps

and

Ip (−)  {Subsets of Pn } −→ Homogeneous ideals of S(n) Y → Ip (Y )

(3.5)

Zp (−)  {Subsets of Pn } ←− Homogeneous ideals of S(n) Zp (I) ← I.

(3.6)

To better understand their behaviour on (Zariski) closed subsets and to prove the homogeneous version of Hilbert ”Nullstellensatz”, we need to first introduce some useful definitions. Take notation as in (3.1). Definition 3.2.1 For any subset Y ⊆ Pn , let  Ca (Y ) := π −1 (Y ) ∪ {0} = (p0 , . . . , pn ) ∈ Kn+1 \ 0 | [p0 , . . . , pn ] ∈ Y ∪ {0}. This subset of An+1 is called the affine cone over Y .

Remark 3.2.2 (i) It is clear from the definition that Ca (Y ) = Ca (Y ′ ) ⇔ Y = Y ′ . Moreover, for any subset Y ⊆ Pn , by definition of affine cone one has

(3.7)

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3 Algebraic Projective Sets

Ia (Ca (Y )) = Ip (Y ).

(3.8)

(ii) By Proposition 2.1.25 and by (3.8), one has a

Ca (Y ) = Za (Ia (Ca (Y ))) = Za (Ip (Y )), where

a

Ca (Y )

denotes the (affine) closure in An+1 . Since Ip (Y ) is a homogeneous ideal, then p

Za (Ip (Y )) = Ca (Zp (Ip (Y )) = Ca (Y ), where Y

(3.9)

p

denotes the (projective) closure of Y ⊆ Pn . In particular, a

p

Ca (Y ) = Ca (Y ). From this equality and from (3.7), it follows that Ca (Y ) ∈ Cn+1 if and only if a Y ∈ Cnp and that the map π in (3.1) is continuous. As in the affine case, one has

√ Corollary 3.2.3 For any homogeneous ideal I ⊆ S(n) , Zp (I) = Zp ( I). Proof. It directly follows from Remark 3.2.2 and from Proposition 2.1.11 (v). ⊓ ⊔ Differently from the affine case (cf. (2.15)), even if K is algebraically closed and even if one restricts the maps (3.5) and (3.6) to the sets of homogeneous radical ideals and of (Zariski) closed subsets of Pn , one still does not have bijective correspondences. Indeed, one always has Zp ((1)) = ∅; on the other hand the following result, which can be considered as the homogeneous analogue of the Hilbert ”Nullstellensatz”-weak form Theorem 2.2.1, shows that this is not the only case even on radical ideals. Theorem 3.2.4 (Homogeneous Hilbert ”Nullstellenstaz”-weak form) Let I ⊂ S(n) be a proper, homogeneous ideal. Then √ I = S+ , (3.10) Zp (I) = ∅ ⇔ where S+ is the irrelevant maximal (so radical) ideal of S(n) (cf. Def. 1.10.3). Proof. First √ notice that, by the maximality of S+ and by Lemma 1.1.2, condition I = S+ is equivalent to the fact that there exists an integer t > 1 s.t. St+ ⊆ I. Thus (3.10) reduces to showing:

3.2 Homogenoeus ”Hilbert Nullstellensatz”

Zp (I) = ∅ ⇔ ∃ t > 0 s.t. S+ t ⊆ I.

81

(3.11)

(⇐) Suppose St+ ⊆ I, for some positive integer t. Since (X0t , X1t , . . . , Xnt ) ⊆ St+ ⊆ I, by Proposition 3.1.5 (i), we have ∅ ⊇ Zp (St+ ) ⊇ Zp (I). (⇒) Assume Zp (I) = ∅, with I = (F1 , . . . , Fm ), any Fi a homogeneous polynomial. One can assume that not all polynomials are linear forms, otherwise the assumption Zp (I) = ∅ would give I = S+ and we are done. Using notation as in Definition 1.10.14, consider fi := δ0 (Fi ), 1 6 i 6 m; Zp (F1 , . . . , Fm ) = ∅ implies Za (f1 , . . . , fm ) = ∅. Since K is algebraically closed, by Theorem 2.2.1, one has therefore (f1 , . . . , fm ) = (1) ⊂ A(n) . Thus there exists polynomials g1 , . . . , gm ∈ A(n) such that 1=

m X

fi gi .

(3.12)

i=1

Recall that h0 (fi )|Fi , for any 1 6 i 6 m (more precisely the two polynomials coincide if and only if X0 does not divide Fi ). Hence, there exists a nonnegative integer si s.t. X0si h0 (fi ) = Fi , for any 1 6 i 6 m. This implies there exists a positive integer N0 such that (3.12) becomes X0N0 =

m X i=1

Fi G i ∈ I

(3.13)

where Gi = h0 (gi ). Reasoning in the same way for all the other indices j = 1, . . . , n, one deduces that there also exist positive integers N1 , . . . , Nn such that also X1N1 , X2N2 , . . . , XnNn ∈ I. Let N := Max{N0 , N1 , . . . , Nn }. By the assumptions on the degrees of the Fi ’s, one has N > 2. Take t := (n + 1)(N − 1) + 1 > n + 1.

Pn Thus, any monomial of the form X0a0 X1a1 · · · Xnan , with i=0 ai > t, is such that there exists at least one aj for which aj > N > Nj , for some a1 t an j ∈ {0, . . . , n}, i.e. X0a0 X 1 · · · Xn ∈ I. Since S+ is generated by monomials P n a0 a1 X0 X1 · · · Xnan , where i=0 ai = t, one can conclude. ⊓ ⊔

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3 Algebraic Projective Sets

To sum-up the ideals (1) and S+ both map to the empty-set via n o Z (−) p Homogeneous ideals of S(n) −→ Cnp ,

whereas S+ is not in the image of the map n o Ip (−) Cnp −→ Homogeneous ideals of S(n) ,

since Ip (∅) = (1). On the other hand, one has

Theorem 3.2.5 (Homogeneous Hilbert ”Nullstellenstaz”-strong form) Let I ⊂ S(n) be a homogeneous ideal such that Zp (I) 6= ∅. Then √ Ip (Zp (I)) = I. Proof. From Remark 3.2.2, we have Ip (Zp (I)) = Ia (Ca (Zp (I))) = Ia (Za (I)). Since K is algebraically closed, from Theorem 2.2.3, the latter equals desired.

√

I, as ⊓ ⊔

From the previous results, as for Corollary 2.2.5 (ii), one has Corollary 3.2.6 The maps (3.5) and (3.6) induce bijections: o n 1−1 Radical ideals of S(n) \ {S+ }. Cnp ←→

3.3 Fundamental examples and remarks 3.3.1 Points When K is algebraically closed, (2.15) gave bijective correspondence between points in An and maximal ideals mP ⊂ A(n) ; the situation is different for points in Pn . As in the affine case, points are still closed sets in Zarnp ; indeed, if P = [p0 . . . . , pn ] ∈ Pn with e.g. pi 6= 0, then {P } = Zp (pi X0 − p0 Xi , . . . , pi Xn − pn Xi ). On the other hand, the homogeneous ideal I := (pi X0 − p0 Xi , . . . , pi Xn − (n) pn Xi ) ⊂ S(n) is prime but not maximal, as S I ∼ = K[Xi ] is an integral domain but not a field. Using Theorem 3.2.5 and Example 2.1.18, one can easily show the following more precise result.

3.3 Fundamental examples and remarks

83

Proposition 3.3.1 Let I ⊂ S(n) be a radical ideal. Then Zp (I) = {P }, for some point P ∈ Pn , if and only if I is generated by linearly independent linear (n) forms L0 , · · · , Ln−1 ∈ S1 . In particular, Zarnp is T1 and Cnp contains all finite subsets of Pn . Moreover, from Proposition 1.10.17, when n = 1 these (together with the empty–set) are the only proper closed subsets of P1 . As in the affine case Zar1p is not T2 ; as for the affine case, this will more generally hold for any Zarnp , n > 1 (cf. Remark 4.1.1). 3.3.2 Coordinate linear subspaces Let 0 6 m 6 n be integers. For any sequence of integers 0 < i0 < i1 < · · · < im 6 n, one has an injective map ϕ := ϕi1 ,i2 ,i3 ,··· ,im : Pm −→ Pn defined as follows: [p0 , p1 , . . . , pm ] −→ [0, . . . , 0, p0 , 0, . . . , 0, p2 , 0, . . . , 0, pm , 0, . . . , 0) i0 i1 im . As in Example 2.1.17, ϕ is a homeomorphism between (Pm , Zarm p ) and Im(ϕ) ⊂ Pn endowed with the induced topology by Zarnp . This image is the closed subset defined by the homogeneous linear system Xj = 0, ∀j 6= i0 , . . . , im which is called a m-dimensional coordinate linear subspace of Pn . For m = 0 we get the points Pi which are the vertices of the fundamental pyramid of Pn ; for m = 1 we get coordinate axes of Pn . For m = n − 1, we get fundamental hyperplanes of Pn . The hyperplane given by Xj = 0 will be also denoted by Hj , 0 6 j 6 n. 3.3.3 Hyperplanes and the dual projective space Let V be a (n + 1)-dimensional K-vector space. If we consider V ∗ := Hom(V, K), then P(V ∗ ) is called the dual projective space of P(V ). Recalling Example 1.10.18 and Remark 3.1.1, the choice of an (ordered) basis (e0 , . . . , en ) for V allows one to identify P(V ) with Pn and V ∗ with (n) S1 = K[X0 , . . . , Xn ]1 , where (X0 , . . . , Xn ) the dual basis of (e0 , . . . , en ). For (n) simplicity, P(S1 ) is denoted by (Pn )∗ . Since any hyperplane of Pn is of the form Ha := Zp (a0 X0 + . . . + an Xn ), for some a ∈ Kn+1 \ {0}

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3 Algebraic Projective Sets

and since Ha = Hta , for any t ∈ K∗ , it is clear that (Pn )∗ is identified with the set of hyperplanes in Pn , the correspondence given by [a] ↔ Ha . In this correspondence, the fundamental hyperplane Hi ⊂ Pn corresponds to the vertex Pi∗ of the fundamental pyramid of (Pn )∗ , 0 6 i 6 n. 3.3.4 Fundamental affine open sets (or affine charts) of Pn Consider the principal open set Up (Xi ) = Hic = Pn \ Hi , 0 6 i 6 n. We denote it by Ui := {[p0 , . . . , pn ] ∈ Pn | pi 6= 0}. (3.14) S n n Since P = i=0 Ui , then {U0 , . . . , Un } is a (finite) open covering of Pn , where each open set is principal (cf. (3.4)). Similarly, Pn can be also written as a disjoint union Pn = Ui ∪ Hi , 0 6 i 6 n, where each Hi is homemorphic to Pn−1 . For any i ∈ {0, . . . , n}, consider now the map   pi−1 pi+1 pn p0 φi ,..., , ,..., ∈ An , P = [p0 , . . . , pn ] ∈ Ui −→ pi pi pi pi

(3.15)

which is well-defined. Proposition 3.3.2 For any i ∈ {0, . . . , n}, the map φi is a homemorphism. Proof. We refer to the case i = 0, since the case i > 0 can be proved similarly. Put φ0 = φ and U0 = U . It is clear that φ is bijective, whose inverse is given by φ−1 (c1 , . . . , cn ) = [1, c1 , . . . , cn ]. It suffices to prove that φ and φ−1 are closed maps. For the map φ, notice that for any closed subset Y ⊆ Pn , Y ∩U is closed in U . Therefore, to show that φ is closed, it suffices to prove that φ(Zp (F ) ∩ U ) is closed in An , for any F ∈ H(S(n) ); this is obvious since, by definition of φ, φ(Zp (F ) ∩ U ) = Za (δ0 (F )). Similarly, to show that φ−1 is closed it suffices to show that for any nonconstant polynomial g ∈ A(n) , the set φ−1 (Za (g)) is closed in U . By the definition of φ one observes that φ−1 (Za (g)) = Zp (ho (g)) ∩ U which is closed in U , so φ−1 is closed. ⊓ ⊔

3.3 Fundamental examples and remarks

85

Remark 3.3.3 (i) Since any Ui is homeomorphic to An , the principal open sets Ui are also called principal open affine sets, or even affine charts, of Pn . Thus, Pn has a finite affine open covering. (ii) The projective space Pn can be therefore viewed as an extension of An , by identifying An with any of the affine charts Ui , i.e. An ֒→ Pn (c1 , c2 , . . . , cn ) −→ [c1 , c2 , . . . ci , 1, ci+1 , . . . cn ] (in the sequel, for simplicity, we will usually identify An with the affine chart U0 ). In this identification, the fundamental hyperplane Hi ⊂ Pn will be called the hyperplane at infinity of Ui and each point P ∈ Hi is called a point at infinity (or improper point) of Ui , 1 6 i 6 n. (iii) More generally, for any subset Y ⊂ Pn , Yi := Y ∩ Ui is an open subset of Y , 0 6 i 6 n, and {Y0 , . . . , Yn } is an open covering of Y . If in particular Y is moreover closed in Pn , say Y = Zp (I) for some homogeneous ideal I, then Yi is closed in Ui , 0 6 i 6 n. Indeed, by Noetherianity I = (F1 , . . . , Fm ), for some Fj ∈ H(S(n) ). From the proof of Proposition 3.3.2, one has Yi = Za (δi (F1 ), . . . , δi (Fm )). (3.16) In particular any Yi is a fundamental affine open set of Y (since it is open in Y and homemorphic to an affine closed set) and contemporarly it is locally closed in Pn , i.e. it is the intersection of a closed set and of an open set of Pn . 3.3.5 Projective closure of affine sets As already mentioned above, we will usually identify An with U0 via the map φ0 . In this case, for any subset Y ⊂ An , its closure in Pn , denoted by Y

p

will be called the projective closure of Y . Moreover, one poses p

Y∞ := Y ∩ H0

(3.17)

and call it the set of points at infinity (or improper points) of Y ⊂ An . Suppose that Y ∈ Cna ; let therefore I = (f1 , . . . , fm ) ⊂ A(n) be such that Y = Za (I). Differently from (3.16), in general the m polynomials obtained by p homogenizing the generators of I are not sufficient to determine Y , as the following easy example shows.

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3 Algebraic Projective Sets

Example 3.3.4 Let A2 = U0 and consider Y = Za (I) ⊂ A2 , where I = (x1 , x2 + x21 ). One has h0 (x1 ) = X1 , h0 (x2 + x21 ) = X0 X2 + X12 and let I = (h0 (x1 ), h0 (x2 +x21 )) = (X1 , X0 X2 +X12 ), which is a homogeneous ideal. Notice that Y is the origin O ∈ A2 , indeed I = mO , as x2 = (x2 + x21 ) − x1 (x1 ) ∈ I. The point O is also closed in P2 (corresponding to the fundamental vertex p P0 = [1, 0, 0] of P2 ). Thus, O = O = P0 whereas Zp (I) = {[0, 0, 1], [1, 0, 0]}. p In particular, Za (I) ( Zp (I). Proposition 3.3.5 Let Y = Za (J) ⊆ An be a closed subset, for some ideal J ⊂ A(n) . Identify An with the affine chart U0 ⊂ Pn and define the homogeneous ideal
J∗ := h0 (f ), ∀ f ∈ J ⊆ S(n) (3.18) p

Then Y = Zp (J∗ ).

Proof. By definition of J∗ , Zp (J∗ ) is a closed set of Pn containing Y . Then p Y ⊆ Zp (J∗ ). To prove the opposite inclusion, let W ⊆ Pn be any closed subset containing Y . For any F ∈ H(Ip (W )), the polynomial δ0 (F ) vanishes along Y , i.e. Y ⊆ Za (δ √0 (F )). By Theorem 2.2.3, since K is algebraically closed, one has δ0 (F ) ∈ J, i.e. there exists an integer r > 1 such that δ0 (F )r ∈ J. Assume deg(F ) = d and let deg(δ0 (F )) := d − s, for some integer 0 6 s < d (recall that s = 0 if and only if X0 does not divide F ). Then F = X0s h0 (δ0 (F )) and so F r = X0sr h0 (δ0 (F ))r . r From Lemma 1.10.15 (ii), h0 is multiplicative so h0 (δ0 (F ))r = h√ 0 (δ0 (F ) ). ∗ r ∗ ∗ By definition of J , this implies that F ∈ J . Thus Ip (W ) ⊆ J ; from Proposition 3.1.5 (i) and Corollary 3.2.3, it follows that p W = Zp (Ip (W )) ⊇ Zp ( J∗ ) = Zp (J∗ );

in other words, any projective closed subset containing Y contains also Zp (J∗ ). p ⊓ ⊔ Since this must hold also for Y , one has proved the other inclusion. Notice the following nice consequences of the previous result.

Remark 3.3.6 (i) With same notation and assumptions as in Proposition 3.3.5, take J := (f1 , . . . , fm ) ⊆ A(n) and let Fi := h0 (fi ) ∈ H(S(n) ), 1 6 i 6 p m. Then, even if in general Y ⊆ Zp (F1 , . . . , Fm ), one has p

Y = Y ∩ U0 = Zp (F1 , . . . , Fm ) ∩ U0 ,

(3.19)

as it follows from the proof of Proposition 3.3.2 (therein we showed that φ−1 0 (Y ) = Zp (F1 , . . . , Fm ) ∩ U0 ). (ii) If Y ⊆ An is closed, then

p

Y = Y ∪ Y∞ .

(3.20)

3.3 Fundamental examples and remarks

87

Proof. By the induced topology Zarnp,An , the closure of Y in U0 = An is p given by Y ∩ U0 . On the other hand, since Y is already closed in U0 = An , p one has Y = Y ∩ U0 . ⊓ ⊔ Notice that the previous proof is another way to see that any closed subset of An is locally closed in Pn (cf. Remark 3.3.3 (iii)) (iii) For any subset Y ⊆ Pn , one has

p

Ip (Y ) = Ip (Y )

(3.21) p

Proof. Let indeed F ∈ H(S(n) ) such that Y ⊆ Zp (F ); then Y ⊆ Zp (F ) p p p so Y = Zp (Ip (Y )). On the other q hand, one has also Y = Zp (Ip (Y )). By p p p Theorem 3.2.5, one has Ip (Y ) = Ip (Y ) and so Ip (Y ) = Ip (Y ), since by definition both ideals are radicals. ⊓ ⊔ A different proof follows by Remark 3.2.2 and Corollary 2.2.5 (i); indeed from Remark 3.2.2 one has a

p

p

Ip (Y ) = Ia (Ca (Y )) = Ia (Ca (Y ) ) = Ia (Ca (Y )) = Ip (Y ). (iv) If Y ⊂ U0 is a finite set of points, then clearly Y∞ = ∅. When K is not algebraically closed, it may happen that Y∞ = ∅ even if Y is not reduced to a finite sets of points. Consider e.g. the circle C = Za (x21 + x22 − 1) ⊂ A2R ; its projective closure is the non-degenerate conic Zp (X12 + X22 − X02 ) ⊂ P2R so that C∞ = Zp (X0 , X12 + X22 ) = ∅. The same conic, considered instead in A2C , is such that C∞ = {[0, 1, i], [0, 1, −i]}, with i2 = −1 (cf. Corollary 6.2.6 for more general results). p

(v) Identifying An with U0 , let An denote the projective closure of An in Pn . Then p A n = Pn , i.e. An is dense in Pn . (n)

Proof. For any integer d > 1, let F ∈ Sd be any homogeneous polynomial such that An ⊆ Zp (F ). Up to a permutation of the indeterminates, we can assume that F contains a monomial proportional to X0d . Therefore, we can write F as F = X0d F0 + X0d−1 F1 + . . . + Fd , (n−1)

where Fi ∈ Si

= K[X1 , . . . , Xn−1 ]i , 0 6 i 6 d. We get δ0 (F ) = F0 + F1 + . . . Fd ∈ A(n) ,

(3.22)

which is the decomposition of δ0 (F ) in its homogeneous components. By the assumptions on F , one has An = U0 ⊆ Za (δ0 (F )). Since K is infinite (because

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3 Algebraic Projective Sets

algebraically closed), by Theorem 1.3.14, we get that δ0 (F ) = 0. Thus, Fi = 0, for any 0 6 i 6 d, and so also F = 0. Therefore Zp (F ) = Zp ((0)) = Pn , as desired. ⊓ ⊔ When K is finite, the previous statement is false. Indeed, for any prime p and for any integer n > 1, AnZp is a union of finitely many points. 3.3.6 Projective subspaces and their ideals Let V be a (n + 1)-dimensional K-vector space and let W ⊂ V be a sub-vector space of dimension m + 1 > 0, with m < n. Then P(W ) is called (projective) subspace of P(V ) of dimension m and codimension c := n − m. The empty–set is the only subspace of dimension −1; points in P(V ) are subspaces of dimension 0; codimension-1 subspaces are hyperplanes (cf. § 3.3.3). The following statements are easy consequences of standard linear algebra: (i) Z ⊆ P(V ) is a subspace if and only if Z = Zp (L0 , . . . , Lh ), where L0 , . . . , Lh ∈ V ∗ = S(V ∗ )1 ; (ii) If L, L0 , . . . , Lh ∈ V ∗ , then Zp (L0 , . . . , Lh ) ⊆ Zp (L) if and only if L is linearly dependent from L0 , . . . , Lh in V ∗ ; (iii) Zp (L0 , . . . , Lh ) = Zp (G0 , . . . , Gk ) if and only if L0 , . . . , Lh and G0 , . . . , Gk span the same sub-vector space of V ∗ ; (iv) Zp (L0 , . . . , Lh ) = Zp (Fi1 , . . . , Fic ), where Fi1 , . . . , Fic is a basis of W ∗ := Span{L0 , . . . , Lh } ⊆ V ∗ and where c is the codimension of W ∗ in V ∗ ; (v) if Λ1 , Λ2 ⊆ P(V ) are subspaces s.t. Λi := P(Wi ) for i = 1, 2, then Λ1 ∩ Λ2 ⊂ P(V ) is a subspace, which is called the intersection subspace of the two subspaces; more precisely Λ1 ∩ Λ2 = P(W1 ∩ W2 ). (vi) Let S ⊂ P(V ) be any subset. Since the family LS of projective subspaces contaninig S is not empty, one can consider \ Λ. hSi := Λ∈LS

By (v), this is a subspace of P(V ); more precisely, it is the smallest subspace of P(V ) containing S. This is called the subspace generated by S (or even the linear envelope of S) in P(V ). S is said to be non-degenerate in P(V ) if hSi = P(V ), i.e. if S is not contained in any proper projective subspace. If Λ1 , . . . , Λh are subspaces, one uses the symbol Λ1 ∨ . . . ∨ Λh instead of hΛ1 ∪ . . . ∪ Λh i;

3.3 Fundamental examples and remarks

89

(vii) One has the (projective) Grassmann formula: for any subspaces Λ1 , Λ2 in P(V ) one has dim(Λ1 ) + dim(Λ2 ) = dim(Λ1 ∨ Λ2 ) + dim(Λ1 ∩ Λ2 ).

(3.23)

The proof is a straightforward application of linear Grassmann formula for subspaces in V . (viii) A direct consequence of (vii) is the following result. Let dim(V ) = n + 1. If Λ1 , Λ2 are projective subspaces in P(V ) such that dim(Λ1 ) + dim(Λ2 ) > n then Λ1 ∩ Λ2 6= ∅. This is a natural generalization of the well-known fact that two lines in the projective plane always intersect. It is moreover clear that, by the existence of parallelism, no Grassmann formula can exist for affine subspaces in An defined by non-homogeneous linear systems (cf. Example 2.1.18 and Exercise 3.4.5) ). (ix) Given (m + 1)-points P0 , P1 , . . . , Pm ∈ P(V ), by Grassmann formula, one has dim(P0 ∨ P1 ∨ . . . Pm ) 6 m. The points are said to be linearly independent (or even in general linear position) if equality holds. It is then clear that, for any m-dimensional subspace Λ in P(V ), with m 6 n, there exist (m+1) points P0 , P1 , . . . , Pm ∈ Λ which are linearly independent, i.e. such that Λ = P0 ∨ P1 ∨ . . . Pm . Fixing an (ordered) basis of V induces a choice of homogeneous coordinates on P(V ), which identifies P(V ) with Pn and Λ = P0 ∨ P1 ∨ . . . Pm with an APS of the form Zp (L0 , . . . Lc−1 ) ⊂ Pn , where c is the codimen(n) sion of Λ in Pn and where L0 , . . . , Lc−1 ∈ S1 . It is easy to see that the natural map φ

Pm −→ Λ ⊂ Pn ,

defined by

φ

[λ0 , . . . , λm ] −→

m X

λi Pi ,

(3.24)

i=0

n is a homeomorphism between (Pm , Zarm p ) and (Λ, Zarp,Λ ) (this extends what discuss in § 3.3.2). The map φ is called a parametric representation of Λ. (x) If Λ ⊆ P(V ) is a subspace, one defines

Λ⊥ := {[L] ∈ P(V )∗ | Λ ⊆ Zp (L)}. This is a subspace of P(V )∗ , which is called the orthogonal to Λ. One has that dim(Λ⊥ ) = codim(Λ) = c. (xi) One easily shows the following properties: (Λ⊥ )⊥ = Λ,

⊥ (Λ1 ∨ Λ2 )⊥ = Λ⊥ 1 ∩ Λ2 ,

⊥ (Λ1 ∩ Λ2 )⊥ = Λ⊥ 1 ∩ Λ2 .

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3 Algebraic Projective Sets

As for homogeneous ideals defining subspaces, let Λ be a subspace of codimension c in the projective space Pn . By (i) and (iv) above, Λ = (n) Zp (L0 , . . . , Lc−1 ), where L0 , . . . , Lc−1 linearly independent in S1 . Claim 3.3.7 The ideal I := (L0 , . . . , Lc−1 ) is prime. Assuming for a moment the content of the claim, one gets that I is radical (cf. Lemma 1.1.2) thus it coincides with Ip (Λ). In other words, subspaces Λ ⊂ Pn , of codimension c, are APS’s of Pn whose homogeneous ideal Ip (Λ) is generated by c linearly independent linear forms (cf. Example 2.1.18 for the affine case). Proof (Proof of Claim 3.3.7). Since L0 , . . . , Lc−1 are linearly indepen(n) dent linear forms, there exist Lc , . . . , Ln ∈ S1 such that {L0 , . . . , Lc−1 , Lc , . . . , Ln } (n)

is a basis for S1

(cf. (1.28)). The map (n)

ϕ : S1

(n)

→ S1 , defined by Xi → Li , (n)

is an automorphism of the vector space S1 which extends to an authomorphism of the K-algebra S(n) , always denote by ϕ, by the rule: ϕ(f (X0 , . . . Xn )) = f (L0 (X), . . . , Ln (X)), where we posed for brevity X = (X0 , . . . Xn ) (notice that ϕ preserves the graduation of S(n) , cf. Exercise 1.12.16). In particular, ϕ bijectively maps the maximal ideal S+ to the ideal (L0 (X), . . . , Ln (X)) and, consequently, the ideal (X0 , . . . , Xc−1 ) to I. Since S(n) /(X0 , . . . , Xc−1 ) ∼ ⊓ ⊔ = S(n−c) , it follows that I is prime. By using Example 1.10.18 and Remark 3.1.1, one can more generally state previous result for linear subspaces in P(V ), replacing S(n) with S(V ∗ ). 3.3.7 Projective and affine subspaces. Projective closure of affine subspaces As usual, we will identify An with the affine open set U0 of Pn and we will denote by H0 the hyperplane at infinity for U0 . Let Λ ⊂ Pn be any non-empty subspace of codimension c > 0 and denote by Λ0 := Λ ∩ U0 . If Λ ⊆ H0 , it is clear that Λ0 = ∅; therefore, we may assume that Λ is not contained in H0 . In such a case, from § 3.3.6, Λ is defined by c linearly independent, homogeneus, linear equations of the form:

3.3 Fundamental examples and remarks

a10 X0 + . . . + a1n Xn = 0 ...... ...... ...... ac0 X0 + . . . + acn Xn = 0

91

(3.25)

The previous linear system can be written as A∗ X t = 0, where A∗ is a c×(n+ 1) matrix with entries in K, of maximal rank c, whereas X := (X0 , . . . , Xn ) and 0 the (c × 1)-matrix with zero entries. From (3.16), Λ0 is defined by a11 x1 + . . . + a1n xn + a10 = 0 ...... ...... ...... ac1 x0 + . . . + acn xn + ac0 = 0 where as customary xi = (3.26) can be written as

Xi X0 ,

(3.26)

1 6 i 6 n. The non-homogeneous linear system A · xt + a = 0,

where A is the (c × n)-matrix obtained by selecting the last n columns and all the rows of A∗ , a the (c × 1)-matrix defined by the first column of A∗ , whereas x := (x1 , . . . , xn ). From the assumptions on A∗ , the rank of A is c; in particular, (3.26) is compatible and Λ0 6= ∅ is an affine subspace in the sense of Example 2.1.18. The homogeneous linear system obtained by adding the equation X0 = 0 to (3.25) defines the subspace Λ ∩ H0 of dimensione m − 1, where m = n − c. This subspace is called the direction of Λ0 . Let ξ 0 be a solution of (3.26), i.e. a point P0 ∈ Λ0 . Let moreover ξ 1 , . . . , ξ m be independent solutions of the homogeneous linear system associated to (3.26). The bijective map φ0 : (λ1 , . . . , λm ) ∈ Am → ξ 0 + λ1 ξ 1 + . . . + λm ξ m ∈ Λ0 is the restriction to Am = U0 ⊂ Pm of the map φ : Pm → Λ as in (3.24), when in Λ we choose the m + 1 points P0 , P1 , . . . , Pm with relative homogeneous coordinates [1, ξ 0 ], [0, ξ i ], 1 ≤ i ≤ m,

n respectively. Thus φ0 is a homeomorphism between (Am , Zarm a ) and (Λ0 , Zarp,Λ0 ) and it is a parametric representation of Λ0 (cf. Example 2.1.18). The map φ0 can be interpreted as a way to introduce a system of coordinates in Λ0 , in such a way that the point P0 coincides with its origin. Conversely, from § 3.3.5, for any affine subspace Λ0 as in (3.26) it follows that Λ as in (3.25) is its projective closure. As for § 3.3.6 (vi), one has therefore a natural notion of non-degenerate subset S of an affine space An .

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3 Algebraic Projective Sets

3.3.8 Homographies, projectivities, affinities and subspaces Let V and W be two K-vector spaces. A map ϕ : P(V ) → P(W ) is called a homography if there exists an injective linear map f : V → W s.t. πW ◦ f = ϕ ◦ πV , where πV and πW as in (3.1); i.e. ϕ([v]) = [f (v)] for any v ∈ V \ {0}. Notice that the existence of a homography ϕ : P(V ) → P(W ) implies dim(P(V )) 6 dim(P(W )); moreover, it is straighforward to verify that the composition of homographies is still a homography. As a matter of notation, we will put ϕ = ϕf when we want to stress that the homography ϕ depends on the linear map f . Let Hom(V, W ) be the vector space of linear maps from V to W . It is easy to see that ϕf = ϕg ⇐⇒ [f ] = [g]

in P(Hom(V, W )).

(3.27)

Thus the set of homographies from P(V ) to P(W ), denoted by O(P(V ), P(W )), identifies with the subset of P(Hom(V, W )) whose elements represent equivalence classes of injective linear maps from V to W . A homography ϕ = ϕf is called projectivity if f : V → W is an isomorphism; in such a case, P(V ) and P(W ) are said to be projectively isomorphic. Notice that homographies in O(P(V ), P(V )) are all projectivities of P(V ) onto itself. In particular, O(P(V ), P(V )) forms a group with respect to the composition, which is denoted by PGL(V ). This group is the image of the linear group GL(V ) under the canonical quotient map πEnd(V ) as in (3.1). If in particular P(V ) and P(W ) are projectively isomorphic, then GL(V ) ∼ = GL(W ) and PGL(V ) ∼ = PGL(W ). If V = Kn+1 , then GL(V ) and PGL(V ) will be denoted by GL(n + 1, K) and PGL(n + 1, K), respectively; the first group identifies with the group of non-degenerate, (n + 1) × (n + 1) matrices with entries in K whereas the second one with the quotient of GL(n + 1, K) modulo its center, i.e. modulo the group of scalar matrices of the form tIn+1 , where t ∈ K∗ and In+1 the identity matirx of order n + 1. If V has dimension n + 1, a projectivity ϕ : Pn → P(V ) assigns to any P ∈ P(V ) a proportionality class [p0 , . . . , pn ] of a numerical vector in Kn+1 ; in other words, ϕ : Pn → P(V ) can be viewed as introducing a system of homogeneous coordinates or a projective frame on P(V ). In this correspondence, fundamental points in P(V ) are the images via ϕ of the fundamental points in Pn . In the projective frame on P(V ) given by ϕ, the fact that P has homogeneous coordinates [p0 , . . . , pn ] will be denoted by P =ϕ [p0 , . . . , pn ], or simply by P = [p0 , . . . , pn ] if no confusion arises. With the choice of two different projective frames ϕ : Pn → P(V ) and ψ : Pn → P(V ) on P(V ), there exists a non-degenerate, (n + 1) × (n + 1) matrix A with entries in K such that, for any P ∈ P(V ) with

3.3 Fundamental examples and remarks

93

P =ϕ [x] = [x0 , . . . , xn ] and P =ψ [y] = [y0 , . . . , yn ], then y t = A · xt , where A denotes any representative of the proportionality class [A] of matrices determined by A. Notice that [A] determines the projectivity ψ −1 ◦ ϕ ∈ PGL(n + 1, k). Let ϕ : P(V ) → P(W ) be a homography and assume dim(P(V )) = n, dim(P(W )) = m. If we introduce projective frames on P(V ) and on P(W ), the previous discussion shows there exists a (m + 1) × (n + 1) matrix A, of rank m + 1, which is defined up to an element of K∗ , such that ϕ(P ) = P ′ ⇔ y t = A · xt , where P = [x] e P ′ = [y]. Theorem 3.3.8 (Fundamental theorem of projectivities) Let P1 and P2 be two projective spaces, both of dimension n. Consider (n + 2)–tuples of points (P0 , . . . , Pn+1 ) and (Q0 , . . . , Qn+1 ) in P1 and P2 , respectively, which are in general position (i.e. any (n + 1)–tuple of points contained in them are formed by linearly independent points). Then there exists a unique projectivity ϕ : P1 → P2 such that ϕ(Pi ) = Qi , 0 ≤ i ≤ n + 2. Proof. Left to the reader.

⊓ ⊔

Corollary 3.3.9 (Fundamental theorem of projective frames) Let P(V ) be a projective space of dimension n and let P0 , . . . , Pn+1 ∈ P(V ) be points in general position. Then, there exists a unique projective frame on P(V ) in such a way that the ordered (n + 2)–tuple (P0 , . . . , Pn+1 ) corresponds to the ordered (n+2)–tuple formed by fundamental points and the unit point of the projective frame. Let P(V ) be a projective space of dimension n. If Z := P(W ) is a mdimensional linear subspace of P(V ), to explicitely construct a homography ϕ : Pm → Z one can proceed as follows. Consider any basis w0 , . . . , wm of W and then the linearly independent points Pi = [wi ] ∈ P(V ), 0 ≤ i ≤ m. One has Z = P0 ∨ . . . ∨ Pm and one can construct the map ϕ : [λ0 , . . . , λm ] ∈ Pm → [λ0 w0 + . . . + λm wm ] ∈ Z which is a projectivity sending the (natural) fundamental points of Pm to P0 , . . . , Pm , respectively, and the unit point to [w0 +. . .+wm ]; the homography ϕ ∈ O(Pm , P(V )) as above is said to be a parametric representation of Z. In particular, if Z1 , Z2 are two m-dimensional linear subspaces of P(V ), the

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3 Algebraic Projective Sets

previous discussion shows there exists a projectivity of P(V ) into itself which sends Z1 onto Z2 . This also implies that, given any m-dimensional linear subspace Z of P(V ), there always exists a projective frame on P(V ) in such a way that Z = Zp (Xm+1 , . . . , Xn ). Proposition 3.3.10 Any homography ϕ : P(V ) → P(W ) is a continuous map in the Zariski topologies of the projective spaces P(V ) and P(W ), respectively. If in particular ϕ is a projectivity, then it is a homeomorphism. Proof. Let f : V → W be the injective linear map inducing ϕ; its transpose lienar map f t : W ∗ → V ∗ extends to a degree-zero, surjective, graded algebra homomorphism f t : S(W ∗ ) → S(V ∗ ).

For any G ∈ H(S(W ∗ )), one has ϕ−1 (Zp (G)) = Zp (f t (G)), which implies that ϕ is continuous. ⊓ ⊔

Proposition 3.3.11 A homography ϕ : P(V ) → P(W ) is a homeomorphism of P(V ) onto its image, which is a linear subspace of P(W ) of the same dimension of P(V ). Proof. The image of ϕ = ϕf is the linear subspace Z := P(f (V )) ⊆ P( W ). From Proposition 3.3.10, it therefore suffices to show that ϕ is closed. Consider the surjective algebra homomorphism f t : S(W ∗ ) → S(V ∗ ) introduced in the proof of Proposition 3.3.10. For any F ∈ H(S(V ∗ )), there exists G ∈ H(S(W ∗ )) such that f t (G) = F . Since ϕ−1 (Zp (G)) = Zp (f t (G)) = Zp (F ) (cf.the proof of Proposition 3.3.10), one has ϕ(Zp (F )) = Zp (G) ∩ Z, which implies that ϕ is closed. ⊓ ⊔

A map ψ : An → Am , with n 6 m, is called an affinity if there exists a homography Ψ : Pn → Pm such that, identifying An with the affine chart U0 , the restriction of Ψ to U0 coincides with ψ. In particular, any affinity is a homeomorphism onto its image and it sends affine subspaces of An to affine subspaces of Am of the same dimension. From § 3.3.7 and what discussed above, ψ : An → Am is an affinity if and only if there exist a (n×m)-matrix A of maximal rank n and a (m×1)-matrix a such that (3.28) ψ(xt ) = A · xt + a,

where x = (x1 , . . . , xm ). In particular, ψ is a homomorphism of K-vector space if and only if ψ(0t ) = 0t . Affinities of An onto itself form a group under composition, which is called the affine group of An and which is denoted by Aff(An ). It is then clear that elements of Aff(An ) transform affine subspaces to affine subspaces of the same dimension. In particular, if Z is any affine subspace of dimension m of An , there always exists an affinity in Aff(An ) which maps Z to the affine subspace Za (xm+1 , . . . , xn ).

3.3 Fundamental examples and remarks

95

3.3.9 Projective cones Let Z ⊂ Pn be a (Zariski) closed subset and let Ca (Z) ⊂ An+1 be its affine cone. We can always identify An+1 as an affine open subset of Pn+1 in such a way that the given Pn coincides with the hyperplane at infinity of An+1 . Thus, we can consider the projective closure of Ca (Z) in Pn+1 , which we will denote by Cp (Z) and call the projective cone over Z with vertex O, where O = [1, 0, . . . , 0] ∈ U0 = An+1 ⊂ Pn+1 . From (3.17) we have that Cp (Z)∩Pn = Ca (Z)∞ = Z ⊂ Pn . Therefore Cp (Z) = Ca (Z) ∪ Ca (Z)∞ = Ca (Z) ∪ Z, so Cp (Z) is the union of (projective) lines passing through O and through a point of Z. Finally one has Ip (Cp (Z)) = Ip (Z), (3.29) where the latter ideal is considered as a homogeneous ideal in S(n+1) . 3.3.10 Projective hypersurfaces and projective closure of affine hypersurfaces Similarly to Example 2.1.19, for any non-constant polynomial F ∈ H(S(n) ) we can consider its decomposition F = F1r1 F2r2 . . . Fℓrℓ as in (2.5), where F1 , . . . , Fℓ ∈ S(n) are all its non-proportional, irreducible factors (all homogeneous, from Prop. 1.10.13 (iii)) and where r1 , . . . , rℓ are positive integers. Then Y := Zp (F ) is called the projective hypersurface determined by F and the polynomial Fred = F1 F2 · · · Fℓ (3.30) is the reduced equation of Y . As in the affine case, deg(Fred ) is the degree of Y and Ip (Y ) = (Fred ) is its radical ideal. One can consider principal open (projective) sets Up (F ) and show that they form a basis for Zarnp (the proof is identical to that of Lemma 2.1.21). Moreover, using the Homogeneous Hilbert ”Nullstellensatz”-strong form (cf. Theorem 3.2.5), one can easily adapt the proof of Theorem 2.3.1 to prove: Theorem 3.3.12 (Homogeneous Study’s principle) Let F, G ∈ H(S(n) ) be non-constant polynomials and let Y = Zp (F ) the projective hypersurface defined by F . If Zp (G) ⊇ Y , then G is divided by Fred in S(n) . The previous result allows us to consider projective closures of affine hypersurfaces. Indeed one has:

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3 Algebraic Projective Sets

Proposition 3.3.13 Let Y = Za (f ) ⊂ An be an affine hypersurface, for some non-constant polynomial f ∈ A(n) . Identifying An with the affine chart U0 of Pn , one has
p Ip (Y ) = h0 (fred ) , where Y

p

the projective closure of Y and fred the reduced equation of Y .

Proof. Since Ia (Y ) = (fred ) (cf. Remark 2.2.9 (d)), we can directly assume that f = fred . From Proposition (3.3.5), p

Y = Zp (J∗ ),
where J∗ := h0 (g)| g ∈ (f ) . Since any g ∈ (f ) is of the form g = f h, for some h ∈ A(n) , and since h0 is multiplicative (cf. Lemma 1.10.15 (ii)), then
J∗ = h0 (f ) ,
p i.e. h0 (f ) ⊆ Ip (Y ). Notice that if f factors as in (2.7), the fact that h0 is multiplicative implies that h0 (f ) = h0 (f1 )h0 (f2 ) · · · h0 (fℓ ) where all the factors are square-free homogeneous polynomials, i.e. h0 (f ) is a reduced homogeneous polynomial. On the other hand, for any homogeneous polynomial G ∈ Ip (Y p ), one p has Zp (G) ⊇ Y . Thus, from the homogeneous Study’s principle, we have

p ⊓ ⊔ G ∈ h0 (f ) , i.e. Ip (Y ) ⊆ h0 (f ) and we are done. 3.3.11 Proper closed subsets of P2

As in Corollary 2.3.4, we can classify all proper closed subsets of the projective plane. In what follows, the term curve is always meant to be any projective hyspersurface in P2 . Proposition 3.3.14 Proper closed subsets in Zar2p consist only of finitely many points, curves and unions of a curve and finitely many points. Proof. If Z = ∅, there is nothing to prove. Therefore, let Z ⊂ P2 be any proper, non-empty closed subset. We can proceed as follows. First proof (using affine curves in A2 ). One has Z = Z1 ∪ Z2 , where Z1 := Z ∩U0 is a AAS in U0 ∼ = A2 whereas Z2 := Z ∩H0 = (Z1 )∞ , with H0 = Zp (X0 ) the line at infinity of U0 . Since Z2 ⊆ H0 and since H0 is homeomorphic to P1 , Z2 is either empty, or a finite number of points or it is the whole line H0 . As for Z1 , since it is closed in U0 ∼ = A2 , we can apply Corollary 2.3.4. Then ′ ′ ′ Z1 = Z1 ∪ Z2 , where Z1 either consists of finitely many points or it is empty and Z2′ either consists of finitely many affine curves or it is empty. Then one concludes by using §’s 3.3.1, 3.3.7 and Proposition 3.3.13. Second proof (using resultants and elimination). The proof follows the lines of that of Corollary 2.3.4. Indeed, Z = Zp (I), for some homogeneous ideal I ⊂

3.3 Fundamental examples and remarks

97

S(2) . By the Hilbert basis theorem, I = (F1 , . . . , Fm ) for some Fi ∈ H(S(2) ), 1 6 i 6 m, since I is homogeneous (cf. Proposition 1.10.6). If m = 1, Z is a projective curve and we are done. If m = 2 and F1 , F2 have a non-constant greatest common divisor G ∈ H(S(2) , then Fi = GAi , for some Ai ∈ H(S(2) ), 1 6 i 6 2. Then Zp (I) = Zp (G) ∪ Zp (A1 , A2 ), where Zp (G) is a projective curve, as above, and where A1 and A2 have no non-constant common factor. Therefore, we can reduce to F1 and F2 with no non-constant common factor. In such a case, a necessary condition for a point P = [p0 , p1 , p2 ] ∈ P2 to belong to Zp (F1 , F2 ) is that [p0 , p1 ] ∈ Zp (RX2 (F1 , F2 )), where RX2 (F1 , F2 ) is the resultant polynomial of F1 and F2 as in Theorem 1.10.16. By the assumption on F1 and F2 , this is a homogeneous polynomial of degree deg(F1 ) deg(F2 ) in the indeterminates X0 and X1 . By Proposition 1.10.17, one has only finitely many choices for [p0 , p1 ]. Applying the same procedure also with respect to the other indeterminates, one deduces that Zp (F1 , F2 ) consists of at most finitely many points. Recursively applying the previous arguments, one proves the general case for any possible m. ⊓ ⊔ 3.3.12 Affine and projective twisted cubics Let n > 2 be an integer, t an indeterminate over K and h2 (t), . . . , hn (t) ∈ K[t], not all of them constant polynomials. Consider C := {(t, h2 (t), . . . , hn (t)) ∈ An | t ∈ K} ⊂ An .

(3.31)

It is clear that C is the AAS defined by C = Za (x2 − h2 (x1 ), . . . , xn − hn (x1 )).


Lemma 3.3.15 One has Ia (C) = x2 − h2 (x1 ), . . . , xn − hn (x1 ) , which is a prime ideal in A(n) . Proof. Consider the K-algebra homomorphism ρn : A(n) → K[t]

defined by ρn (x1 ) = t, ρn (xi ) = hi (t), 2 6 i 6 n,

which is obviously surjective, so Ker(ρn ) is a prime ideal. Now g ∈ Ker(ρn ) if and only if 0 = ρn (g(x1 , . . . , xn )) = g(t, h2 (t), . . . , hn (t)), ∀t ∈ K, i.e. if and only if g ∈ Ia (C), which means Ia (C) = Ker(ρ n ).
The ideal J := x2 − h2 (x1 ), . . . , xn − hn (x1 ) is contained in Ker(ρn ) and is such that A(n) /J ∼ = K[x1 ] ∼ = K[t]. This implies that J = Ker(ρn ), as desired. ⊓ ⊔

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3 Algebraic Projective Sets

C is an affine rational curve with polynomial parametrization. In what follows, we will study a particular case of the previous set–up. Consider the injective map φ : t ∈ A1 → (t, t2 , t3 ) ∈ A3 .

(3.32)

From (3.31), C := Im(φ) is a rational curve with polynomial parametrization, which is called affine twisted cubic. From Lemma 3.3.15, the ideal Ia (C) = (f1 , g1 ), where f1 := x21 − x2 , g1 := x31 − x3 , is prime so radical (cf. Lemma 1.1.2). By straightforword computations, if one takes f2 := g1 − x1 f1 , one has Ia (C) = (f1 , f2 ) := (x2 − x21 , x3 − x1 x2 ).

(3.33)

Consider now the map ψ : [λ, µ] ∈ P1 → [λ3 , λ2 µ, λµ2 , µ3 ] ∈ P3

(3.34)

which is well-defined, injective and whose image Z is called projective twisted cubic. Notice that φ is the restriction of ψ to A1 ∼ = U0 ⊂ P1 , thus C ⊂ Z. More precisely Z = C ∪ {P }

where P = [0, 0, 0, 1] = ψ([0, 1]). The set Z is an APS in P3 ; indeed, one finds Z = Zp (F1 , F2 , F3 ), where F1 := X0 X2 − X12 , F2 := X0 X3 − X1 X2 , F3 := X1 X3 − X22 ,

(3.35)

are three irreducible, homogeneous polynomials which are linearly indepen(3) dent in S2 . These polynomials are determined by the maximal minors of the matrix of linear forms   X0 X1 X2 . (3.36) A := X1 X2 X3 For this reason Z is said to be determinantal and the condition rg(A) = 1 is said to be a matrix equation of Z. p

p

Claim 3.3.16 One has C = Z, where as usual C denotes the projective closure of C in P3 . Even if this is a consequence of a more general fact (cf. Remark 10.2.2 (iii)), we give here a constructive proof.

3.3 Fundamental examples and remarks

99

p

Proof (Proof of Claim 3.3.16). The inclusion C ⊆ Z is trivial. On the p other hand, if we consider any homogeneous G ∈ Ip (C ) and if we identify A3 with U0 ⊂ P3 , one has δ0 (G) ∈ Ia (C), i.e. G(1, t, t2 , t3 ) = 0 for any t ∈ K. Thus G(λ3 , λ2 µ, λµ2 , µ3 ) = 0, for any µ ∈ K and for any λ ∈ K∗ , which implies therefore that G(λ3 , λ2 µ, λµ2 , µ3 ) is identically zero, i.e. G ∈ Ip (Z). ⊓ ⊔ From (3.33) and (3.35), we notice that δ0 (F1 ) = f1 , δ0 (F2 ) = f2 , δ0 (F3 ) = f3 := x1 f2 − x2 f1 ∈ Ia (C). and viceversa h0 (f1 ) = F1 , h0 (f2 ) = F2 , h0 (f3 ) = F3 . Even if Ia (C) is generated by f1 and f2 , the polynomials F1 and F2 do not generate Ip (Z). More precisely, they do not even define Z as an APS, since Zp (F1 , F2 ) = Zp (h0 (f1 ), h0 (f2 )) = Z ∪ Zp (X0 , X1 ) Claim 3.3.17 Ip C

p


(3.37)

= (F1 , F2 , F3 ).

Proof. To prove the claim, we use the following notation B1 := F1 , B2 := −F2 , B3 := F3 , i.e. we consider the maximal minors of the matrix A in (3.36) with a suitable sign. Let us consider the free, graded S(3) -module
⊕3 M := S(3) (−2) ,

whose homogeneous degree-d part Md is the K-vector space ⊕3


S(3) (−2)

(3)

d

= (Sd−2 )⊕3

(cf. § 1.10.2). One is therefore reduced to proving that the homogeneous, degree-0 homomorphism φ

(G1 , G2 , G3 ) ∈ M −→ G1 B1 + G2 B2 + G3 B3 ∈ Ip C

p


is surjective. This is equivalent to showing that the induced K-vector space homomorphism φd

(G1 , G2 , G3 ) ∈ Md −→ G1 B1 + G2 B2 + G3 B3 ∈ Ip C

p


d

is surjective for any integer d. Let K := Ker(φ), which is still a graded S(3) -module. Its homogeneous, (3) degree-d piece Kd consists of triples (G1 , G2 , G3 ) ∈ (Sd−2 )⊕3 such that

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3 Algebraic Projective Sets

P3

i=1 Gi Bi = 0, i.e. Kd = Ker(φd ). These triples are called syzygies of (F1 , F2 , F3 ) and K is said to be the syzygy module. One trivially has Kd = {0} if d < 2; moreover K2 = {0} since, as observed (3) above, the polynomials F1 , F2 and F3 are linearly independent in S2 . On the other hand, a1 := (X0 , X1 , X2 ), a2 := (X1 , X2 , X3 )

are two linearly independent elements in K3 , as it clearly follows from the matrix A in (3.36). Consider the free, graded S(3) -module
⊕2 N := S(3) (−3)

and the degree-0, homogeneous homomorphism ψ

(D1 , D2 ) ∈ N −→ D1 a1 + D2 a2 ∈ K. We first show that ψ is an isomorphism of S(3) - graded modules. To prove this it suffices to show that, for any positive integer d, the induced K-vector space homomorphism φd

(3)

(3)

(D1 , D2 ) ∈ (Sd−3 )⊕2 −→ D1 a1 + D2 a2 ∈ Kd ⊆ (Sd−2 )⊕3 is an isomorphism. To prove this, consider any (G1 , G2 , G3 ) ∈ Kd . By Laplace’s rule and the definition of Bi , 1 6 i 6 3, this is equivalent to   G3 G2 G1 (3.38) det X0 X1 X2  = 0, X1 X2 X3

which implies that the three rows in (3.38) are linearly dependent over the field K(X0 , X1 , X2 , X3 ), whereas the rows of A are linearly independent, since the polynomials F1 = B1 , F2 = −B2 and F3 = B3 are non-zero. There exist therefore rational functions a1 , a0

b1 ∈ K(X0 , X1 , X2 , X3 ), b0

with a0 , a1 (respectively, b0 , b1 ) relatively prime elements, such that a1 X2 + a0 a1 G2 = X 1 + a0 a1 G2 = X 0 + a0 G1 =

b1 a1 b0 X2 + a0 b1 X3 X3 = b0 a0 b0 b1 a1 b0 X1 + a0 b1 X2 X2 = b0 a0 b0 b1 a1 b0 X0 + a0 b1 X1 X1 = b0 a0 b0

(3.39)

Since the Gi ’s are polynomials in K[X0 , X1 , X2 , X3 ], if p is any prime factor of a0 , then it must divide b0 X0 , b0 X1 , b0 X2 i.e. p must divide b0 . Recursively

3.3 Fundamental examples and remarks

101

applying the same argument and changing the roles between a0 and b0 , one can assume a0 = b0 , so that (3.39) become a1 X2 + b1 X3 b0 a1 X1 + b1 X2 G2 = b0 a1 X0 + b1 X1 G2 = b0

G1 =

Thus, b0 divides the polynomials α1 := a1 X2 + b1 X3 α2 := a1 X1 + b1 X2 α3 := a1 X0 + b1 X1 , which implies that b0 divides therefore the polynomials X1 α3 − X0 α2 = −b1 B1

X2 α3 − X0 α1 = b1 B2 X2 α2 − X1 α1 = −b1 B3 . Since B1 , B2 , B3 are all irreducible and distinct, this implies that b0 divides b1 . Since, by assumption, b0 and b1 are relatively prime elements, one concludes that b0 ∈ K∗ . Therefore we can assume a0 = b0 = 1. For simplicity of notation, let a := a1 , b := b1 . We want to show that (3) (a, b) ∈ (Sd−3 )⊕2 . Indeed, if j 6= d − 3 and if Aj , Bj are the homogeneous, degree-j components of a, b, from (3.39) one gets Aj X2 + Bj X3 = 0 Aj X1 + Bj X2 = 0 Aj X0 + Bj X1 = 0 so, reasoning as above, one deduces Aj = Bj = 0, for any integer j 6= d − 3. This proves that ψ is bijective. In particular, if I := (F1 , F2 , F3 ), we proved the existence of an exact sequence (cf. Exercise ??) 0 → N → M → I → 0, where M and N are free, graded S(3) -modules and the arrows are given by degree-0, homogeneous homomorphisms. This exact sequence is called a homogeneous free resolution of the homogeneous ideal I (cf. Exercise ??). From Lemma 1.10.11 and exactness, for any integer d one has       d+1 d d+3 dim(Id ) = 3 −2 = − (3d + 1) 3 2 3

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3 Algebraic Projective Sets

(cf. also Exercise ??, ??). p We now compute the dimension of Ip (C )d , for any integer d > 0 (for d < 0 this dimension is obviously 0). Consider the homogeneous substitution of indeterminates of degree 3 ν : f (X0 , X1 , X2 , X3 ) ∈ S(3) → f (λ3 , λ2 µ, λµ2 , µ3 ) ∈ S(1) which induces surjective K-vector space homomorphisms (3)

(1)

νd : Sd → S3d , p

for any non–negative integer d. One has Ker(νd ) = Ip (C )d , for any d > 0, so   d+3 p
− (3d + 1) = dim(Id ). dim Ip (C )d = 3 p

This proves that I = Ip (C ) as desired.

⊓ ⊔

3.4 Exercises Exercise 3.4.1. Let P = P(V ) be a projective space of dimension n and let H ⊂ P be any hyperplane. Prove that P \ H is homeomorphic to the affine space An . Exercise 3.4.2. Let Z ⊂ Pn be a closed subset. Prove that Ca (Z)∞ = Z. Exercise 3.4.3. Let P := P(V ) be a projective space of dimension r. Consider in P two projective subspaces Λ1 and Λ2 of dimensions n and r − n − 1, respectively, and such that Λ1 ∩ Λ2 = ∅. Let Z ⊂ Λ1 be a non-empty, closed subset. Consider the set [ CΛ2 (Z) := hP, Λ2 i, P ∈Z

where hP, Λ2 i = P ∨ Λ2 as in § 3.3.6 (vi). Show that CΛ2 (Z) is a closed subset of P, which is called the cone over Z with vertex Λ2 . Determine Ip (CΛ2 (Z)), assuming that you know Ip (Z), when Z is considered as a closed subset of Λ1 . Exercise 3.4.4. Let P := P(V ) be a projective space. Consider Λ = P′ a projective subspace and Z ⊂ P a closed subset, both of them assumed to be non–empty. Show that Z is a cone of vertex Λ if and only if for any point P ∈ Z one has (P ∨ Λ) ⊆ Z. Exercise 3.4.5. If Z1 , Z2 are affine subspaces of An , prove that Z1 ∩ Z2 is an affine subspace of An (possibly the empty set). If Zi = Zi′ ∩ An , where Zi′ projective subspace of Pn , 1 6 i 6 m, one defines the linear envelope of the affine subspaces as ′ ) ∩ An Z1 ∨ . . . ∨ Zm := (Z1′ ∨ . . . ∨ Zm

(cf. § 3.3.6 (vi)). Show that in general (affine) Grassmann formula as in (3.23) does not hold. Find sufficient conditions in order that (affine) Grassmann formula holds.

3.4 Exercises

103

Exercise 3.4.6. Let ϕ : P(V ) → P(W ) be a bijective homography, determined by the isomorphism f : V → W . The homography φt : P(W )∗ → P(V )∗ determined by the transposed map f t : W ∗ → V ∗ is said to be the transposed homography of ϕ. Verify that, for any linear subspace Z ⊂ P(W ), one has ϕt (Z ⊥ ) = (ϕ−1 (Z))⊥ . Exercise 3.4.7. Let P := P(V ) be a projective space of dimension n and let Z ⊂ P be a projective subspace of codimension c. Prove that, once one introduces a system of homogeneous coordinates X0 , . . . Xn on P(V ), there exists a (c × (n + 1))–matrix A, with entries in K and of rank c, such that Z is the set of points P ∈ P(V ) whose homogeneous coordinates [P ] = [p0 , . . . , pn ], in the chosen frame, satisfy A · P t = 0, where 0 the (c×1)–zero matrix. This gives a homogeneous linear system of c linearly independent, linear forms in the indeterminates X0 , . . . Xn which is called a system of equations of Z in the given frame. Conversely, show that any such subset of Pn is a projective subspace of codimension c. When P(V ) = Pn , one usually consider the natural frame induced by the canonical basis of Kn+1 . Exercise 3.4.8. Let P := P(V ) be a projective space of dimension n and let Λ be any projective subspace of dimension m < n. Once one introduces in P(V ) a system of homogeneous coordinates, show that there exists a ((m + 1) × (n + 1))–matrix A, with entries in K and of rank m + 1, such that Z is the set of points P ∈ P(V ) whose homogeneous coordinates [P ] = [p0 , . . . , pn ], in the chosen frame, are of the form P = Q · A, where [Q] varying in Pm . Conversely, show that any such subset of Pn is a projective subspace of dimension m. Exercise 3.4.9. Consider distinct points P1 , . . . , P4 on a projective line P p(V ). From Corollary 3.3.9, there exists a unique projective frame such that P1 = [1, 0], P2 = [0, 1], P3 = [1, 1] and so one has P4 = [p, q], for some p, q 6= 0 such that p 6= q. In this set-up, one defines the cross ratio of the (ordered) quadruple of points, denoted by (P1 , P2 , P3 , P4 ), as [p, q] ∈ P1 or equivalently pq ∈ K∗ . Given (ordered) quadruples of points P1 , P2 , P3 , P4 of the projective line P(V ) and Q1 , Q2 , Q3 , Q4 of the projective line P(W ), prove that there exists a projectivity ϕ : P(V ) → P(W ) such that ϕ(Pi ) = Qi , 1 6 i 6 4, if and only if (P1 , P2 , P3 , P4 ) = (Q1 , Q2 , Q3 , Q4 ). Exercise 3.4.10. Prove that the map φ in (3.32) is a homeomorphism from (A1 , Zar1a ) and the (affine) twisted-cubic (C, Zar3a,C ). Prove moreover that φ defines a natural K-algebra isomorphism φ∗ : A(C) → K[t], where A(C) := A(3) /Ia (C).

Exercise 3.4.11. An (affine) twisted cubic is defined to be any closed subset Z ⊂ A3 for which there exists an affine transformation Φ of A3 such that Φ(Z) coincides with the image of (3.32). ProveP that Z is an (affine) twisted cubic if and only if there 3 j exist polynomials fi (t) = j=1 aij t + ai s.t. the matrix A := (aij )1≤i≤j≤3 is of maximal rank and Z coincides with the image of the map φ : t ∈ A1 → (f1 (t), f2 (t), f3 (t)) ∈ A3 , which is called a parametric representation of the (affine) twisted cubic Z.

(3.40)

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3 Algebraic Projective Sets

Exercise 3.4.12. Given Z, the twisted cubic of parametric representation (3.40), determine the ideal Ia (Z) and show that it does not contain any polynomial of degree 1. From this, deduce from this that Z is non-degenerate in A3 , i.e. it is not contained in any plane (cf. § 3.3.7). Exercise 3.4.13. Notation and assumptions as in Exercise 3.4.12. Determine all degree–two polynomials in Ia (Z), i.e. all affine quadrics of A3 containing Z. Prove moreover that A(Z) := A(3) /Ia (Z) is isomorphic to A(1) . Exercise 3.4.14. Notation and assumptions as in Exercise 3.4.12. Prove that any surface of degree d not containing Z intersects Z in at most 3d points and that there exist surfaces of degree d intersecting Z in exactly 3d distinct points. Exercise 3.4.15. Given Z, the twisted cubic of parametric representation (3.40), determine the subgroup of Aff(A3 ) formed by affinities sending Z in itself. Exercise 3.4.16. Prove that the map ψ in (3.34) is a homeomorphism from (P1 , Zar1p ) and the (projective) twisted-cubic (C, Zar3a,C ), where C := C p . Exercise 3.4.17. A (projective) twisted cubic is any closed subset Z ⊂ P3 for which there exists a projectivity Φ of P3 such that Φ(Z) coincides with the image of (3.34). Show that Z is a (projective) twisted cubic if and only if there exists a basis Fi (λ, µ) (1) of S3 , 0 6 i 6 3, such that Z is the image via the map ψ : [λ, µ] ∈ P1 → [F0 (λ, µ), F1 (λ, µ), F2 (λ, µ), F3 (λ, µ)] ∈ P3

(3.41)

which is called a parametric representation of the (projective) twisted cubic Z. Exercise 3.4.18. Given the twisted cubic Z of parametric representation (3.41), determine Ip (Z). Prove that Z is non-degenerate in P3 (in the sense of § 3.3.6 (vi)). (3) Prove that, for any integer d > 0, the codimension of Ip (Z)d in Sd is 3d + 1. Prove (3) that the graded ring S(Z) := S /Ip (Z) is not isomorphic to the graded ring S(1) .

4 Some topological properties

4.1 Irreducible topological spaces Let (Y, TY ) be any non-empty topological space, where TY denotes a given topology on Y . Y is said to be irriducibile if Y cannot be expressed as Y = Y1 ∪ Y2 , where Y1 , Y2 are proper closed subsets of Y . Equivalently, Y is irreducible if and only if any two non-empty, open subsets U1 , U2 ∈ TY are such that U1 ∩ U2 6= ∅. When Y is not irreducible, it is called reducibile. Remark 4.1.1 (i) If Y contains more than one point and it is irreducible, then it cannot be Hausdorff (i.e. TY is not T2 ). (ii) If K = R or C, AnK endowed with the euclidean topology is reducible for any n > 1. (iii) The property of being irreducible is topological, i.e. it is invariant under homeomorphisms. (iv) If Y is irreducible, then it is connected. The converse is not true in general (cf. Example 4.1.5 (iii) below). In what follows, any subset W ⊆ Y will be considered as endowed with the induced topology by that of Y . Proposition 4.1.2 Let Y be a topological space and let W ⊆ Y be any subset.

(i) W is irreducible if and only if for any pair of distinct points P1 , P2 ∈ W there exists an irreducible subset Z ⊆ W such that P1 , P2 ∈ Z; (ii) W is irreducible if and only if any non-empty, open subset U ⊆ W is dense, i.e. U = W where U denotes the closure of U in W ; (iii) W is irreducible if and only if W is irreducible, where W denotes the closure of W in Y ; (iv) W is irreducible if and only if any non-empty, open subset U ⊆ W is irreducible.

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4 Topological properties

Proof. (i) The implication (⇒) is trivial. For (⇐), suppose by contradiction that W is reducible and let W = W1 ∪ W2 , where Wi proper closed subsets of W , 1 6 i 6 2. Let Pi ∈ Wi \ W3−i , 1 6 i 6 2, and let Z be any irreducible subset Z ⊆ W s.t. P1 , P2 ∈ Z. Then Z = Z1 ∪ Z2 , where Zi = Wi ∩ Z, 1 6 i 6 2, and Z1 , Z2 irreducible, proper closed subsets of Z, since Pi 6∈ Z3−i . This contradicts the irreducibility of Z. (ii) Let us prove the non–trivial implication of (ii). Let W be irriducible and let U be any non-empty open subset of W . If U were not dense in W , then one would have W = (U ∩ W ) ∪ (W \ U ), where U ∩ W and W \ U are both proper closed subsets of W ; this is against the irreducibility assumption on W. (iii) Let Z be any closed subset of Y and denote by Z ′ := Z ∩ W and Z ′′ := Z ∩ W , where W denotes the closure of W in Y . One has Z ′ = W ⇔ W ⊆ Z ⇔ W ⊆ Z ⇔ Z ′′ = W In other words, Z ′ is a proper closed subset of W if and only if Z ′′ is a proper closed subset of W . From (ii) we know that W is irreducible if and only if any of its proper closed subsets have empty interior set. Thus, if the interior of Z ′′ is empty, the same occurs for the interior of Z ′ , so that if W is irreducible then W is. Viceversa, if W is irriducibile, let W = W1 ∪ W2 , where Wi closed subsets of W , 1 6 i 6 2. The irreducibility of W implies that W ⊆ Wi , for either i = 1 or i = 2. Consequently, W ⊆ Wi , i.e. W = Wi . Thus, W is irriducible. (iv) The implication (⇒) is a direct consequence of (ii) and (iii), whereas (⇐) is trivial. ⊓ ⊔ Corollary 4.1.3 For any integer n > 1: (i) (Pn , Zarnp ), (ii) any non-empty, open subset U ⊆ Pn , (iii) (An , Zarna ), (iv) any non-empty, open subset U ⊆ An , are irreducible. Proof. From Proposition 4.1.2 (iv), it suffices to prove (i), which is a direct consequence of Proposition 4.1.2 (i). Indeed, for any pair of distinct points P1 , P2 ∈ Pn , the line P1 ∨ P2 is irreducible, since homemorphic to P1 (recall (3.24) and Remark 4.1.1 (iii)). ⊓ ⊔ Remark 4.1.4 From Remark 4.1.1 (i) and Corollary 4.1.3, Zarna and Zarnp are not T2 for any integer n > 1 (this was already observed for n = 1 in Remark 2.1.14 (i) and § 3.3.1). Notice that irreducibility in Corollary 4.1.3 holds since K is infinite; if otherwise K = Zp , for p ∈ Z a prime, then all the topological spaces listed in Corollary 4.1.3 are reducible and both Zarna , Zarnp are the discrete topology.

4.1 Irreducible topological spaces

107

Example 4.1.5 (i) Any affine subspace of An is irreducible, since homeomorphic to Am for some m 6 n. Same conclusion for any projective subspace of Pn . (ii) The affine twisted cubic C in § 3.3.12 is irreducible, since homemorphic to A1 (cf. Exercise 3.4.10). Therefore, the projective twisted cubic is irreducible too. (iii) The affine hypersurface Y = Za (x1 x2 ) ⊂ A2 is reducible. Indeed, Ia (Y ) = (x1 x2 ) = (x1 ) ∩ (x2 ) so Y = Za (x1 ) ∪ Za (x2 ) = Y1 ∪ Y2 , where Y1 , Y2 ⊂ Y proper closed subsets of Y . In particular, Y is connected but not irreducible. Recalling Proposition 4.1.2, notice moreover that: (a) there exist pairs of distinct points in Y not contained in any irreducible subset of Y ; (b) U := Za (x1 )c ∩ Y is an open subset of Y whose closure in Y is U = Za (x2 ) = Y2 , so not dense in Y ; (c) U1 := Za (x1 , x2 )c ∩ Y is a non-connected open subset of Y which is reducible. Corollary 4.1.6 Let Y, W be topological spaces such that Y is irreducible. Assume there exists a continuous map f : Y → W . Then Im(f ) ⊆ W is irreducible. Proof. Since Im(f ) ⊆ W is endowed with the induced topology of W , it suffices to show that when f is assumed to be also surjective then W is irreducible. To prove this, if W = W1 ∪ W2 , where W1 , W2 closed subsets of W , then Y = f −1 (W1 ) ∪ f −1 (W2 ). Since Y is irreducible and f is continuous, one must have Y = f −1 (Wi ), for some i ∈ {1, 2}. Consequently, by the surjectivity of f , W = f (Y ) = f (f −1 (Wi )) = Wi . ⊓ ⊔ 4.1.1 Coordinate rings, ideals and irreducibility Here we focus on the case of Zariski topology. Definition 4.1.7 For any subset Y ⊆ An , consider Ia (Y ) ⊆ A(n) . The Kalgebra of finite type (cf. Def. 1.5.3) A(Y ) := A(n) /Ia (Y ),

(4.1)

is called the (affine) coordinate ring of Y . Similiarly, for any subset Y ⊆ Pn , with homogeneous ideal Ip (Y ) ⊆ S(n) , the graded K-algebra of finite type (recall Prop. 1.10.8 (ii)) S(Y ) := S(n) /Ip (Y ) (4.2) is called the homogeneous coordinate ring of Y .

108

4 Topological properties

Proposition 4.1.8 Let Y ⊆ Pn be any irreducible subset and let U ⊆ Y be any non-empty open subset of Y , then Ip (Y ) = Ip (U ). The same holds replacing Pn with An and Ip (−) with Ia (−). Proof. We prove the first part, the proof in the affine case being identical. p Let Y be the projective closure of Y in Pn ; then (3.21) gives p

Ip (Y ) = Ip (Y ). e = Y , where U e Since Y is irreducible, from Proposition 4.1.2 (ii), we have U denotes the closure of U in Y . Thus p

p e =Yp U =U

⊓ ⊔

and, from (3.21), we get Ip (U ) = Ip (Y p ) = Ip (Y ). Corollary 4.1.9 For any subset Y ⊆ An , let Y Then a A(Y ) = A(Y ). Similarly, for any subset Y ⊆ Pn , let Y

p

a

denote its closure in An .

denote its projective closure. Then p

S(Y ) = S(Y ). When it is clear from the context, in what follows we will sometimes use for brevity the symbol I(Y ) instead of Ia (Y ) or Ip (Y ), if no confusion arises. The rings A(Y ) and S(Y ) are always reduced rings (cf. Def. 1.1.1) since I(Y ) is always radical. On the other hand, in some cases these rings are not integral domains (cf. Exercise 2.5.8). An algebraic criterion for irreducibility is given by the following result. Proposition 4.1.10 A subset Y ⊆ An [resp. Y ⊆ Pn ] is irreducible if and only if I(Y ) is a prime ideal, i.e. if and only if A(Y ) [resp. S(Y )] is an integral domain. Proof. The second part of the statement is obvious by definition of A(Y ) [resp. S(Y )]; therefore we prove the first equivalence. From Corollary 4.1.9 it suffices to consider the case with Y closed in the Zariski topology. Assume Y to be irriducible. If f g ∈ I(Y ) then (f g) ⊆ I(Y ) so Y = Z(I(Y )) ⊆ Z(f g) = Z(f ) ∪ Z(g) (cf. Propositions 2.1.11, 3.1.5). Thus, either Y ⊆ Z(f ) or Y ⊆ Z(g). This means that either f ∈ I(Y ) or g ∈ I(Y ), i.e. I(Y ) is prime. Let now I(Y ) be a prime ideal and let Y = Y1 ∪ Y2 , where Y1 , Y2 closed subsets of Y . Then I(Y ) = I(Y1 ) ∩ I(Y2 ) (cf. Propositions 2.2.8, 3.1.7). Claim 4.1.11 In the above assumptions, I(Y ) is either I(Y1 ) or I(Y2 ).

4.1 Irreducible topological spaces

109

Proof (Proof of Claim 4.1.11). If it were I(Y ) ( I(Yi ) for both i = 1, 2, there would exist fi ∈ I(Yi ) \ I(Y ), 1 6 i 6 2, such that f1 f2 ∈ I(Y1 ) · I(Y2 ) ⊆ I(Y1 ) ∩ I(Y2 ) = I(Y ), contradicting that I(Y ) is prime (cf. also Exercise 1.12.1).

⊓ ⊔

From the claim, we have either Y = Y1 or Y = Y2 and so Y is irriducible.

⊓ ⊔

Recalling bijective correspondences in Corollaries 2.2.5, 3.2.6, we have: Corollary 4.1.12 The maps (2.11) and (2.12) induce bijections: 1−1

{irreducible closed subsets of An } ←→



prime ideals of A(n)

Similarly, the maps (3.5) and (3.6) induce bijections: 1−1

{irreducible closed subsets of Pn } ←→

n



o homogenoeus prime ideals of S(n) \{S+ }.

Example 4.1.13 (i) We find in this way an alternative proof of Corollary 4.1.3, via purely algebraic approach. Indeed A(An ) = A(n) and S(Pn ) = S(n) which are integral domains; so one concludes by Proposition 4.1.10 and, for the open sets, by Corollary 4.1.9. (ii) For any subset Z ⊂ Pn we have that Z irreducible ⇔ Ca (Z) irreducible ⇔ Cp (Z) irreducible.

(4.3)

This directly follows from what discussed above and (3.8), (3.29). (iii) If Z ⊂ Pn is any hypersurface, then Z is irreducible if and only if its reduced equation F ∈ S(n) is an irreducible polynomial, as it follows from (3.30) and Claim 4.1.11. Same occurs in the affine case (cf.Example 2.1.19). (iv) Any affine rational curve with polynomial representation as in (3.31) is irreducible, as it follows from Lemma 3.3.15. (v) Recalling (3.37), the closed subset K := Zp (F1 , F2 ) is reducible consisting of the (projective) twisted cubic union the line at infinity of the affine plane Za (x1 ) ⊂ A3 . From Proposition 4.1.10, I = (F1 , F2 ) cannot be a prime ideal. We can in fact directly show that S(K) is not an integral domain. Denote by F3 , X0 ∈ S(K) the images of F3 , X0 ∈ S(3) in S(K), respectively. Since Ip (K) = (F1 , F2 ), then F3 , X0 ∈ S(K) \ {0} (recall that F3 is linearly independent from F1 , F2 ). On the other hand X0 F3 = 0 in S(K), since X0 F3 = X1 F2 − X2 F1 ∈ Ip (K). (vi) From Corollary 2.3.4 and Proposition 3.3.14, it follows that any nonempty, irreducible, proper closed subset W of A2 and of P2 is either one point or an irreducible curve.

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4 Topological properties

4.1.2 Algebraic varieties In the sequel we will always endowed topological space with Zariski topology and use the following terminology. Definition 4.1.14 Y is said to be a projective variety [resp. affine variety] if Y is irreducible and closed in a projective [resp. affine] space. Y is said to be a quasi-projective variety [resp. quasi–affine variety] if Y is irreducible and locally closed (i.e. it is the intersection of a closed set and of an open set) in a projective [resp. affine] space. Remark 4.1.15 From the previous definition, we notice that Y is a quasiprojective [resp. quasi–affine] variety if and only if it is an open subset of a projective [resp. affine] variety, namely its closure in Pn [resp. in An ]. Proposition 4.1.16 Let Z ∈ Cnp . Then Z is a projective variety if and only if for any affine chart Ui of Pn , 0 6 i 6 n, Zi := Z ∩ Ui is an affine variety. Proof. (⇒) is obvious. To prove (⇐), one is reduced to showing that Ip (Z) is a prime ideal. Since Ip (Z) is a homogeneous ideal, the primality of Ip (Z) can be proved by using Proposition 1.10.7 (ii) and the fact that, by assumptions, Ia (Zi ) is prime for any 0 6 i 6 n (cf. also Exercise 4.3.10). ⊓ ⊔ From Definition 4.1.14, any projective [resp. affine, quasi-affine] variety is also a quasi-projective variety, in other words the notion of quasi-projective variety is the most general one. Definition 4.1.17 With the term algebraic variety we will intend any quasiprojective variety Y . Similarly, with the term closed algebraic set we will indicate any AAS or any APS, whereas with algebraic locally-closed set any locally-closed subset of either an affine or a projective space.

4.2 Noetherian spaces. Irreducible components Definition 4.2.1 A non-empty topological space (Y, TY ) is said to be noetherian if it verifies the following condition: (*) for any sequence {Yn }n∈N of closed subsets of Y such that, for any n ∈ N one has Yn ⊇ Yn+1 , there exists m ∈ N s.t., for any n > m, one has Yn = Ym . Condition (*) is called the descending chain conditions [denoted by d.c.c. for short] on closed subsets of Y .

4.2 Noetherian spaces. Irreducible components

111

Remark 4.2.2 By passing to complementary sets, one clearly has that Y is noetherian if and only if it verifies the ascending chain conditions [a.c.c. for short] on its open sets, that is: (**) for any sequence {Un }n∈N of open subsets of Y such that, for any n ∈ N one has Un ⊆ Un+1 , there exists m ∈ N s.t., for any n > m, one has Un = Um . In what follows, any subset W ⊆ Y of a topological space Y will be considered endowed with the induced topology, i.e. with TY,W . Proposition 4.2.3 Let Y be a noetherian topological space. (i) Any non-empty subset W ⊆ Y is noetherian; (ii) Y , as well as any non-empty subset W ⊆ Y , is compact. Proof. Part (i) is an easy consequence of the following observation: if W ⊆ Y and if W1 ⊇ W2 are closed subsets of W , there exist Yi closed subsets of Y such that Wi = Yi ∩ W , 1 6 i 6 2, and Y1 ⊇ Y2 ; indeed it suffices to replace Y2 with Y1 ∩ Y2 if necessary. As for (ii), by contradiction let {Ui }i∈I be an open covering of Y from which one cannot extract a finite sub-covering. Thus, there exist sequences {in }n∈N of indices in I and of distinct points {Pn }n∈N of Y such that, for Un := ∪nh=1 Uih , one has Pn−1 ∈ Un but Pn 6∈ Un for any n ∈ N. This would contradict the a.c.c. on open subsets of Y . ⊓ ⊔ Example 4.2.4 An easy example of notherian topological space is the following: consider Y an infinite set, P(Y ) the power-set of Y and let CY ⊂ P(Y ) be the subset consisting of ∅, Y and all finite subsets of Y . Then CY can be taken as a family of closed subsets for a topology on Y , which turns out to be noetherian. The next result shows that also Zar is notherian. Corollary 4.2.5 (i) For any integer n > 1, (An , Zarna ) and (Pn , Zarnp ) are noetherian. (ii) Any locally-closed algebraic set is notherian. Proof. (i) It directly follows from reversing inclusions (cf. (2.10) and Prop. 3.1.7 (i)) and the noetherianity of A(n) and of S(n) (cf. Prop. 1.4.4). (ii) From Proposition 4.2.3 (i), statement (i) implies (ii). ⊓ ⊔ The following result bridges noetherianity with irreducibility. Theorem 4.2.6 (Irredundant decomposition) Let Y be a noetherian topological space and let W ⊆ Y be a non-empty, closed subset. Then:

112

4 Topological properties

(i) W can be expressed as a finite union W := W1 ∪ . . . ∪ Wn ,

(4.4)

where each Wi is a closed, irreducible subset of W ; (ii) decomposition (4.4) is uniquely determined (up to reordering the Wi ’s) if it is irredundant, i.e. if Wi 6⊆ Wj for any i 6= j ∈ {1, . . . , n}. Proof. (i) If W is irreducible, we are done. Otherwise let W = W1 ∪ W ′ , where W1 , W ′ proper closed subsets of W . If both of them are irreducible, we are done (observe moreover that neither W1 ⊆ W ′ nor W ′ ⊆ W1 , otherwise W would be irreducible against the assumption). Otherwise, at least one of W1 , W ′ has to be reducible. Assume that W ′ is. Repeating the same argument, we can costruct a sequence {Wn }n∈N of closed subsets of W s.t., for any n ∈ N, Wn+1 ( Wn . By noetherianity of Y and by Proposition 4.2.3, we conclude. ′ (ii) Let W = W1′ ∪ . . . ∪ Wm be another decomposition, satisfying the same assumptionss of that in (4.4); assume moreover that both of them are irredundant. One has ′ Wi = (Wi ∩ W1′ ) ∪ . . . ∪ (Wi ∩ Wm ),

for any

i ∈ {1, . . . , n}.

Since Wi is irreducible, there exists j ∈ {1, . . . , m} s.t. Wi = Wi ∩ Wj′ i.e. Wi ⊆ Wj′ . Similarly, there exists h ∈ {1, . . . , n} s.t. Wj′ ⊆ Wh i.e. Wi ⊆ Wh . Since (4.4) is irredundant, then i = h and Wi = Wj′ . Repeating the same arguments, one can conclude. ⊓ ⊔ Definition 4.2.7 When (4.4) is irredundant, W1 , . . . , Wn are called the irreducible components of W and (4.4) is said to be the decomposition of W into its irreducible components. Corollary 4.2.8 If Y is notherian and Hausdorff (i.e. TY is T2 ), then Y is finite and TY is the discrete topology. Proof. By Theorem 4.2.6, Y can be decomposed as a finite union of its irreducible components. Therefore, to prove the statement, we can assume Y to be irreducible. In this case, one concludes by using Remark 4.1.1 (i). ⊓ ⊔ In the case of Zariski topology we have: Corollary 4.2.9 (i) Any algebraic locally-closed subset can be decomposed into finitely many algebraic varieties. (ii) Any radical ideal I ⊂ A(n) [resp., homogeneous radical ideal I ⊂ S(n) ] can be uniquely written as n \ Ij , I= i=1

(n)

where Ij prime ideals in A

[resp., homogeneous prime ideals in S(n) ]

4.3 Exercises

113

Proof. (i) is a consequence of Theorem 4.2.6, whereas (ii) follows from the bijective correspondences proved in Corollary 2.2.5, in the non-homogeneous case, and in Corollary 3.2.6 in the homogeneous one; in this latter situation, the only left case deals with the ideal S+ , which obviously is S+ = (X0 ) ∩ (X1 ) ∩ . . . ∩ (Xn ). ⊓ ⊔ We conclude with an important observation. Remark 4.2.10 In general, the intersection of algebraic varieties is not necessarily an algebraic variety. Take e.g. F1 and F2 as in (3.35). These two polynomials are irreducible in S(3) so the quadric (hyper)surfaces Zp (F1 ), Zp (F2 ) are projective varieties (cf. Example 4.1.13 (iii)). On the other hand, Zp (F1 )∩ Zp (F2 ) = Zp (F1 , F2 ) is reducible (cf. Example 4.1.13 (iii)).

4.3 Exercises (2)

Exercise 4.3.1. Give an example of an irreducible polynomial f ∈ AR whose zero set Za (f ) ⊂ A2R is not irreducible. Exercise 4.3.2. Let K be algebraically closed. Let V ⊂ An be a non-empty affine variety. Prove that the following are equivalent: (a) V is a point; (b) A(V ) = K; (c) A(V ) is a K-vector space of finite dimension. Exercise 4.3.3. Let K be algebraically closed. Let V and W be AAS’s in An , with W ⊂ V . Show that each irreducible component of W is contained in some irreducible component of V . Exercise 4.3.4. Find the irreducible components of Za (x22 − x1 x2 − x21 x2 + x31 ) in A2R and in A2C , respectively. Do the same for Za (x22 − x1 (x21 − 1)) and for Za (x31 + x1 − x21 x2 − x2 ) (cf. [12, Ex. 1.31, p.10]). Exercise 4.3.5. Decompose Za (x21 + x22 − 1, x21 − x23 − 1) ⊂ A3C into irreducible components (cf. [12, Ex. 1.32, p.11]).

Exercise 4.3.6. Let K be algebraically closed and let U be an open subset of An (or of Pn ). Let f ∈ A(n) (respectively, f ∈ S(n) ) be any non constant polynomial. Show that if U ⊆ Z(f ) then f is the zero polynomial (here Z(−) stands for Za (−) in the affine case and for Zp (−) in the projective case). Exercise 4.3.7. Let K be algebraically closed and let I = (x21 − x32 , x22 − x33 ) ⊂ A(3) . Define the K-algebra homomorphism α : A(3) → K[t] by α(x1 ) = t9 , α(x2 ) = t6 , α(x3 ) = t4 . (3)

(a) Show that any element of the quotient ring AI is the residue of an element a + bx1 + cx2 + dx1 x2 , for some a, b, c, d ∈ K[x3 ]. (b) If f = a + bx1 + cx2 + dx1 x2 , for some a, b, c, d ∈ K[x3 ], such that α(f ) = 0 compare like powers of t to conclude that f = 0. (c) Show that Ker(α) = I. Deduce that I is prime, so Za (I) is irreducible and Ia (Za (I)) = I (cf. [12, Ex. 1.40, p.12]).

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4 Topological properties

Exercise 4.3.8. Referring to Exercise 3.4.3, prove that Z is irreducible if and only if CΛ2 (Z) is. Exercise 4.3.9. Prove that any affine and any projective twisted cubic as in Exercises 3.4.11 and 3.4.17, respectively, is irreducible. Exercise 4.3.10. Prove in full details Proposition 4.1.16. On the other hand, for Z either Zp (X0 X1 ) ⊂ P2 or Z ⊂ P3 as in (3.37), show that Ia (Zi ) fails to be prime, for some i = 0, 1, 2. Exercise 4.3.11. Let Y be any algebraic variety. Show that Y admits a finite covering Y = ∪n i=0 Yi , where each Yi can be identified to a quasi-affine variety. Exercise 4.3.12. Let Y be a quasi-projective variety in Pn , for some n, and let W ⊂ Y be any locally closed subset of Y . Show that W is a quasi-projective variety in Pn . Exercise 4.3.13. Let Y, W ⊂ Pn be quasi-projective varieties. If W ⊂ Y , then show that W is locally closed in Y . Exercise 4.3.14. Let Z = Za (f ) be an affine hypersurface, with f = f1 . . . fh reduced and f1 , . . . , fh irriducible. Prove that Za (f1 )∪. . .∪Za (fh ) is the decomposition of Z into its irreducible components. Exercise 4.3.15. As customary, identify AnS with the affine chart U0 of Pn . Let Z ⊂ An be a closed subset and let Z = m i=1 Zi be its decomposition into its of the affine variety irreducible components. Let Z i denote the projective closure S Zi ⊂ An . Show that the projective closure of Z coincides with m i=1 Z i Exercise 4.3.16. Let Z ⊂ An = U0 be a non-empty, closed subset. Let Z be its projective closure. Show that no irreducible components of Z either containes or is contained in H0 = Zp (X0 ). Exercise 4.3.17. Let Z ⊂ Pn be a non-empty, closed subset such that none of its irreducible components either containes or is contained in H0 = Zp (X0 ). Let Z0 := Z ∩ U0 . Show that Z coincides with the projective closure of Z0 . Exercise 4.3.18. Let D be the set parametrizing degenerate conics in P2R . Show that D can be identified with an irreducible cubic hypersurface in P5R .

5 Regular and rational functions

5.1 Regular functions If we take a polynomial f ∈ A(n) , we can consider the associated function f : An → K (recall (2.1)). Accordingly, if g1 , g2 ∈ A(n) are polynomials, with g2 6= 0, the rational function g1 g2 can be considered as a function defined on the principal open subset Ua (g2 ) := An \ Za (g2 ) and with values in K. By contrast, given a projective space P(V ), an element g ∈ S(V ∗ ) cannot be considered as function on P(V ), with values in K, even if g is homogeneous. On the other hand, if we take homogeneous elements G1 , G2 ∈ H(S(V ∗ )) of the same degree and P = [p] ∈ Up (G2 ) := P(V ) \ Zp (G2 ), where p = (p0 , . . . , pn ) ∈ V , it makes sense to consider G1 (p) G1 (tp) G1 (P ) := = , G2 G2 (p) G2 (tp)

∀t ∈ K \ {0}

G1 so that, as in the affine case, G can be considered as a function defined on the 2 principal open subset Up (G2 ) and with values in K. We sometimes will drop the indices a and p in Ua (g2 ), Up (G2 ), when this does not create ambiguity.

Remark 5.1.1 In what follows, the field of fractions Q(A(n) ) will be simply denoted by Q(n) and its elements will be simply called rational functions in the ideterminates x1 , . . . , xn (or even rational functions on An ) Similarly, the field Q(S(V ∗ )) contains the subfield Q(V ∗ ) which consists of ∗ 1 all fractions G G2 such that G1 , G2 ∈ H(S(V )) are of the same degree, where G2 6= 0. These will be called degree-zero rational functions. If we introduce in P(V ) homogeneous coordinates [X0 , . . . , Xn ], then Q(V ∗ ) identifies with the G1 where G1 , G2 are hosubfield Q0 (S(n) ) of Q(n+1) consisting of all fractions G 2 mogeneous polynomials of the same degree in the indeterminates X0 , . . . , Xn ,

116

5 Regular and rational functions

with G2 non–zero (recall Def.1.10.9). This sub-field will be simply denoted by (n) Q0 . Let now Y ⊂ P(V ) be a locally closed subset. Let f : Y → K be a function and let P ∈ Y be a point. The function f is said to be regular at P if there exist an open neighborhood U ⊆ Y of P and G1 , G2 ∈ H(S(V ∗ )) of the same degree, such that Zp (G2 ) ∩ U = ∅, i.e. U ⊆ Up (G2 ), and such that the restriction of f to U 1 coincides with the restriction of G G2 to U . f : Y → K is said to be regular on Y , if it is regular at any point of Y (in the above sense). Remark 5.1.2 For a function f : Y → K, being regular at a point P is both a local and open property, i.e. it depends only on what happens in a neighborood of P and, if it holds, it holds for all points in a neighborood of P. If, as usual, we identify An with the affine open set U0 of Pn a straightforward application of the maps in Definition 1.10.14 gives the following: Lemma 5.1.3 Let Y ⊆ An be any locally closed subset and let f : Y → K be a function. Then f is regular at P ∈ Y (in the sense of the above definition) if and only if there exist an open neighborhood U ⊂ Y of P and polynomials g1 , g2 ∈ A(n) , such that Za (g2 ) ∩ U = ∅ and the restriction of f to U coincides with the restriction of gg21 to U . The same strategy also shows that (n)

Q0

= Q0 (S(n) ) ∼ = Q(n) = Q(A(n) ).

(5.1)

Let therefore Y be any algebraic variety. For any non-empty, open subset U of Y we denote by OY (U ) the set of regular functions on U . Constant functions on Y are obviously regular functions on any open set U ⊆ Y . If moreover f, g ∈ OY (U ), then the functions f + g : P ∈ U → f (P ) + g(P ),

f · g : P ∈ U → f (P ) · g(P )

(5.2)

are also regular on U . The previous easy observations show that, for any algebraic variety Y and for any non-empty open subset U ⊆ Y , OY (U ) is a K–algebra with respect to the operations (+, ·) as in (5.2), which is called the algebra of regular functions on U . If moreover U, U ′ ⊆ Y are non-empty open subsets, with U ′ ⊆ U , one has an obvious restriction map

5.1 Regular functions ′ ρU U ′ : f ∈ OY (U ) → f |U ′ ∈ OY (U ),

117

(5.3)

which is well-defined and which clearly is a K–algebra homomorphism with respect to the operations (5.2). We will sometimes abuse notation and denote ρU U ′ (f ) simply by f |U ′ . From Definition 5.27 in Appendix 5.2.2 and what observed above about the Since K-algebras OY (U ) and homomorphisms (5.3) satisfy conditions (F1) and (F2) in Definition 5.27 (cf. Appendix 5.2.2 below), we will denote by OY the pre-sheaf of regular functions on Y as the datum of these K-algebras and morphisms (in Remark 5.1.5 below we deduce that it is a pre-sheaf of integral K-algebras and in Proposition 5.2.4 we will prove that OY is actually a sheaf). For any non-empty open subset U ⊆ Y , if f ∈ OY (U ) is non–zero we will denote by ZU (f ) the zero locus of f in U , i.e. the set ZU (f ) := f −1 (0).

(5.4)

Proposition 5.1.4 Let Y be any algebraic variety. (i) Any f ∈ OY (Y ) is continuous, if f is considered as a function with values in (A1 , Zar1a ). (ii) If f, g ∈ OY (Y ) are such that there exists a non-empty open subset U of Y such that f |U = g|U , then f = g on Y . Proof. (i) One has to show that ZY (f − a) is closed for all a ∈ K. By replacing f with f − a, it suffices to verify that ZY (f ) is closed; this can be done locally. If ZY (f ) 6= ∅, let P ∈ ZY (f ) be any point. Since f is regular at P , G1 |UP , there exists an open neighborhood UP ⊆ Y of P such that f |UP = G 2 for suitable G1 , G2 ∈ H(S(n) of the same degree, G2 6= 0, such that UP ⊆ Up (G2 ) = Zp (G2 )c . Thus ZY (f ) ∩ UP = Zp (G1 ) ∩ UP , where Zp (G1 ) ⊂ Pn a projective hypersurface. In other words we have shown that, for any P ∈ ZY (f ), there exists an open neighborhood UP of P in Y s.t. ZY (f ) ∩ UP is closed in UP . This implies that ZY (f ) is closed in Y . Indeed, denote by ZY (f ) the closure in Y of ZY (f ); if we had ZY (f ) ( ZY (f ), any point Q ∈ ZY (f ) \ ZY (f ) would give that, for / V ∩ ZY (f ). any open neighborhood V of Q in ZY (f ), V ∩ ZY (f ) 6= ∅ but Q ∈ On the other hand, all the open sets of ZY (f ) are induced by opens sets of ZY (f ) thus we would contradict that ZY (f ) ∩ UP is closed in UP , for some P ∈ V ∩ ZY (f ). (ii) Note that ZY (f − g) is closed in Y and contains U , which is dense (see Proposition 4.1.2). ⊓ ⊔

118

5 Regular and rational functions

Remark 5.1.5 (a) Point (i) of Proposition 5.1.4 more generally holds for any locally closed subset in P(V ), on the other hand we will not use it. (b) If f ∈ OY (Y ) is non–zero, then ZY (f ) is a proper closed subset of Y . Therefore UY (f ) := Y \ ZY (f ), also denoted by U (f ) for brevity, is a non-empty open subset of Y which is called the principal open subset of Y associated to f . Notice that, by construction, in U (f ) one can consider the function f1 which is regular therein. (c) A similar proof of Proposition 5.1.4 (ii) shows that, for any non-empty open subset U ⊆ Y , the K-algebra OY (U ) is integral. Indeed if 0 = f · g ∈ OY (U ) then ZU (f · g) = U , i.e. U = ZU (f ) ∪ ZU (g). Since U is irreducible, then either U = ZU (f ) or U = ZU (g), i.e. either f = 0 on U or g = 0 on U . (d) If f : Y → K is a function which is regular at a point P ∈ Y , then the value f (P ) depends only on f and P , i.e. it does not depend on the choice of the open neighborhood U = UP of P and on the choice of representative G1 |U . homogeneous polynomials G1 , G2 of the same degree such that f |U = G 2 (n)

1 (e) For any point P ∈ Y , let ΦP := G be a rational function and let G2 ∈ Q0 UP ⊆ Y be an open neighborhood of P s.t. UP ⊆ Up (G2 ) = Zp (G2 )c . Then ΦP |UP ∈ OY (UP ). For any pair of points P 6= Q ∈ Y , and for any choice of open neighborhoods UP and UQ respectively, one has UP ∩ UQ 6= ∅ since Y is irreducible. Thus, if one has ΦP |UP ∩UQ = ΦQ |UP ∩UQ , the datum  (UP , ΦP |UP ), (UQ , ΦQ |UQ )

defines an element in OY (U ), where U := UP ∪ UQ open set in Y .

5.2 Rational functions For Y any algebraic variety, we let H(Y ) := {(U, f ) | U ⊂ Y non-empty, open subset, f ∈ OY (U )} ; we define a relation R on H(Y ) in this way: (U, f ) R (U ′ , f ′ ) ⇔ f |U ∩U ′ = f ′ |U ∩U ′ . Since Y is irreducible, R makes sense and it is an equivalence relation. Definition 5.2.1 The quotient set H(Y )/R will be denoted by K(Y ) and the equivalence class of (U, f ) will be denoted by [U, f ] (sometimes also by Φf ) and called a rational function on Y .

5.2 Rational functions

119

Lemma 5.2.2 K(Y ) is endowed with a structure of field and moreover it is a field extension of the ground field K. Proof. One poses (i) [U, f ] + [V, g] := [U ∩ V, f + g], (ii) [U, f ] · [V, g] := [U ∩ V, f g], (iii) one has an injective map K ֒→ K(Y ) defined by λ → [Y, λ], for any λ ∈ K, (iv) if [V, g] is such that g 6= 0, then [V, g]−1 := [V \ ZV (g), g1 ] = [UV (g), g1 ], as in Remark 5.1.5. It is straightforward to check that the above conditions are well-defined, i.e. they do not depend on the representatives, and moreover that in (ii) [U ∩ V, f g] 6= 0 if and only if [U, f ] and [V, g] are both non zero (cf. Remark 5.1.5 (c)). ⊓ ⊔ For this reason, K(Y ) is called the field of rational functions on Y Lemma 5.2.3 For any non-empty, open subset U ⊆ Y , one has an injective homomorphism of integral K-algebras jU : OY (U ) ֒→ K(Y ), defined by f → [U, f ].

(5.5) ⊓ ⊔

Proof. Obvious. In particular, one has OY (Y ) ⊆ K(Y );

(5.6)

however the previous inclusion in general is strict: e.g. x1 ∈ K(A1 ) is not regular on A1 , on the other hand x1 ∈ OW (W ), where W = A1 \ {0} ⊂ A1 . Proposition 5.2.4 For any algebraic variety Y , OY is a sheaf (of integral K-algebras). This is called the structural sheaf of Y Proof. We already know that OY is a pre-sheaf of integral K-algebras on Y . We are left to show that condition (F3) in Definition 5.28 holds (cf. Appendix 5.2.2). Let {Ui }i∈I be any family of non-empty, open subset of Y and correspondingly, consider any collection of regular functions (fi ) ∈ Πi∈I OY (Ui ) s.t. U

j i ρU Ui ∩Uj (fi ) = ρUi ∩Uj (fj ), ∀ i, j ∈ I.

This condition implies jUi (fi ) = jUj (fj ), i.e. there exists a rational function Φf ∈ K(Y ) such that Φf = [Ui , fi ] = [Uj , fj ], On the other hand, since its restriction to S any Ui is fi ∈ OY (Ui ), for any i ∈ I, then Φf |U = f ∈ OY (U ), where U = i∈I Ui . The fact that f ∈ OY (U ) is uniquely determined follows from the fact that restriction maps ρU Ui are injective, for any i ∈ I. Indeed, assume there exist f, g ∈ OY (U ) s.t., for any i ∈ I, one has

120

5 Regular and rational functions U ρU Ui (f ) = f |Ui = fi = g|Ui = ρUi (g).

Since f and g are both regular on U and since Ui ⊂ U is open in U , then one concludes by Proposition 5.1.4 (ii). ⊓ ⊔ Definition 5.2.5 For any rational function Φ ∈ K(Y ), we consider the open set [ Ui , UΦ := i

where the union is over all the representatives [Ui , fi ] of Φ in K(Y ). This is called the domain of Φ (or even the open-set of definition of Φ) and it is the biggest open subset of Y over which Φ is regular. With this set-up, one simply has OY (U ) = {Φ ∈ K(Y ) | U ⊆ UΦ } . Definition 5.2.6 Let Y be any algebraic variety. Any locally-closed and irreducible subset W ⊆ Y is said to be a (algebraic) subvariety of Y . In other words, W is an algebraic variety on its own which is a locally-closed subset of Y . Remark 5.2.7 (i) Take any algebraic variety Y ⊆ An and consider its (affine) coordinate ring A(Y ), as in Def. 4.1.7. For any subvariety W ⊆ Y , the ideal Ia,Y (W ) := Ia (W )/Ia (Y ),

(5.7)

defined as the image of the ideal Ia (W ) ⊆ A(n) via the canonical quotient morphism A(n) ։ A(Y ), is a prime ideal of A(Y ) (when otherwise W = Y , one obviously has Ia,Y (W ) = (0)). (ii) Similarly, let Y ⊆ Pn be any algebraic variety and let S(Y ) be its (homogeneous) coordinate ring. For any subvariety W ⊆ Y , the homogeneous ideal Ip,Y (W ) := Ip (W )/Ip (Y ), (5.8) defined as the image of the homogeneous ideal Ip (W ) ⊆ S(n) via S(n) ։ S(Y ), is a homogeneous prime ideal of S(Y ). Definition 5.2.8 Let W be a subvariety of an algebraic variety Y . A rational function Φ ∈ K(Y ) is said to be defined in W if there exists a representative (U, f ) of Φ = [U, f ] s.t. U ∩ W 6= ∅, i.e. Φ is regular in an open, dense subset of W . The set of rational functions defined in W will be denoted by OY,W ; in symbols OY,W := {Φ ∈ K(Y ) | UΦ ∩ W 6= ∅} .

5.2 Rational functions

121

Operations (i)-(iii) as in the proof of Lemma 5.2.2 make OY,W an integral K-algebra, for any subvariety W . The following properties are easy consequences of the previous definitions. Proposition 5.2.9 Let Y be any algebraic variety and let W ⊆ Y be any subvariety. Then (i) OY,Y = K(Y ); (ii) OY,W is a subring of the field K(Y ); (iii) OY,W contains OY (Y ) as a subring; (iv) the set mY,W := {Φ ∈ OY,W | Φ(P ) = 0, ∀ P ∈ UΦ ∩ W } is an ideal of OY,W . When in particular W = {P } is a point, OY,P is called the ring of germs of regular functions at P and mY,P is the ideal of germs of regular functions vanishing at P . For any point P ∈ Y , one has OY (Y ) ⊆ OY,P ⊆ K(Y ). Thus, for any non-empty, open set U ⊆ Y , OY (U ) = {Φ ∈ K(Y ) | Φ regular at P , ∀ P ∈ U } =

(5.9) \

OY,P ,

(5.10)

P ∈U

whereas K(Y ) = {Φ ∈ K(Y ) | Φ regular at some point P ∈ Y } =

[

OY,P . (5.11)

P ∈Y

Recalling definitions as in § 1.11, we can prove the following result. Theorem 5.2.10 Let Y be any algebraic variety and let W be any subvariety of Y . Then (OY,W , mY,W ) is a local ring of residue field K(W ). Proof. Take any Φ := [U, f ] ∈ OY,W \ mY,W where, up to a change of representative (U, f ) for Φ, we can assume U ∩ W 6= ∅. Since f ∈ OY (U ), ZU (f ) is a proper closed subset of U (cf. the proof of Prop. 5.1.4). Moreover, since f ∈ / mY,W , then also ZU (f ) ∩ W is a proper, closed subset of U ∩ W . Let U ′ := UU (f ) = U \ ZU (f ) be the principal open set in U associated to f . From Remark 5.1.5 (b), one has f ′ := f1 ∈ OY (U ′ ); moreover U ′ ∩ W = (U ∩ W ) \ (ZU (f ) ∩ W ) 6= ∅. Thus Φ is invertible with Φ−1 := [U ′ , f ′ ] ∈ OY,W . From Proposition 1.11.10 we conclude that (OY,W , mY,W ) is a local ring. Let now KY,W := OY,W /mY,W be the residue field of (OY,W , mY,W ). Consider the map

122

5 Regular and rational functions

ϕY,W :

KY,W −→ K(W ) [U, f ] + mY,W −→ [U ∩ W, f |U ∩W ].

(5.12)

This is well-defined and it is easy to check that it is a (non-zero) homomorphism of K-algebras; since KY,W and K(W ) are boths fields, ϕY,W is injective. We now prove that ϕY,W is also surjective. Let [U ′ , f ′ ] ∈ K(W ) and let P ∈ U ′ be any point. The fact that f ′ is regular at P implies there exists an open neighborhood U ′′ of P in U ′ and two homogeneous polynomials G1 , G2 ∈ H(S(V ∗ )) of the same degree, with G2 6= 0, such that f ′ |U ′′ =

G1 |U ′′ G2

(5.13)

and U ′′ ⊆ Up (G2 ) (recall definitions as in § 5.1). e ⊂ P(V ) s.t. One can always find an open subset U

e ∩ W = U ′′ and U e ∩ Zp (H) = ∅ U

e with U e ∩ Up (H)). Let U := U e ∩ Y . By (otherwise one can always replace U e the choice of U , one has U ⊆ Up (H). Define f :=

G1 |U ∈ OY (U ) ⊆ K(Y ). G2

(5.14)

e ∩ W = U ′′ ⊂ W , then [U, f ] ∈ OY,W . From (5.13) and (5.14) Since U ∩ W = U it follows that [U ′′ , f |U ′′ ] R [U ′ , f ′ ], i.e. ϕY,W ([U, f ] + mY,W ) = [U ′ , f ′ ],

proving the surjectivity of ϕY,W .

⊓ ⊔

Definition 5.2.11 For any algebraic variety Y and any subvariety W ⊆ Y , the ring (OY,W , mY,W ) is called the local ring of W in Y . For any open subset U ⊆ Y s.t. U ∩ W 6= ∅ the map jU as in (5.5) induces an injective K-algebra homomorphism γU : OY (U ) ֒→ OY,W , defined by f → [U, f ].

(5.15)

Definition 5.2.12 The ideal IU (W ) := γU−1 (mY,W ) = {f ∈ OY (U ) | ZU (f ) ⊇ U ∩ W } will be called the ideal of W in U . Lemma 5.2.13 Let Y be any algebraic variety and let U ⊆ Y be any nonempty, open subset. (i) For any subvariety W ⊆ Y s.t. W ′ := U ∩ W 6= ∅, one has OU,W ′ ∼ = OY,W . ∼ (ii) In particular, K(U ) = K(Y ).

5.2 Rational functions

123

Proof. It is clear that (ii) is a particular case of (i), taking W = Y . Therefore, we only need to prove (i). Let [U ′ , f ′ ] ∈ OU,W ′ , where U ′ ⊆ U an open set s.t. U ′ ∩W ′ = U ′ ∩U ∩W 6= ∅. Since U ′ is open in Y , f ′ ∈ OY (U ′ ) and U ′ ∩ W 6= ∅, then [U ′ , f ′ ] ∈ OY,W . Thus, one has an injective K-algebra homomorphism ψ : OU,W ′ ֒→ OY,W , defined by [U ′ , f ′ ] → [U ′ , f ′ ]. We notice that ψ is also surjective; indeed for any [U ′′ , f ′′ ] ∈ OY,W , where the representative (U ′′ , f ′′ ) is s.t. U ′′ ∩ W 6= ∅, one has [U ′′ , f ′′ ] = ψ ([U ′′ ∩ U, f ′′ |U ′′ ∩U ]) since U ′′ ∩ U ∩ W 6= ∅.

⊓ ⊔

Recalling notation as in Remarks 1.11.11, 5.2.7, we can prove the following fundamental result. Theorem 5.2.14 Let Y ⊆ An be any affine variety. Then (a) OY (Y ) = A(Y ). (b) For any subvariety W ⊆ Y , the ideal IY (W ) (cf. Def. 5.2.12) is a prime ideal in OY (Y ) which is isomorphic to the prime ideal Ia,Y (W ) of A(Y ) as in (5.7). Conversely any prime ideal of OY (Y ) is of this type; furthermore, IY (W ) is a maximal ideal if and only if W is a point P ∈ Y . (c) For any subvariety W ⊆ Y , OY,W ∼ = OY (Y )IY (W ) ∼ = A(Y )Ia,Y (W ) . (d) K(Y ) = Q(A(Y )) whereas, for any subvariety W ⊆ Y , K(W ) ∼ =

OY (W )IY (W ) IY (W )OY (W )IY (W )

∼ =

A(Y )Ia,Y (W ) . Ia,Y (W )A(Y )Ia,Y (W )

Let Y ⊆ Pn be any projective variety. Then (e) OY (Y ) = K. (f ) For any subvariety W ⊆ Y , the ideal Ip,Y (W ) as in (5.8), is a homogeneous, prime ideal of S(Y ). Conversely any homogeneous, non-irrelevant, prime ideal of S(Y ) is of this type. (g) For any subvariety W ⊆ Y , OY,W ∼ = S(Y )(Ip,Y (W )) . (h) K(Y ) = S(Y )((0)) whereas, for any subvariety W ⊆ Y , K(W ) ∼ =

S(Y )(Ip,Y (W )) . Ip,Y (W )S(Y )(Ip,Y (W ))

124

5 Regular and rational functions

Proof. Let us first focus on the affine case. (a) By definition of A(Y ), there is a natural injective K-algebra homomorphism α : A(Y ) ֒→ OY (Y ). From the Hilbert ”Nullstellensatz”-weak form in An (cf. Theorem 2.2.1) and (5.7), there is a bijective correspondence between points P ∈ Y and maximal ideals mY (P ) := Ia,Y (P ) of A(Y ); identifying A(Y ) with its image α(A(Y )), we can therefore interpret mY (P ) = {f ∈ A(Y ) ⊆ OY (Y ) | f (P ) = 0} ⊂ OY (Y ), i.e. mY (P ) ⊆ IY (P ), where the latter ideal is as in Def. 5.2.12. Claim 5.2.15 For any P ∈ Y , one has A(Y )mY (P ) ∼ = OY,P . Proof (Proof of Claim 5.2.15). For any fg ∈ A(Y )mY (P ) , the set UY (g) := Y \ ZY (g) is a non empty, open subset of Y containing P , indeed g ∈ A(Y ) \ mY (P ) by definition of A(Y )mY (P ) so fg is regular over UY (g) (cf. the proof of Prop. 5.1.4 (i) and Remark 5.1.5 (b)). We can therefore consider the following map   f f . (5.16) αP : A(Y )mY (P ) −→ OY,P , defined by −→ UY (g), g g This is a K-algebra homomorphism.

    ′ Notice that αP is an isomorphism. First UY (g), fg = UY (g ′ ), fg′ if and

only if on UY (g) ∩ UY (g ′ ) 6= ∅ one has f g ′ − f ′ g = 0. Since UY (g) ∩ UY (g ′ ) is an open, dense subset of Y , by Proposition 5.1.4 (ii), one has f g ′ − f ′ g = 0 on Y , i.e. f g ′ − f ′ g = 0 ∈ A(Y ). Since g, g ′ ∈ A(Y ) \ mY (P ), from (1.34) we ′ get that fg = fg′ ∈ A(Y )mY (P ) , so αP is injective. At last, the surjectivity of αP directly follows from Lemma 5.1.3. ⊓ ⊔ By construction, αP is a local isomorphism of local rings, i.e. the maximal ideals mY (P )A(Y )mY (P ) and mY,P = IY (P ) bijectively correspond under αP . In particular, this proves points (b), (c) and (d) for the case W = {P }. From (5.10) and Claim 5.2.15 we get \ \ OY,P ∼ A(Y ) ∼ A(Y )mY (P ) , = = α(A(Y )) ⊆ OY (Y ) = P ∈Y

mY (P )∈Specm(A(Y ))

where Specm(A(Y )) := {maximal ideals of A(Y )}. Since A(Y ) is integral, then any A(Y )mY (P ) is a subring of Q(A(Y )) and, from the very definition of localization (cf. also [22, Lemma 2, p.8]) one has

5.2 Rational functions

\

125

A(Y )mY (P ) = A(Y )

mY (P )∈Specm(A(Y ))

proving (a). (b) Since Ia (W ) = Ia (W ), we can consider W a closed subvariety. By definition of A(Y ), we have a bijective correspondence between prime ideals p ⊆ A(Y ) and prime ideals p ⊂ A(n) containing Ia (Y ). From Proposition 2.1.11 (i) and (2.15), any such prime ideal p corresponds to a closed subvariety W := Za (p) ⊆ Za (Ia (Y )) = Y ⊆ An . Since p is prime, it is radical so p = Ia (W ) and p = Ia,Y (W ). Using (a), the prime ideal Ia,Y (W ) by definition coincides with the ideal IY (W ) ⊂ OY (Y ). At last, from the proof of part (a), Ia,Y (W ) ∼ = IY (W ) is maximal if and only if W is a point. (c) On the one hand, for any subvariety W ⊆ Y , the ring OY,W is local, with maximal ideal mY,W (cf. Theorem 5.2.10). On the other, from (a) above, we have A(Y ) ∼ = OY (Y ) and Ia,Y (W ) ∼ = IY (W ). Thus, A(Y )Ia,Y (W ) ∼ = OY (Y )IY (W ) is a local ring of maximal ideal Ia,Y (W )A(Y )Ia,Y (W ) ∼ = IY (W )OY (Y )IY (W ) (cf. Prop. 1.11.12). Since A(Y )Ia,Y (W ) ⊆ K(Y ), as in the proof of Claim 5.2.15, one can easily construct a map αW : A(Y )Ia,Y (W ) −→ OY,W and prove that it is a local isomorphism of local K-algebras. (d) From Proposition 5.2.9 (i) and from (c), we get K(Y ) = OY,Y ∼ = A(Y )(0) = Q(A(Y )), the last equality following from Remark 1.11.11. Similarly, from Theorem 5.2.10, Proposition 1.11.12 (i) and from (c) above, one gets the assertions on K(W ). We now consider the case of Y ⊆ Pn any projective variety. Recalling notation as in (1.39) and (3.14), a preliminar step is given by the following result. Lemma 5.2.16 Let i ∈ {0, . . . , n} be any integer for which Yi := Y ∩ Ui 6= ∅. Then A(Yi ) ∼ (5.17) = S(Y )([Xi ]) , where Xi ∈ S(Y ) denotes the image via the quotient morphism S(n) ։ S(Y ) of the indeterminate Xi ∈ S(n) .

126

5 Regular and rational functions

Proof (Proof of Lemma 5.2.16). Assuming that the affine variety Y0 6= ∅, for simplicity we consider only the case i = 0, the other cases being similar. One has a natural K-algebra isomorphism   Xn X1 (n) (n) ϕ0 ,..., . A −→ S([X0 ]) , defined by f (x1 , . . . , xn ) → f X0 X0 If f ∈ Ia (Y0 ), by (1.33) and the definition of Ip (Y ), we get that h0 (f ) = deg(f ) X0 ϕ0 (f ) ∈ Ip (Y ), i.e. (n)

ϕ0 (f ) ∈ Ip (Y )S([X0 ]) . (n)

(n)

Notice that Ip (Y )S([X0 ]) is a proper ideal of S([X0 ]) : indeed if we denote by S the multiplicative system given by 1 and by all the powers of the indeterminate X0 , by the fact that Y0 6= ∅ it follows that Ip (Y ) ∩ S = ∅, so one concludes by Proposition 1.11.6 (ii). In particular, ϕ0 induces a homomorphism of K-algebras (n)

ϕ0 : A(Y ) −→

S([X0 ]) (n)

Ip (Y )S([X0 ])

∼ = S(Y )([X0 ])

where the isomorphism on the right directly follows from the fact that S(Y ) = S(n) /Ip (Y ) and from the definition of localization with respect to the multiplicative system S = {1, X0 , X02 , X03 , . . .}. Notice that ϕ0 is an isomorphism: it is injective since ϕ0 (f ) = 0 if and only if h0 (f ) = 0 i.e. if and only if f = 0; the surjectivity follows from the fact that, for any positive integer n and for any F ∈ S(Y )n , there always exists a ⊓ ⊔ polynomial f ∈ A(Y ) such that XFn = ϕ0 (f ). 0

We can proceed with the proof of the ”projective” part of the statement. (g) Let W ⊆ Y be any (closed) subvariety and let i ∈ {0, . . . , n} be any integer for which Yi 6= ∅ and Wi := W ∩ Ui 6= ∅. For simplicity, assume that it occurs for e.g. i = 0. From Lemma 5.2.13 (i), we have OY,W ∼ = OY0 ,W0 . Since Y0 and W0 are affine varieties, from (c) above we have OY0 ,W0 ∼ = OY0 (Y0 )IY0 (W0 ) . Moreover, from (a), one has also OY0 (Y0 ) ∼ = Ia,Y0 (W0 ). = A(Y0 ) and IY0 (W0 ) ∼ From the proof of Lemma 5.2.16, the isomorphism ϕ0 bijectively maps the ideal Ia,Y0 (W0 ) to the ideal Ip,Y (W )S(Y )([X0 ]) so we get
OY,W ∼ . = S(Y )([X0 ]) I (W )S(Y ) = A(Y0 )Ia,Y0 (W0 ) ∼ p,Y

([X0 ])

5.2 Rational functions

127

Since X0 ∈ / Ip,Y (W ) (because W0 6= ∅ by assumption), reasoning as in Lemma 5.1.3, one easily notices that
∼ S(Y )([X0 ]) I (W )S(Y ) = S(Y )(Ip,Y (W )) p,Y

([X0 ])

proving (g).

(h) The statement about K(Y ) follows from Proposition 5.2.9 (i) and from (g), whereas statement about K(W ) follows from Theorem 5.2.10, Proposition 1.11.12 (ii) and from (g) above. (f) As in the affine case, it directly follows from bijective correspondence as in Corollary 4.1.12 for Pn . (e) Let f ∈ OY (Y ) be any non-zero element. Since for any algebraic variety we have OY (Y ) ⊆ K(Y ), from (h) above any such f can be considered as an element of K(Y ) ∼ = S(Y )((0)) = Q0 (S(Y )) ⊂ Q(S(Y )), where Q0 (S(Y )) as in Definition 1.10.9. Notice first that an index j ∈ {0, . . . , n} is such that Yj = ∅ if and only if Y ⊆ Zp (Xj ) = Hj , where Hj ∼ = Pn−1 the hyperplane at infinity of the affine open set Uj ; this occurs if and only if Xj ∈ Ip (Y ), in which case S(Y ) =

S(n) ∼ S(n−1) = Ip (Y ) Ip,Hj (Y )

where S(n−1) ∼ = S(n) /(Xj ) and Ip,Hj (Y ) = Ip (Y )/(Xj ). In other words, for any such j, we can replace S(n) with S(n−1) , i.e. Y ⊂ Pn is degenerate as it is contained into the hyperplane Hj ∼ = Pn−1 and one can work directly therein. From now on we will therefore assume Yi 6= ∅, for any i ∈ {0, . . . , n}. Since f ∈ OY (Y ) then f |Yi ∈ OYi (Yi ), for any i. Moreover, as Yi is affine, from (a) and Lemma 5.2.16, f |Yi ∈ OYi (Yi ) ∼ = A(Yi ) ∼ = S(Y )([Xi ]) . Thus, for any i ∈ {0, . . . , n}, there exist a positive integer Ni and a polynomial Gi ∈ S(Y )Ni such that f |Yi := GNii , i.e. Xi

XiNi f |Yi ∈ S(Y )Ni , ∀ i ∈ {0, . . . , n}. Since

f ∈ OY (Y ) ⊂ K(Y ) ∼ = S(Y )((0)) = Q0 (S(Y )) ⊂ Q(S(Y ))

and f |Yi ∈ S(Y )([Xi ]) ⊂ Q0 (S(Y )) ⊂ Q(S(Y )),

for any i ∈ {0, . . . , n}, considering f as an element of Q(S(Y )) we have XiNi f ∈ S(Y )Ni , ∀ i ∈ {0, . . . , n}.

(5.18)

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5 Regular and rational functions

Let N be any positive integer such that N > generated as a K-vector space by monomials X0α0 X1α1 · · · Xnαn s.t.

n X

Pn

i=1

Ni . Since S(Y )N is

αi = N,

i=0

from (5.18) we get S(Y )N f ⊆ S(Y )N .

(5.19)

Recursively applying (5.19), for any positive integer q one has S(Y )N f q ⊆ S(Y )N . In particular, X0N f q ∈ S(Y )N , for any integer q > 1, i.e. S(Y )[f ] ⊆ S(Y ) ·

1 . X0N

(5.20)

Since S(Y ) is a noetherian ring and since S(Y )· X1N is a finitely generated S(Y )0 module, then S(Y )[f ] is a finitely generated S(Y )-module (cf. [1, Propositions 6.2, 6.5]). From Proposition 1.6.3, it follows that f ∈ Q0 (S(Y )) is therefore integral over S(Y ), i.e. there exist a1 , . . . , am ∈ S(Y ) s.t. f m + a1 f m−1 + · · · + am = 0.

(5.21)

Since f ∈ Q0 (S(Y )) there exist homogeneous polynomials G1 , G2 ∈ H(S(Y )) G1 , so (5.21) becomes of the same degree such that f = G 2 m−1 Gm G2 + · · · + am Gm 1 + a1 G1 2 = 0.

Let aj = Aj0 + Aj1 + . . . + Ajℓj be the decomposition of the coefficients aj ∈ S(Y ) in their homogeneous components, 1 6 j 6 n. Decomposing the previous equality into homogeneous pieces we get in particular Gm + A10 Gm−1 H + · · · + Am0 H m = 0, i.e. f m + A10 f m−1 + · · · + Am0 = 0,

(5.22)

where Aj0 ∈ S(Y )0 = K, for any 1 6 j 6 m. In particular, f ∈ K(Y ) is algebraic over K. Since K is algebraically closed, then f ∈ K proving (e). ⊓ ⊔ From Theorem 5.2.14 (a) and (e) it follows that for Y either an affine or a projective variety, OY (Y ) is an integral K-algebra of finite type. This hase already been observed to more generally hold for any algebraic variety Y (cf. Remark 5.1.5 (c)).

5.2 Rational functions

129

5.2.1 Consequences of Theorem 5.2.14 We want to discuss some fundamental consequences of Theorem 5.2.14. Corollary 5.2.17 For any integer n > 0, one has OAn (An ) = A(n) , OPn (Pn ) = K, and K(An ) ∼ = K(Pn ) ∼ = Q(n) . Proof. The first two equality are given by (a) and (e) of Theorem 5.2.14. The fact that K(An ) ∼ = Q(n) is point (d) of the same result. The fact that n ∼ (n) also K(P ) = Q follows either from Lemma 5.2.13 (ii) or from point (h) of Theorem 5.2.14, Proposition 1.11.12 (ii) and isomorphism (5.1), namely (n) K(Pn ) ∼ ⊔ = S((0)) ∼ = Q0 (S(n) ) ∼ = Q(A(n) ) = Q(n) . ⊓

Corollary 5.2.18 If Y is any affine variety, the ring A(Y ) completely determines the ringed space (Y, OY ), i.e. it determines the topological space Y together with all integral, K-algebras OY (U ) and OY,P , for any open set U ⊆ Y and for any point P ∈ Y . Proof. From the bijective correspondence of Theorem 5.2.14 (b) between points P ∈ Y and maximal ideals of A(Y ), Y can be identified with Specm(A(Y )) := {maximal ideals of A(Y )}. In this identification, the correspondence between prime ideals p of A(Y ) and closed subvarieties ZY (p) ⊆ Y in Theorem 5.2.14 (b), reads as ZY (p) = {m ∈ Specm(A(Y )) | p ⊆ m}.

(5.23)

Since Y is a noetherian topological space, any closed subset Z is a finite union of its irreducible components, say Z1 , . . . , Zk . From Proposition 2.2.8 applied i) to ideals Ia,Y (Zi ) = IIaa(Z (Y ) of A(Y ), we get Ia,Y (Z) =

k \

Ia,Y (Zi )

i=1

so identification (5.23) can be extended to any radical ideal of A(Y ). This implies that A(Y ) recover the whole topology of Y , since it determines the family of all its closed subsets. From (5.24) and (5.10), where any OY (U ) and any OY,P is viewed as a subring of K(Y ) as in (5.11), one realizes that A(Y ) determines also the structural sheaf OY . ⊓ ⊔ Corollary 5.2.19 Let Y be any algebraic variety, and let U ⊂ Y be any non empty open subset. Then OU ∼ = OY |U

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5 Regular and rational functions

(cf. Remark 5.29 below). In particular, if Y is any affine variety and if Y denotes its projective closure, then for any P ∈ Y one has OY,P ∼ = mY ,P . = OY ,P and mY,P ∼ Proof. From Definition 5.2.5 and (5.10) one has that, for any non empty open set V ⊆ U ⊂ Y , OU (V ) = {Φ ∈ K(U ) | V ⊂ UΦ } and OY (V ) = {Φ ∈ K(Y ) | V ⊂ UΦ }. Then, one concludes by Lemma 5.2.13 (ii).

⊓ ⊔

Recalling Definition 1.5.5, one has also the following fundamental result. Corollary 5.2.20 Let Y be any algebraic variety. Then K(Y ) is finitely generated field extension of the base field K. Proof. From Lemma 5.2.13 (ii), we can assume Y to be projective. Thus, from Theorem 5.2.14 (h), K(Y ) ∼ = S(Y )((0)) = Q0 (S(Y )). Since S(Y ) is a quotient of S(n) , it is a K-algebra of finite type generated by the images of the indeterminates X0 , . . . , Xn via the quotient morphism. Thus, Q(S(Y )) is a finitely generated field extension of K. Since Q0 (Y ) is the subfield of degree-zero fractions in Q(S(Y )), one can conclude. ⊓ ⊔ Corollary 5.2.21 Let Y be any algebraic variety and let W ⊆ Y be any subvariety. (i) There is a bijiective correspondence between prime ideals OY,W and closed subvarieties of Y containing W . (ii) Q(OY,W ) = K(Y ). Proof. Since OY,W ⊆ K(Y ), from Lemma 5.2.13 (ii) we can assume Y to be affine. (i) Since Y affine, the statement follows from (b) and (c) of Theorem 5.2.14 and from Remark 1.11.9. (ii) By (a) and (d) of Theorem 5.2.14 one has OY (Y ) = A(Y ) ⊆ OY,W ⊆ K(Y ) and Q(A(Y )) = K(Y ), which proves the statement.

⊓ ⊔

5.2 Rational functions

131

5.2.2 Examples We discuss here some applications of the previous results. Example 5.2.22 (Points) (i) If Y = {P } is a point, from the first part of Theorem 5.2.14 we get OP (P ) ∼ = K(P ) ∼ = K. (ii) More generally, if Y is any algebraic variety and P ∈ Y is any point, one has K(P ) ∼ = K, i.e. the field of rational functions of a point is of intrinsic nature for P , i.e. it is independent from the inclusion of P as a subvariety of a given variety Y . To prove this, one can assume Y to be affine: indeed, if Y is any quasiprojective variety, consider Y ⊆ Pn be its projective closure and, for any index i ∈ {0, . . . , n} s.t. Y ∩Ui 6= ∅, take the affine variety Yi := Y ∩Ui ; from Lemma 5.2.13 (ii), K(Y ) = K(Y ) = K(Yi ) so we can replace Y by Yi . Since we can consider Y to be affine, from Theorem 5.2.14 (c), OY,P ∼ = A(Y )mY,P ,

(5.24)

where mY,P denotes both the ideal of germs of regular functions vanishing at P and the ideal mP /Ia (Y ), via the identification in (a) of Theorem 5.2.14. Then, from (d) and Proposition 1.11.12 (i), we get K(P ) ∼ =

A(Y )mY,P ∼ = Q (A(Y )/mY,P ) ∼ = K, mY,P A(Y )mY,P

since A(Y )/mY,P ∼ = K. (iii) If Y ⊂ Pn is any projective variety and if P ∈ Y is any point, from Theorem 5.2.14 (f), one has that Ip,Y (P ) is a homogeneous prime ideal of S(Y ) which does not coincide with the irrelevant ideal of S(Y ); in particular it is not maximal (as it occurs also for Ip (P ) ⊂ S(n) , cf. § 3.3.1). The ideal Ip,Y (P ) is the prime ideal generated by all homogeneous elements F ∈ H(S(Y )) vanishing at P . Example 5.2.23 (Irreducible conics) (i) Let Y := Za (x1 x2 − 1) ⊂ A2 , which is a hyperbola. Since x1 x2 − 1 ∈ A(2) is irreducible, Ia (Y ) ⊂ A(2) is a prime ideal. The (affine) coordinate ring A(Y ) is s.t. (1) A(Y ) ∼ = K[x1 ][x−1 1 ] = Ax1

(recall notation as in (1.38)). From Theorem 5.2.14, we have OY (Y ) ∼ = A(1) x1 . In particular OA1 (A1 ) ∼ = A(1) is identified with a (proper) subring of A(Y ). On the other hand, always from Theorem 5.2.14, one has

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5 Regular and rational functions

K(A1 ) ∼ = K(Y ) ∼ = Q(1) . To give geometric motivations of the behaviours of these function rings, consider the quasi-affine variety W := A1 \ {0} ⊂ A1 and φ1

A1 ⊃ W −→ Y ⊂ A2 1 t → (t, t ) x1 ← (x1 , x2 ) . It is clear that φ1 is a homeomorphism. Moreover, one can easily observe that (1) Ax1 can be identified with a subring of OW (W ) (in Example 6.2.5 (iv), we (1) will more precisely show that Ax1 ∼ = OW (W )). As for function fields, one has 1 K(W ) = K(A ), since W a principal open set of A1 (cf. Lemma 5.2.13). Thus Y and W are not only homeomorphic as topological spaces, but moreover the function rings OY (Y ) = A(Y ) and K(Y ) are isomorphic to OW (W ) and K(W ), respectively. In Remark 6.1.4 we shall see that φ1 is indeed much more than a homemorphism between topological spaces; it is actually an isomorphism of algebraic varieties, which in particular garantees that for any t ∈ W one has OW,t ∼ = OY,Pt , where Pt := φ1 (t) ∈ Y (cf. (6.3)). From (5.10) and (5.11), the previous isomorphism gives also that ∼ K(Y ) ∼ = K(W ), OY (U ) ∼ = OW (φ−1 1 (U )), OW (V ) = OY (φ1 (V )), for any open sets U ⊂ Y and V ⊂ W (cf. Definition 6.1.1 and (6.4)) The map φ1 is called a rational parametrization of the hyperbola Y whereas φ−1 is nothing but the restriction to Y of the first projection of A2 , i.e. the 1 projection of the affine plane onto its x1 -axis.

Fig. 5.1. Affine hyperbola with its parametrization (real part)

Notice moreover that φ1 is the restriction of a natural projective homeomorphism. Indeed, identifying A2 with the affine chart U0 of P2 , the projective closure Y of Y is given by Zp (X1 X2 −X02 ). The x1 axis becomes Zp (X2 ), which is homeomorphic to P1 . In this way, one has the homemorphism Φ

1 P1 −→ Y ⊂ P2 2 2 [λ, µ] → [λµ, µ , λ ] [a0 , a1 ] ← [a0 , a1 , a2 ] ,

5.2 Rational functions

133

where [a0 , a1 ] = [λµ, µ2 ] = [λ, µ] as it occurs on points of Y ⊂ U0 (cf. Example 8.1.11 for more general motivations). The map Φ−1 naturally restricts to φ−1 and moreover it sends the two 1 1 points at infinity of Y , i.e. Q1 = [0, 0, 1] and Q2 = [0, 1, 0], respectively to [1, 0] ∈ A1 \ W and to [0, 1], the point at infinity of A1 . In the next chapter, we will show that Φ1 is an isomorphism of projective varieties, which ensures as above that regular and rational functions on Y will be the same as those on P1 . One can easily give a geometric interpretation of Φ−1 1 , observing that this map is obtained by simply projecting Y to the line Zp (X2 ) ⊂ P2 from the point Q1 = [0, 0, 1] ∈ Y . To see this, consider the pencil of lines in P2 through the point Q1 ; equation for this pencil is given by Zp (λX1 − µX0 ) ⊂ P2 , for [λ, µ] ∈ P1 . For any point P ∈ Y , there is only one line of the pencil passing through P and conversely, since Y is a conic, points P ∈ Y are in bijiective correspondence with the subset of lines in the pencil consisting of all lines except for that passing through Q2 and for the tangent line to Y at Q1 . The line of the pencil passing through Q2 is given by Zp (X0 ), whose trace over the X1 -axis in P2 is the point [0, 1, 0], identified with [0, 1] ∈ P1 via Φ−1 , i.e. Q2 corresponds via this projection to the point at infinity of the originary A1 . Finally, the tangent line to Y at Q1 is Zp (X1 ), which itersects Y only at the point Q1 with multiplicity 2. The trace of this tangent line over the X1 -axis is the point [0, 0, 1], identified with [1, 0] ∈ P1 via Φ−1 1 , i.e. Q1 corresponds via the projection to the origin O ∈ A1 \ W , where the originary parametrization φ1 was not defined.

Fig. 5.2. Projective conic with its parametrization (real part)

(ii) Let now V := Za (x2 − x21 ) ⊂ A2 , which is a parabola. This conic has ”more similarity” to A1 than what the hyperbola above does. Indeed, the (affine) coordinate ring A(V ) is A(V ) ∼ = K[x1 ] = A(1) = A(A1 ). From Theorem 5.2.14, K(V ) ∼ = K(x1 ) ∼ = Q(1) ∼ = K(A1 ),

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5 Regular and rational functions

as it also occurred for the hyperbola. As in (i), one has the following homeomorphism φ2 A1 −→ V ⊂ A2 2 t → (t, t ) x1 ← (x1 , x2 ) , i.e. V is homeomorphic to A1 (φ2 will be more precisely an isomorphism of algebraic varieties, cf. Remark 6.1.4). The map φ2 is a polynomial parametrization of V (recall (3.31)) and φ−1 is as in (i) the restriction to V of the first 2 projection of A2 .

Fig. 5.3. Affine parabola with its parametrization (real part)

As in case (i), φ2 is the restriction of the projective homemorphism Φ

2 V ⊂ P2 P1 −→ [λ, µ] → [λ2 , λµ, µ2 ] [a0 , a1 ] ← [a0 , a1 , a2 ] ,

where [a0 , a1 ] = [λ2 , λµ] = [λ, µ] as it occurs on points of V .

Fig. 5.4. Projective conic with its parametrization (real part) −1 The map Φ−1 2 naturally restricts to φ2 and, similarly as in (i), it is nothing 1 but the projection of V to P , as one can show by using the pencil of lines through [0, 0, 1] which is the point at infinity of V , i.e. where V is tangent to the line Zp (X0 ). As in (i), Φ2 will be an isomorphism of projective varieties (cf. Chapter 6). (iii) At last, consider Z := Za (x21 + x22 − 1) ⊂ A2 , which is a circle. Consider K = C; similar computations as in Remark 2.1.16 (iii) show that one has a rational parametrization

A1 ⊃

φ3

W −→ Z ⊂ A2 t2 +1 t2 −1 t → ( 2it , 2t ) x2 − ix1 ← (x1 , x2 ) ,

where i2 = −1 and W := A1 \ {0}. The map φ−1 is given by the pencil of 3 (parallel) lines in A2

5.2 Rational functions

135

ℓt : x2 − ix1 − t = 0, t ∈ W. This is actually given by the pencil of (projective) lines through the cyclic point [0, i, 1], i.e. through one of the two points at infinity of Z. One easily computes that 1 ∼ (1) ∼ A(Z) ∼ = A(1) x1 = A(W ) and k(Z) = K(W ) = K(A ) = Q

as it occurred in the case of hyperbola. The map φ3 will be indeed an isomorphism of algebraic varieties between Z and W (cf. Remark 6.1.4). Example 5.2.24 (Semi-cubic parabola) Let Y := Za (x31 − x22 ) ⊂ A2 , which is called a semi-cubic parabola (or also a cuspidal plane cubic). In this case, Y has a polynomial parametrization as in (3.31) given by φ

A1 −→ Y ⊂ A2 t → (t2 , t3 ) , and A(Y ) ∼ = K[t2 , t2 ] ⊂ K[t]. In particular, A(Y ) is not isomorphic to A(A1 ). On the other hand, since Y is affine, from Theorem 5.2.14 one has K(Y ) ∼ = K(t) ∼ = K(A1 ). As in Example 5.2.23, the pencil of lines x2 = t x1 through the origin gives a bijective correspondence between Y and A1 by the following rule: ψ

Y −→ A1 (0, 0) 6= (x1 , x2 ) → xx12 (0, 0) = (x1 , x2 ) → 0 The map ψ is bijective so it is a homeomorphism between (Y, Zar2a,Y ) and (A1 , Zar1a ); moreover ψ −1 = φ, i.e. φ is a homomorphism.

Fig. 5.5. Semi-cubic parabola or cuspidal plane cubic (real part)

In Example 6.2.5 (v) we will show that φ on the other hand is not an isomorphism of algebraic varieties. Recall that Corollary 5.2.20 ensures that, for any algebraic variety Y , K(Y ) is a finitely generated field extension of K. Notice that all fields of rational functions appearing in Examples 5.2.23 and 5.2.24 are more precisely isomorphic to K(t), where t an indeterminate, i.e. they all are purely trascendental extensions of K, with trascendence degree 1 (cf. (1.17)). The following example shows that this is not always the case, i.e. not all algebraic varieties have fields of rational functions which are purely trascendental extension of K.

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5 Regular and rational functions

Example 5.2.25 (Some elliptic affine plane cubics) Let K = C and Ya := Za (x22 − x1 (x1 − 1)(x1 − a)) ⊂ A2 , a ∈ C. When a is either 0 or 1, Ya has a node (i.e. an ordinary double point) at the origin and for this reason it has a rational parametrization. To see this, for simplicity, consider the case a = 0 the other being similar.

Fig. 5.6. Nodal plane cubic (real part)

Y0 is an affine rational cubic with polynomial parametrization φ

A1 −→ Y0 , defined by t → (t2 + 1, t3 + t). Notice that the map φ is injective only on W := A1 \{i, −i}, i2 = −1, whereas φ(i) = φ(−i) = (0, 0). The non-injectivity of φ produces the nodal singularity of Y0 at the origin. Conversely, the pencil of lines x2 = tx1 through the origin gives bijective correspondence between Y0 \ {(0, 0)} and W . Using Theorem 5.2.14 one can compute that K(Y0 ) ∼ = K(t), as we expected from the existence of the polynomial parametrization φ and the fact that φ|W will be an isomorphism of algebraic varieties (cf. Chapter 6). To sum up, in terms of fields of rational functions, Y0 and Y1 behave exactly as the previous examples of conics and cuspidal plane cubics. Let us consider therefore the case a 6= 0, 1. Since Ya is affine, from Theorem 5.2.14 (a) we have that OYa (Ya ) ∼ = A(Ya ), where   (1) 2 ∼ A(Ya ) = f + x2 g | f, g ∈ A and x2 = x1 (x1 − 1)(x1 − a) , where A(1) = C[x1 ]. From Theorem 5.2.14 (d), we get   (1) 2 ∼ K(Ya ) = φ + x2 ψ | φ, ψ ∈ Q and x2 = x1 (x1 − 1)(x1 − a) , +x2 a1 , where Q(1) = C(x1 ); indeed, any element of Q(A(Ya )) is of the form ab00 +x 2 b1 (1) 2 with a0 , a1 , b0 , b1 ∈ A and x2 = x1 (x1 − 1)(x1 − a), which can be therefore written as a0 + x2 a1 b0 − x2 b1 = φ(x1 ) + x2 ψ(x2 ) b0 + x2 b1 b0 − x2 b1

for some φ, ψ ∈ Q(1) , as claimed. To sum up, the rational function x2 ∈ K(Ya ) is a root of the polynomial Px2 (t) := t2 − x1 (x1 − 1)(x1 − a) ∈ Q(1) [t], where t an indeterminate over Q(1) . A key remark is now the following:

5.2 Rational functions

137

Theorem 5.2.26 Let K be a field of characteristic different from 2 and let λ ∈ K, with λ 6= 0, 1. Let t be an indeterminate and Φ, Ψ ∈ K(t) rational functions s.t. Φ2 = Ψ (Ψ − 1)(Ψ − λ). Then Φ, Ψ ∈ K. The proof of this result is of pure algebraic nature and uses the fact that K[t] is a UFD and Fermat’s method of infinite descent. The interested reader is referred to [27, Ch. 2,§ 2.2]. In our set-up, Theorem 5.2.26 ensures it cannot exist a non-constant map from a non-empty open set U ⊆ A1 to Ya given by rational functions in the coordinate t ∈ U (i.e. for a 6= 0, 1, Ya admits no rational parametrization). Notice moreover that it also implies that the polynomial Px2 (t) is irreducible over Q(1) (otherwise, it would admit a root in Q(1) = C(x1 ) contradicting Theorem 5.2.26). Thus for a 6= 0, 1, the rational function x2 ∈ K(Ya ) is algebraic over Q(1) (but not in Q(1) ) and Q(1) [t] K(Ya ) ∼ = (Px2 (t)) is an algebraic extension of degree 2 of Q(1) , i.e. K(Ya ) is a finitely generated field extension of C of trascendental degree 1 over C, but not purely trascendental over C.

Appendix to Chapter 5: Basics on sheaves Pre-sheaves and sheaves on topological spaces. Let Y be any topological space. Definition 5.27 A pre-sheaf F (of sets) on Y is the datum of: (i) a non-empty set F(U ), for any open set U ⊆ Y , and (ii) a map ρU V : F(U ) → F(V ), for any V ⊆ U ⊆ Y open sets, such that the following hold: (F1) ρU U = IdF(U ) , for any open set U ⊆ Y ; V V (F2) for any open sets W ⊆ U ⊆ V , one has ρU W ρU = ρW . The maps ρU V are called restrictions and, for any open set U ⊆ Y , the elements of F(U ) are called sections of F over U . In particular, elements of F(Y ) are called global sections of F. In the above definition, to any open set U of a topological space it is associated a non-empty set F(U ); in particular, for any pre-sheaf F, we have F(∅) 6= ∅. In the sequel, we shall always consider pre-sheaves F such that F(∅) = {·} is a fixed singleton. This convention will be taken for granted and not mentioned anymore.

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5 Regular and rational functions

Definition 5.28 A sheaf (of sets) on Y is the datum of a pre-sheaf F on Y as above, satisfying moreover the following condition: (F3) for any family {Ui }i∈I of open sets of Y , where I a set of indexes, and for any collection of sections (fi ) ∈ Πi∈I F(Ui ) s.t. U

j i ρU Ui ∩Uj (fi ) = ρUi ∩Uj (fj ), ∀ i, j ∈ I, S there exists a unique f ∈ F(U ), where U := i∈I Ui , such that

ρU Ui (f ) = fi , ∀ i ∈ I.

Not all pre-sheaves are sheaves (cf. Exercises 5.1.7). When, for any nonempty open subset U ⊆ Y , the sets F(U ) are not only sets but more precisely e.g. abelian groups, rings, K-algebras, R-modules for a given ring R, etcetera, we will require restrictions ρU V to be homomorphims of abelian groups, rings, K-algebras, R-modules, etcetera, respectively. In this case, F will be correspondigly called a pre-sheaf (respectively a sheaf) of abelian groups, rings, K-algebras, R-modules, etcetera, on Y . Remark 5.29 Let F be a sheaf on the topological space X and let U ⊂ X be any non empty open subset. One can define the pre-sheaf F|U by F|U (V ) := F(V ), for any non empty open set V of U . Since F is a sheaf, the F|U is a sheaf too. This is called the sheaf obtained by restriction of F to U .

5.1 Exercises Exercise 5.1.1. Let d > 2 be an integer and let V := {(t, t2 , t3 , . . . , td ) | t ∈ C}.

Show that V is an affine variety in Ad ; then show that A(V ) ∼ = Q(1) . = A(1) and K(V ) ∼ Exercise 5.1.2. Let K be algebraically closed and let V := Za (x22 − x21 (x1 + 1)) ⊂ A2 . Consider x1 , x2 ∈ A(V ) (small abuse of notation). Show that the rational function / OV (V ), describing the indeterminacy f := xx12 ∈ K(V ) is not regular on V , i.e. f ∈ locus of f . Exercise 5.1.3. Let K be algebraically closed. Let V be an algebraic variety and let U ⊆ V be a non-empty, open subset. Consider f ∈ K(V ) non constant and suppose that f ∈ OV,P , for any P ∈ U . Show that the set {P ∈ U | f (P ) 6= 0} is an open, non empty subset of V and that the map U → K = A1 , P → f (P ) is continuous.

5.1 Exercises

139

Exercise 5.1.4. Let K be algebraically closed and let C := Za (x32 − x41 − x31 ) ⊂ A2 . Show that C is an affine variety. Then, prove that (x1 , x2 ) → xx21 defines a rational map ψ : C 99K A1 . Exercise 5.1.5. Let K be algebraically closed. Consider the affine variety V := Za (x1 x4 − x2 x3 ) ⊂ A4 and f :=

x1 x2

∈ K(V ). Find Dom(f ) (cf. [27, Ex. 4.9, p. 83]).

Exercise 5.1.6. Let K be algebraically closed and d > 2 be any integer. Consider F ∈ K[X1 , . . . , Xn ]d and G ∈ K[X1 , . . . , Xn ]d−1 and let Z := Zp (F − X0 G) ⊂ Pn . Prove that Z is a projective variety and show that K(V ) ∼ = Q(n) (cf. § 8.2.2 below). Exercise 5.1.7. Let Y be a topological space and let S be any non-empty open set. For any non-empty open set U ⊆ Y , put CS (U ) := S whereas CS (∅) = {·}. Moreover, for any ∅ = 6 V ⊆ U , put ρU V := IdS , whereas if V as above is empty then ρU is the constant map to {·}. CS defined in this way is called the constant pre-sheaf. V Show that if Y is a topological space containing two non-empty open sets U1 , U2 such that U1 ∩ U2 = ∅ (e.g. if Y is a Hausdorff space), then CS is a pre-sheaf but not a sheaf. On the other hand, if Y is an irreducible topological space, show that any constant pre-sheaf is a sheaf. Exercise 5.1.8. If Y is an irreducible topological space, show that any constant pre-sheaf CS is a sheaf.

6 Morphisms of algebraic varieties

In this chapter we will study maps which preserve algebraic variety structure.

6.1 Morphisms Let V and W be algebraic varieties. Definition 6.1.1 A map ϕ : V → W is said to be a morphism of algebraic varieties, or simply a morphism, if: (i) ϕ is a continuous map between the topological spaces V and W , and (ii) for any non empty open set U ⊆ W and for any f ∈ OW (U ), one has fϕ := f ◦ ϕ ∈ OV (ϕ−1 (U )). The set of all morphisms from V to W will be denoted by Morph(V, W ). For any algebraic variety V , IdV ∈ Morph(V, V ). Moreover, if V, W Z are algebraic varieties and if ϕ ∈ Morph(V, W ) and ψ ∈ Morph(W, Z), then ψ ◦ ϕ ∈ Morph(V, Z). Definition 6.1.2 ϕ ∈ Morph(V, W ) is said to be an isomorphism if there exists ψ ∈ Morph(W, V ) such that ϕ ◦ ψ = IdW and ψ ◦ ϕ = IdV . If such a ψ exists, it is uniquely determined and will be denoted by ϕ−1 . The set of all isomorphisms from V to W will be denoted by Isom(V, W ). If Isom(V, W ) 6= ∅, V and W are said to be isomorphic, which will be denoted by V ∼ = W. The set Isom(V, V ) will be denoted by Aut(V ) and called the set of automorphisms of V .

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6 Morphisms

Remark 6.1.3 (i) From the previous definitions, for any ϕ ∈ Morph(V, W ) and for any non empty open subset U ⊆ W , one has a natural map ϕU : OW (U ) −→ OV (ϕ−1 (U )) f −→ fϕ := f ◦ ϕ

(6.1)

which is a homomorphism of integral K-algebras. (ii) When U = W , the morphism ϕW will be simply denoted by ϕ# : OW (W ) −→ OV (V )

(6.2)

(iii) If ϕ ∈ Isom(V, W ) then, for any non empty open subset U ⊆ W , ϕU is an isomorphism of K-algebras; indeed (ϕU )−1 := (ϕ−1 )ϕ In particular

−1

(U )

.

V ∼ = W ⇒ OV (V ) ∼ = OW (W ).

(6.3)

Remark 6.1.4 Recalling Example 5.2.23, (6.3) implies that the hyperbola Y = Za (x1 x2 −1) and the ellipsis Z = Za (x21 +x22 −1) cannot be isomorphic to A1 . Below (cf. Example 6.2.5 (iv)) we will show that they both are isomorphic to the quasi-affine variety W := A1 \ 0 whereas the parabola V = Za (x2 − x21 ) is instead isomorphic to A1 . Definition 6.1.5 If ϕ ∈ Morph(V, W ) is such that Im(ϕ) = W , then ϕ is said to be a dominant morphism, or simply dominant. Parametrizations φ1 and φ3 in Example 5.2.23 (i) and (iii) are examples of dominant (but not surjective) morphisms to A1 . Proposition 6.1.6 If ϕ ∈ Morph(V, W ) is dominant, then K(W ) ⊆ K(V ) is a field extension. In particular V ∼ = W ⇒ K(V ) ∼ = K(W ).

(6.4)

Proof. Since ϕ is dominant, for any non empty open subset U ⊆ W one has U ∩ Im(ϕ) 6= ∅; indeed, if there existed a non empty open set U1 ⊂ W s.t. U1 ∩ Im(ϕ) = ∅ one would have Im(ϕ) ⊆ U1c ⊂ W with U1c a proper closed subset of W , contradicting the dominance of ϕ. Recalling notation as in Definition 5.2.1, ϕ induces a map H(W ) −→ H(V ) (U, f ) −→ (ϕ−1 (U ), fϕ ) which is compatible with the equivalence relation R. Thus, it defines a map ϕ∗ : K(W ) −→ K(V ) [U, f ] −→ [ϕ−1 (U ), fϕ ]

(6.5)

6.2 Morphisms with (quasi) affine target

143

which is easily seen to be a K-algebra homomorphism. Since ϕ∗ is a nonzero field homomorphism it is therefore injective, proving the first part of the statement. When in particular ϕ ∈ Isom(V, W ), ϕ∗ is a field isomorphism between K(W ) and K(V ). ⊓ ⊔ Remark 6.1.7 The converse of (6.4) does not hold: take e.g. the hyperbola Y = Za (x1 x2 − 1), for which we found K(Y ) ∼ = K(A1 ) (cf. Example 5.2.23 1 (i)), even if Y cannot be isomorphic to A as observed in Remark 6.1.4. Definition 6.1.8 Let V be an algebraic variety and let W ⊆ V be any subvariety. The natural inclusion map ιW : W ֒→ V is a morphism which is called an open (closed, locally-closed, respectively) immersion if W is an open (closed, locally closed, respectively) subvariety of V. In the next sections we shall give some criteria ensuring that a given map between algebraic varieties is actually a morphism.

6.2 Morphisms with (quasi) affine target In this section we will focus on the case where the target of a map ϕ : V → W is a quasi-affine (or affine) variety. We start with the following basic result. Proposition 6.2.1 Let V be an algebraic variety. There is a bijective correspondence between Morph(V, A1 ) and OV (V ). Proof. Any f ∈ OV (V ) is a continuous map from V to A1 , as it follows from Proposition 5.1.4 (i). Let now U ⊆ A1 be any non empty open set. From Lemma 5.2.13 (ii) and Corollary 5.2.17, one has K(U ) ∼ = K(A1 ) = Q(1) ∼ = K(t), where t an indeterminate over K. Let (1)

c(t) b(t)

∈ OA1 (U ) be any regular funtion

on U , i.e. c(t), b(t) ∈ A = K[t] and U ⊆ Ua (b) = Za (b)c . Using notation as in (6.1), one has   c(t) c(t) c(f ) U f = ◦f = ∈ OV (f −1 (U )) b(t) b(t) b(f ) i.e. f defines a morphism from V to A1 . Conversely, any ϕ ∈ Morph(V, A1 ) gives rise to ϕ# (IdA1 ) = (IdA1 )ϕ ∈ OV (V ) and, from the previous step, it turns out that ϕ is the morphism defined by this regular function. ⊓ ⊔

144

6 Morphisms

Since An is affine, from Theorem 5.2.14 (a), any indeterminate xi ∈ A(n) = A(An ) = OAn (An ), 1 6 i 6 n, can be identified with the regular function x

i An −→ A1 , (a1 , a2 , . . . , an ) → ai

(6.6)

which is the projection πi of An onto its ith -axis. If W ⊆ An is any quasiaffine variety, by abuse of notation, we will always denote by xi ∈ A(W ) the restriction of the regular function (6.6) to W . Proposition 6.2.2 Let V be an algebraic variety and let W ⊆ An be any quasi-affine variety. Then ϕ ∈ Morph(V, W ) ⇔ ϕi := xi ◦ ϕ ∈ OV (V ), ∀ 1 6 i 6 n. In particular, for any n > 1, there is a bijective correspondence between Morph(V, An ) and OV (V )⊕n . Proof. (⇒) Since xi ∈ A(W ) ⊆ OW (W ), by composition of morphisms we have ϕi = xi ◦ ϕ ∈ Morph(V, A1 ) = OV (V ), as it follows from Lemma 6.2.1. (⇐) Let ϕ : V → W be any map s.t. ϕi = xi ◦ ϕ ∈ OV (V ), for any 1 6 i 6 n. Then, for any f ∈ A(An ) = A(n) , one has f ′ := f (ϕ1 , . . . , ϕn ) ∈ OV (V ), since f ′ is a polynomial expression in the ϕi ’s. From the proof of Proposition 5.1.4 (i), ZV (f ′ ) ⊆ V is a closed subset of V and moreover ϕ−1 (Za (f ) ∩ W ) = ZV (f ′ ), i.e. ϕ is continuous. Let us show that ϕ is also a morphism. Take any non empty open subset U ⊆ W and any f ∈ OW (U ); by definition of regularity and by Lemma 5.1.3, for any P ′ ∈ U there exist an open neighborhood UP ′ ⊆ U and two polynomials gP ′ , hP ′ ∈ A(n) such that hP ′ 6= 0, UP ′ ⊆ Ua (hP ′ ) = Za (hP ′ )c , such that f |UP ′ = hgP ′′ |UP ′ and moreover, for any Q′ ∈ U different from P ′ P one has gP ′ gQ′ (∗) |UP ′ ∩UQ′ = f |UP ′ ∩UQ′ = |U ∩U . hP ′ hQ′ P ′ Q′ Consider ϕU (f ) = f ◦ ϕ as in (6.1); we are left to show that ϕU (f ) is regular on the open set ϕ−1 (U ) ⊆ V . From the facts that: regularity is a local condition, {UP ′ }P ′ ∈U is an open  −1 ′ covering of U , ϕ (UP ) P ′ ∈U is an open covering of ϕ−1 (U ) and from (∗), it suffices to check that for any P ′ ∈ U one has ϕUP ′ (f ) ∈ OV (ϕ−1 (UP ′ )). Notice that

6.2 Morphisms with (quasi) affine target

ϕUP ′ (f ) = f ◦ ϕ|ϕ−1 (UP ′ ) =

145

gP ′ (ϕ1 , . . . , ϕn ) . | −1 hP ′ (ϕ1 , . . . , ϕn ) ϕ (UP ′ )

Since ϕi ∈ OV (V ) for any 1 6 i 6 n and since UP ′ ⊆ Ua (hP ′ ), one has gP ′ (ϕ1 ,...,ϕn ) −1 (UP ′ )), as desired. hP ′ (ϕ1 ,...,ϕn ) ∈ OV (ϕ For what concerns Morph(V, An ), the case n = 1 is Lemma 6.2.1 whereas the case n > 2 follows from the first part of the lemma applied to W = An . ⊓ ⊔ Similarly to (5.4), for any f1 , . . . , fn ∈ OV (V ) one can define ZV (f1 , . . . , fn ) := {P ∈ V | f1 (P ) = f2 (P ) = · · · = fn (P ) = 0} ⊆ V.

(6.7)

Thus, from Prop. 6.2.2, for any ϕ ∈ Morph(V, An ) one can define ZV (ϕ) := ZV (ϕ1 , . . . , ϕn ) = ϕ−1 ((0, . . . , 0)). In the sequel, for any K-algebras R and S, we will denote by HomK (R, S) the set of all K-algebra homomorphisms from R to S. Recalling (6.2), one has: Proposition 6.2.3 Let V be an algebraic variety and let W ⊆ An be any affine variety. The map α

Morph(V, W ) −→ HomK (A(W ), OV (V )) ϕ −→ ϕ#

(6.8)

is bijective. Proof. By Theorem 5.2.14 (i), one has OW (W ) = A(W ) = A(n) /Ia (W ). Take now any h ∈ HomK (A(W ), OV (V )) and let ξi := h(xi ) ∈ OV (V ),

(6.9)

for any xi ∈ A(W ), 1 6 i 6 n. Consider the map ϕh

V −→ An P −→ (ξ1 (P ), . . . , ξn (P )). From Prop. 6.2.2, ϕh ∈ Morph(V, An ). If we show that ϕh (V ) ⊆ W , we will get that ϕh ∈ Morph(V, W ). To prove this, consider any polynomial f ∈ Ia (W ). For any P ∈ V we have ϕh (f )(P ) = (f ◦ ϕh )(P ) = f (ξ1 (P ), . . . , ξn (P )) which means ϕh (f ) = f ◦ ϕh = f (h(x1 ), . . . , h(xn )).

146

6 Morphisms

Since h ∈ HomK (A(W ), OV (V )), the latter equals h(f (x1 , . . . , xn )) = h(0) = 0 since xi ∈ A(W ) and f ∈ Ia (W ). This means ϕh (V ) ⊆ W , as desired. At last, the map HomK (A(W ), OV (V )) −→ Morph(V, W ) h −→ ϕh ⊓ ⊔

is the inverse of α, which is therefore a bijection as desired. Corollary 6.2.4 Let V and W be affine varieties. Then V ∼ = W ⇔ A(V ) ∼ = A(W ). Proof. One implication is (6.3), the other follows from Proposition 6.2.3.

⊓ ⊔ Example 6.2.5 (i) Let 1 6 m < n be integers and consider the sets I := {i1 , i2 , . . . , im } ⊂ {1, 2, . . . , n}. The injective, K-algebra homomorphism K[xi1 , . . . , xim ] ֒→ K[x1 , . . . , xn ] corresponds to the surjective morphism πI : An → Am , (a1 , . . . , an ) → (ai1 , . . . , aim ),

(6.10)

which is called the projection of An onto the coordinates I = {i1 , i2 , . . . , im }. (ii) Let (b1 , . . . , bn ) 6= (0, . . . , 0) be non-negative integers. The morphism ϕ : A1 → An , t → (tb1 , . . . , tbn ) corresponds to the K-algebra homomorphism ϕ# : K[x1 , . . . , xn ] → K[t], xi → tbi , 1 6 i 6 n. The image of ϕ is an affine rational curve with polynomial parametrization (recall (3.31)). (iii) For any non negative integer b, the morphism ϕ : A1 → A2 , t → (t, tb ) defines an isomorphism between A1 and the affine plane curve V = Za (x2 −xb1 ) as ϕ corresponds to the K-algebra isomorphism

6.2 Morphisms with (quasi) affine target

A(V ) =

147

# K[x1 ,x2 ] ϕ −→ (x2 −xb1 )

K[t] = A(A1 ) x1 −→ t x2 −→ tb .

(cf. Corollary 6.2.4). Specifically ϕ−1 is given by π1 |V , where π1 the projection of A2 onto the x1 -axis, which is therefore a morphism since it is a restriction to V of a morphism from A2 to A1 . (iv) The map φ2 in Example 5.2.23 (ii) is an isomorphism. Indeed the parametrization φ2 : A1 → V of the parabola V = Za (x2 − x21 ) ⊂ A2 is given by the two regular functions t, t2 ∈ OA1 (A1 ) so, from Prop. 6.2.2, φ2 is a morphism. The map φ−1 2 is given by the restriction to V of the first projection as in (6.6), thus it is a morphism. Since V and A1 are both affine varieties, Corollary 6.2.4 gives an intrinsic motivation of the isomorphism A(V ) ∼ = A(A1 ) we showed in Example 5.2.23 (ii). Different situation occurs for e.g. the hyperbola Y = Za (x1 x2 − 1) as in Example 5.2.23 (i). From Prop. 6.2.2 the parametrization φ1 : A1 \ {0} := W → Y

is an isomorphism, i.e. Y is isomorphic to the (quasi-affine) variety W ⊂ A1 . From (6.3) it follows that OW (W ) ∼ = OY (Y ) ∼ = A(Y ), the latter isomorphism coming from the fact that Y is affine (cf. Theorem 5.2.14). Recall that in (1) Example 5.2.23 (i) we showed A(Y ) ∼ = Ax1 ; the fact that φ1 is an isomorphism (1) in particular proves that OW (W ) ∼ = Ax1 (as stated without proof in Example 5.2.23 (i)) and so we refind K(W ) ∼ = Q(1) . What observed about the hyperbola holds verbatim for the map φ3 and the circle Z := Za (x21 + x22 − 1) in Example 5.2.23 (iii), i.e. Z is isomorphic to W = A1 \ {0}. (v) By definition of isomorphism, any φ ∈ Isom(V, W ) is also a homeomorphism between V and W as topological spaces; on the other hand, the converse does not hold. To see this, consider e.g. the semi-cubic parabola Y = Za (x31 − x22 ) in Example 5.2.24. The map φ : A1 → Y therein is a morphism, as it follows from Prop. 6.2.2. In Example 5.2.24, we already proved that φ is also a homeomorphism; on the other hand we also showed that A(Y ) ∼ = K[t, t2 ], which is a (proper) subring of A(A1 ) ∼ = K[t]. Since Y and A1 are both affine varieties, from Corollary 6.2.4 it follows that φ cannot be an isomorphism. Notice in fact that the map ψ constructed in Example 5.2.24 is the inverse of φ as a homeomorphism, but it is not a morphism, as ψ is not a polynomial map, i.e. a map given by a collection of polynomials (i.e. regular functions on the affine varieties Y ), as prescribed by Prop. 6.2.2 for any morphism from Y to A1 . Indeed, for any open set U ⊆ A1 containing 0, one has t ∈ OA1 (U ) but / OY (ψ −1 (U )) as (0, 0) ∈ ψ −1 (U ). ψ −1 (t) = xx12 ∈ (vi) If V is any algebraic variety and if φ ∈ Morph(V, W ), in general φ(V ) is neither open nor closed in W . One can in fact show that, in general, φ(V ) is

148

6 Morphisms

a constructible set i.e. a disjoint union of finitely many locally closed subsets of W (cf. e.g. [25] for a proof). To have an example, take e.g. the hyperbolic paraboloid ιV

V := Za (x1 x3 − x2 ) ֒→ A3 . For the multi-index I := (1, 2), let πI : A3 → A2 be the associated projection morphism. If we take W := A2 then φ := πI ◦ ιV ∈ Morph(V, W ) and φ(V ) = (A2 \ Za (x1 )) ∪ {(0, 0)} which is a constructible set in A2 . Interesting consequences of previous results are the following. Corollary 6.2.6 If V is an affine variety which is isomorphic to a projective variety, then V is a point. Proof. From Theorem 5.2.14 (a), if V is an affine variety, then OV (V ) ∼ = A(V ). On the other hand, since V is isomorphic to a projective variety, by (6.3) and Theorem 5.2.14 (e), we have also OV (V ) = K. Thus, for some non negative integer n, one has that Ia (V ) ⊂ A(n) must be a maximal ideal. By Theorem 2.2.1, V is a point. ⊓ ⊔ Corollary 6.2.7 Any morphism ϕ : V → W , where V a projective variety and W an affine variety, is constant. Proof. Since V is projective, OV (V ) = K (cf. Theorem 5.2.14 (e)). At the same time, since W is affine then it is a closed, irreducible subset of some affine space An , for some non-negative integer n, so A(W ) = A(n) /Ia (W ). For any ϕ ∈ Morph(V, W ), consider ϕ# ∈ HomK (A(W ), OV (V )); as in (6.9), for any generator xi ∈ A(W ) one has ξi = ϕ# (xi ) := ai ∈ K, for any 1 6 i 6 n. This implies that for any point P ∈ V one has ξi (P ) := ai so ϕ(P ) = (a1 , . . . , an ), i.e. ϕ is constant. ⊓ ⊔ Corollary 6.2.8 Let V and W be affine varieties and let ϕ ∈ Morph(V, W ). Then, ϕ is dominant if and only if ϕ# ∈ HomK (A(W ), A(V )) is injective. Proof. (⇒) If by contradiction Ker(ϕ# ) 6= 0, for any non-zero polynomial f ∈ Ker(ϕ# ) one has ϕ# (f ) = 0 ∈ A(V ), i.e. ϕ# (f ) ∈ Ia (V ). Since f ∈ A(W ) = OW (W ), by the proof of Proposition 5.1.4 (i), ZW (f ) is a proper closed subset of W (as 0 6= f ∈ A(W )). Now ϕ# (f ) ∈ Ia (V ) is equivalent to f ◦ ϕ identically zero on W , i.e. ϕ(V ) ⊆ ZW (f ) ( W , contradicting the dominance of ϕ. (⇐) Assume by contradiction that ϕ is not dominant, i.e. ϕ(V ) = K ( W , where K is an irreducible, proper closed subset of W (recall Corollary 4.1.6

6.2 Morphisms with (quasi) affine target

149

and Proposition 4.1.2 (iii)). Then, with notation as in (5.7), for any non zero f ∈ Ia,W (K) we would have ϕ(V ) ⊆ ZW (f ). Any such f determines a nonzero regular function f ∈ A(W ) = OW (W ) s.t. f ◦ ϕ(P ) = 0 for any P ∈ V , i.e. a non zero element of Ker(ϕ# ), contradicting the injectivity assumption. ⊓ ⊔ The previous results are also consequences of more general facts which will be discussed later on (cf. Remark 10.2.2). −1 Example 6.2.9 Morphisms φ−1 1 and φ3 of Examples (5.2.23) (i), (iii) and (6.2.5) (iv) are dominant but not surjective over A1 . Indeed, in both cases (1) we have the injection A(A1 ) = A(1) ֒→ Ax1 ∼ = A(Y ) ∼ = A(Z), where Y the hyperbola and Z the circle, respectively.

The use of (iso)morphisms allows one to extend some definitions introduced before. Definition 6.2.10 Let V be any algebraic variety. V is said to be an affine variety if V is isomorphic to an irreducible closed subset in some affine space. If U ⊆ V is an open subset which is an affine variety, then U will be called an affine open set of V . Lemma 6.2.11 For any i ∈ {0, . . . , n}, the principal open set Ui ⊂ Pn is an affine open set of Pn isomorphic to An . In particular, any projective variety V ⊆ Pn has a finite open covering {Vi }i∈{0,...,n} , where Vi := V ∩ Ui , which is formed by affine open sets of V . Proof. One is reduced to show that the map φi as in (3.15) is an isomorphism, for any i ∈ {0, . . . , n}. From Proposition 3.3.2, any such φi is a homeomorphism. The fact that φi is a morphism follows from the expression of φi in (3.15) and from Prop. 6.2.2. To prove that φ−1 is a morphism too, one uses i the definition of regular function and Lemma 5.1.3. ⊓ ⊔ Another easy example of affine open set of an algebraic variety is W := A1 \ {0} ⊂ A1 , as it follows from the isomorphism between W and e.g. the hyperbola Y = Za (x1 x2 − 1) as in Example 6.2.5 (iv). This is a particular case of the following more general result. Lemma 6.2.12 For any positive integer n and for any non constant polynomial f ∈ A(n) , let Z := Za (f ) ⊂ An be the affine hypersurface defined by f . Then U := An \ Z ⊂ An is an affine open set, being isomorphic to the irreducible hypersurface Z ′ := Za (xn+1 f − 1) ⊂ An+1 . Moreover, one has (n) OU (U ) ∼ = Af .

150

6 Morphisms

Proof. First notice that (xn+1 f −1) ∈ A(n+1) is irreducible so Z ′ is an affine variety: this follows from the fact that A(Z ′ ) =

A(n+1) (n) ∼ = Af ⊂ Q(A(n) ) (xn+1 f − 1)

has to be an integral domain and from Proposition 4.1.10. Consider now the map ψ : Z ′ → An \ Z, defined as (a1 , a2 , . . . , an+1 ) → (a1 , a2 , . . . , an ). Since ψ = πI |Z ′ , for the multi-index I := (1, 2, . . . , n), then ψ is a morphism. Conversely, the map   1 n ′ A \ Z → Z , defined as (a1 , a2 , . . . , an ) → a1 , a2 , . . . , an , f (a1 , . . . , an ) is a morphism by Prop. 6.2.2; moreover it is the inverse of ψ proving that Z ′ and U = An \ Z are isomorphic. At last, since Z ′ is an affine hypersurface, by Theorem 5.2.14 (a) and by (6.3) we have (n) OU (U ) ∼ = A(Z ′ ) ∼ = Af , where the last isomorphism has been proved above.

⊓ ⊔

In particular, one has: Corollary 6.2.13 An admits a basis of the topology Zarna consisting of affine open sets. Proof. It directly follows from Lemma 2.1.21 and the previous result.

⊓ ⊔

The next example shows that there actually exist quasi-affine varieties which cannot be affine. Example 6.2.14 From Lemma 6.2.12, if Z ⊂ A2 is any curve then A2 \ Z is an affine open set of A2 . On the contrary, for any point P ∈ A2 , we now show that A2 \ {P } is a quasi-affine variety which cannot be affine. With the use of affine transformations, it is not a restriction to consider P to be the origin O = (0, 0). Then W := A2 \{O} = Za ((x1 , x2 ))c is quasi-affine; let ιW : W ֒→ A2 be the open immersion, so ιW ∈ Morph(W, A2 ). Since A2 is affine, from Proposition 6.2.3, the morphism ιW corresponds to the K-algebra homomorphism 2 ι# W ∈ HomK (A(A ), OW (W )). (2) (viewed as a Since A(A2 ) = A(2) , ι# W simply sends any polynomial f ∈ A 2 regular function on A ) to its restriction to W . Since W is a dense open set in A2 , it is clear that ι# W is an injective homomorphism.

6.3 Morphisms with (quasi) projective target

151

Claim 6.2.15 ι# W is an isomorphism. Proof. From above, we are left to show that ι# W is surjective. From Lemma 5.2.13 and Theorem 5.2.14 (d), we have OW (W ) ⊂ K(W ) ∼ = K(A2 ) = Q(A(2) ) = Q(2) , i.e. any regular function on W can be identified with a rational function φ = g (2) such that g, h ∈ A(2) , h 6= 0 and W ⊆ Ua (h) = Za (h)c . Thus, to h ∈ Q prove the surjectivity of ι# W is equivalent to showing that any φ ∈ OW (W ) is actually a polynomial, i.e. that h ∈ K∗ . Take therefore any φ = hg ∈ OW (W ) and assume by contradiction that h ∈ A(2) \K. Since W ⊆ Ua (h), then one has either Za (h) p = ∅ or Za (h) = {O}. (h) = (1) whereas in From Theorem 2.2.3, in the first case we would have p the second case (h) = mO = (x1 , x2 ), a contradiction in both cases. ⊓ ⊔ If therefore W were affine, from Theorem 5.2.14 (a) we would have OW (W ) ∼ = A(W ). Thus, from Claim 6.2.15 and Corollary 6.2.4, ιW would be an isomorphism which is a contradition, since it is injective but not surjective onto A2 .

6.3 Morphisms with (quasi) projective target We start with the following general result. Proposition 6.3.1 Let n, m be non negative integers and let V ⊆ Pn and W ⊆ Pm be quasi-projective varieties. A map ψ : V → W is a morphism if and only if, for any P ∈ V , there exist an open neighborhood UP of P in V , a (n) non negative integer k and m + 1 homogeneous polynomials F0 , . . . , Fm ∈ Sk such that: • there exists i ∈ {0, . . . , m} for which Fi (P ′ ) 6= 0, for any P ′ ∈ UP , and • ψ(P ′ ) = [F0 (P ′ ), . . . , Fm (P ′ )], for any P ′ ∈ UP . Proof. (⇒) For P ∈ V , let Q := ψ(P ) ∈ W ⊆ Pm . Since W 6= ∅, there exists an index i ∈ {0, . . . , m} for which Wi := W ∩ Ui 6= ∅, Ui the fundamental affine open set of Pm . For simplicity, assume this occurs for i = 0. Thus W0 is an open set of W , so UP := ψ −1 (W0 ) ⊆ V is an open neighborhood of P in V , being ψ continuous. The map ψ ′ := ψ|UP : UP → W0 is a morphism; up to composing with the ιW0

φ0

inclusion W0 ֒→ U0 and with the isomorphism U0 −→ Am , ψ ′ has quasi-affine target W0 . From Proposition 6.2.2, ψ ′ is therefore given by a m-tuple of regular functions over UP ⊆ V ⊆ Pn . Up to suitably restricting the open neighborhood

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UP , this means there exist m non-negative integers di , 1 6 i 6 m, and m pairs of homogeneous polynomials (F1 , F1,0 ), (F2 , F2,0 ), . . . , (Fm , Fm,0 ), (n)

where Fi , Fi,0 ∈ Sdi , 1 6 i 6 m, such that UP ⊆

m \

i=1

Up (Fi,0 ) =

m \

Zp (Fi,0 )c

i=1

(i.e. Fi,0 (P ′ ) 6= 0 for any P ′ ∈ UP and for any i ∈ {1, . . . , m}) and   F1 (P ′ ) F2 (P ′ ) Fm (P ′ ) , ∀ P ′ ∈ UP . , , . . . , ψ ′ (P ′ ) = F1,0 (P ′ ) F2,0 (P ′ ) Fm,0 (P ′ ) If we put Fb := F1,0 F2,0 · · · Fm,0 and Fbi := F1,0 F2,0 · · · Fi−1,0 Fi Fi+1,0 · · · Fm,0 ,

Pm (n) for any i ∈ {1, . . . m} we have Fb, Fbi ∈ Sk , where k := i=1 di . Moreover, for any P ′ ∈ UP , we have Fb(P ′ ) 6= 0 and ! Fbm (P ′ ) Fb1 (P ′ ) Fb2 (P ′ ) ′ ′ ψ (P ) = . , ,..., Fb(P ′ ) Fb(P ′ ) Fb(P ′ )

Thus, we can assume that

(n)

F0 := F1,0 = F2,0 = · · · = Fm,0 ∈ Sk , (n)

and that for any P ′ ∈ UP one has F0 (P ′ ) 6= 0 and   F1 (P ′ ) F2 (P ′ ) Fm (P ′ ) ψ ′ (P ′ ) = . , , . . . , F0 (P ′ ) F0 (P ′ ) F0 (P ′ )

so F1 , F2 , . . . , Fm ∈ Sk

m By the isomorphism φ−1 0 : A → U0 , the previous equality reads as   F1 (P ′ ) F2 (P ′ ) Fm (P ′ ) ′ ′ ψ (P ) = 1, = [F0 (P ′ ), F1 (P ′ ), . . . , Fm (P ′ )], , ,..., F0 (P ′ ) F0 (P ′ ) F0 (P ′ )

for any P ′ ∈ UP , as desired. (⇐) Take P ∈ V and assumptions as in the statement. Composing with the isomorphism φi : Ui → Am , from Prop. 6.2.2 the map ψ|UP is a morphism with target Wi := W ∩ U0 a quasi-affine variety. Since {UP }P ∈V is an open covering of V , it follows that ψ is a morphism. ⊓ ⊔

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Example 6.3.2 For any integer n > 1, consider the projective space Pn . (n)

(i) Let d > 1 be an integer and let F0 , . . . , Fr ∈ Sd be homogeneous polynomials. Put B := Zp (F0 , . . . , Fr ) ⊂ Pn (6.11) and U := Pn \ B.

(6.12)

B is a proper (possibly empty) closed subset of Pn , whereas U is an open (so dense and irreducible) subset of Pn . Define the map ν

ν : U −→ Pr , [P ] = [p0 , . . . , pn ] −→ [F0 (P ), . . . , Fr (P )].

(6.13)

From Proposition 6.3.1, ν is a morphism; U is called the domain of ν, i.e. the open set where ν is defined, whereas B is called the indeterminacy locus of ν. Take homogeneous coordinates [Y0 , . . . , Yr ] in the target and let Hi := Zp (Yi ) ⊂ Pr be the fundamental hyperplane. Notice that Im(ν) is not contained in any of the hyperplanes Hi ’s, 0 6 i 6 r: indeed ν(U ) ⊆ Hi means U ⊆ Zp (Fi ); since U ⊂ Pn is open and dense, the latter condition would imply Fi = 0 as a polynomial, contradicting that any Fi is homogeneous of some positive degree d. Notice moreover that, for any 0 6 i 6 r, one has ν −1 (Hi ) = U ∩ Zp (Fi ) ⊂ Pn ,

(6.14)

i.e. the hypersurface U ∩Zp (Fi ) = ZU (Fi ) is the pre-image via ν of Hi ∩Im(ν), which is called a hyperplane section of Im(ν). (n) The polynomials F0 , . . . , Fr are linearly independent in Sd if and only if Im(ν) is non-degenerate in Pr : any possible hyperplane ! r X ai Yi ⊂ Pr H := Zp i=0

Pr n for Pr which Im(ν) ⊆ H would give U ⊆ Zp ( i=0 ai Fi ) ⊂ P . In such a case i=0 ai Fi is identically zero which, by the linear independence of the Fi ’s, implies (a0 , . . . , ar ) = (0, . . . , 0). If otherwise Pr F0 , . . . , Fr are linearly dependent, any non trivial linear combination i=0 ai Fi = 0 corresponds, via ν, to a hyperplane of Pr containing Im(ν). (n) Thus, if F0 , . . . , Fr are linearly independent in Sd , we pose L := Span{F0 , . . . , Fr }

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and denote by νL : U −→ Pr

(6.15)

the associated morphism as in (6.13). The target Pr of νL can be identified with the linear subspace P(L) of P(Snd ); this linear subspace will be called linear system of hypersurfaces of degree d in Pn , of (projective) dimension r = dim(L) − 1. (n) When L = Sd the associated map will be simply denoted by νn,d (cf. also (n) § 6.3.1 below) and P(Sd ) will be called the complete linear system of hypersurfaces of degree d is Pn ; it has projective dimension N (n, d) := p(n, d) − 1, where p(n, d) as in (1.28). (1)

(ii) Consider P1 and take d = 2. If we take the complete linear system P(S2 ), with F0 = X02 , F1 = X0 X1 , F2 = X12 (1)

the canonical basis of S2 , then B = ∅, U = P1 and ν1,2 : P1 → P2 , where Im(ν1,2 ) = Zp (Y0 Y2 − Y12 ) ⊂ P2 ,

where [Y0 , Y1 , Y2 ] homogeneous coordinates for P2 . Any linear section of the conic Zp (Y0 Y2 − Y12 ) is given by Zp (Y0 Y2 − Y12 , a0 Y0 + a1 Y1 + a2 Y2 ), where (a0 , a1 , a2 ) 6= (0, 0, 0), and it bijectively corresponds to the two roots (counted with multiplicity) in P1 of the homogeneous polynomial (1)

a0 X02 + a1 X0 X1 + a2 X22 ∈ S2

in the sense of Proposition 1.10.17, i.e. to a pair (counted with multiplicity) of points in P1 . Notice that ν1,2 coincides with the map Φ2 considered in Example 5.2.23 (ii), which is an isomorphism from P1 onto the conic Zp (Y0 Y2 − Y12 ), i.e. the isomorphism φ2 between affine varieties in Example 6.2.5 (iv) naturally extends to an isomorphism between their projective closures. If otherwise we take L = Span{F0 = X02 , F1 = X0 X1 } a proper subspace (1) of S2 , then B = {[0, 1]} is a base point of the linear system P(L), U = 1 P \ {[0, 1]} = U0 ⊂ P1 and the morphism νL is given by ν

L U0 −→ P1 , [X0 , X1 ] → [X02 , X0 X1 ].

Since X0 6= 0, on points of U0 one has [X02 , X0 X1 ] = [X0 , X1 ] so νL extends to the identity of P1 (this follows from more general facts which will be discussed later on; cf. Example 8.1.12).

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155

(1)

If we consider instead d = 3, L = S3 , F0 , F1 , F2 , F3 the canonical basis (1) of S3 , one easily deduces that Im(ν1,3 ) is the (projective) twisted cubic as in § 3.3.12 and that ν1,3 is an isomorphism onto its image. Example 6.3.3 With notation as above, consider the case of F0 , . . . Fr lin(n) early independent linear forms in S1 . (n)

(i) If r = n, then F0 , . . . Fn is a basis of S1 projectivity of Pn , so an automorphism of Pn . (ii) If otherwise r < n then

and the morphism νL is a

B = Zp (F0 , . . . , Fr ) ⊂ Pn is a linear subspace of dimension n − r − 1 and the morphism νL : U → Pr is called projection of Pn to Pr with center the linear subspace B. A geometric interpretation of the morphism νL is given by the following construction: the target Pr is identified with any r-dimensional linear subspace Λ ⊂ Pn which is skew with respect to the center of projection B and, for any point P ∈ U , νL (P ) ∈ Λ is given by the intersection of Λ with the linear subspace P ∨B ⊂ Pn (recall § 3.3.6, in particular (3.23)). 6.3.1 The Veronese morphism (n)

With notation as in Example 6.3.2 (i), take F0 , . . . , FN any basis of Sd , with N := N (n, d). Consider the complete linear system of hypersurfaces of degree d in Pn . By Theorem 3.2.4 (n)

(n)

B = Zp (F0 , . . . , FN ) = Zp ((S+ )d ) = Zp (S+ ) = ∅. Therefore, from Proposition 6.3.1 and from (6.13), νn,d : Pn → PN

(6.16)

is a morphism which will be called form now on Veronese morphism of indices (n) n and d induced by the chosen basis of Sd . We also pose Vn,d := Im(νn,d ) which we call Veronese variety of indices n and d; the use of term ”variety” is motivated by the next result. Theorem 6.3.4 Vn,d is a non degenerate, projective variety in PN , which is isomorphic to Pn and which is set-theoretically the intersection of hyperquadrics in PN ; more precisely, Ip (Vn,d ) is generated by homogeneous quadratic polynomials.

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We will not give here the complete proof of the previous result. We limit ourselves to briefly sketch the proof of easiest parts of Theorem 6.3.4 and give only evidences and motivations of the hardest ones; then we treat concretely and in full details the basic case of V1,d . Turning back to the general statement, irreducibility of Vn,d directly follows from Corollary 4.1.6 and from the fact that νn,d is a morphism. More(n) over, Vn,d is non-degenerate in PN since νn,d is defined by a basis of Sd (cf. Example 6.3.2 (i)). The fact that Vn,d is closed in Pn will easily follow from more general fact which will be discussed later on (cf. Theorem 10.2.1 and Remark 10.2.2 (iii)). At last to show that νn,d is actually an isomorphism of Pn onto its image (n) Vn,d one first observes that, from Example 6.3.2 (i), any base change in Sd N reads as a projective transformation of the target space P ; thus, up to a projectivity of PN , any Veronese morphism νn,d can be reduced into its standard form st νn,d : Pn → PN , [x0 , x1 , . . . , xn ] → [xd0 , xd−1 x1 , xd−2 x21 , . . . , xdn ], 0 0

(6.17) (n)

which is the Veronese morphism determined by the canonical basis of Sd . To show that Vn,d is isomorphic to Pn , it suffices to produce an explicit morphism st −1 st ) deals with the quadratic ). The construction of (νn,d which inverts (νn,d st homogeneous equations defining Im(νn,d ), which we call the standard Veronese st ). variety and, by small abuse of notation, denoted always by Vn,d = Im(νn,d We will not dwell on this, preferring to discuss in full details the case V1,d for any d > 1. As a consequence of Theorem 6.3.4, any V1,d has to be a non degenerate curve in Pd isomorphic to P1 . Such a curve is called a rational normal curve of degree d in Pd . Our aim is to prove Theorem 6.3.4 for V1,d . As explained above, up to a projective transformation of Pd , without loss of st generality we can identify any V1,d with Im(ν1,d ). 1 st Notice that V1,1 = P as ν1,1 = IdP1 ; the conic Zp (Y0 Y2 − Y12 ) in Example 6.3.2 (ii) is nothing but V1,2 ⊂ P2 , whereas the projective twisted cubic in § 3.3.12 is V1,3 ⊂ P3 . More generally, as proved above for any Vn,d , V1,d is irreducible and non degenerate in Pd ; by the correspondence among hyperplane sections of V1,d and hypersurfaces of degree d in P1 , V1,d is a curve of st degree d. If we restrict the morphism ν1,d to the affine chart U0 ⊂ P1 we get the morphism 0 0 := V1,d ∩ U0 ⊂ U0 ∼ ν1,d : A1 → V1,d = Ad , t → (t, t2 , t3 , . . . , td ). 0 V1,d is therefore an affine rational curve with polynomial parametrization, as in 0 0 (3.31). V1,d is irreducible and isomorphic to A1 , since ν1,d is given by a d-tuple 1 of regular functions on A (cf. Prop. 6.2.2) and its inverse is the restriction 0 to V1,d of the first projection π1 : Ad → A1 . As done for the projective 0 twisted cubic in (3.34), V1,d is the projective closure in Pd of V1,d . Moreover, 0 st the isomorphism ν1,d in the affine charts extends to ν1,d as an isomorphism

6.3 Morphisms with (quasi) projective target

157

between P1 and V1,d . At last, following the same procedure as in § 3.3.12 for the projective twisted cubic, V1,d is given by the zero-locus of all the quadratic polynomials coming from the maximal minors of the matrix of linear forms   X0 X1 X2 . . . Xd−1 A := (6.18) X1 X2 X3 . . . Xd and Ip (Cd ) is generated by these quadric hypersurfaces. Another well-known classical example is given by V2,2 , which is called Veronese surface. It is a non degenerate, projective surface in P5 which is isomorphic to P2 and of degree 4 since, by the correspondence among conics in P2 and hyperplane sections of V2,2 , two general hyperplane sections of V2,2 intersect along 4 distinct points. The 2-dimensional complete linear system of lines in P2 gives rise to a 2-dimensional family of conics contained in V2,2 whereas the 5-dimensional complete linear system of conics in P2 gives rise to the 5-dimensional linear system of hyperplane section curves in V2,2 , whose general element is a rational normal quartic curve. 6.3.2 Veronese morphism and consequences One can use the Veronese isomorphism νn,d to prove general properties of arbitrary algebraic varieties. For example, the next result is the extension to the projective case of what proved in Lemma 6.2.12 for An . Proposition 6.3.5 For any non negative integers n, d, let W ⊂ Pn be any hypersurface of degree d. Then Pn \ W is an affine variety. Proof. Using the Veronese isomorphism νn,d , one has that Pn \ W ∼ = Vn,d \ (HW ∩ Vn,d ), where HW denotes the hyperplane in PN cutting out the hyperplane section HW ∩ Vn,d corresponding to the hypersurface W ⊂ Pn of degree d. One concludes by observing that Vn,d \ (HW ∩ Vn,d ) ∼ ⊔ = Vn,d ∩ (Pn \ HW ) ∼ = Vn,d ∩ AN . ⊓ Corollary 6.3.6 If V ⊂ Pn is a projective variety, which is not a point, and W ⊂ Pn is any hypersurface then V ∩ W 6= ∅. In particular, any two (projective) curves in P2 intersect. Proof. If by contradiction one had V ∩ W = ∅, choosing d := deg(W ), from the proof of Proposition 6.3.5 we would get V ∼ = νn,d (V ) ⊂ Zp (HW )c ∼ = AN . Then V would be isomorphic to an irreducible, closed subset in AN i.e. to an affine variety. From Corollary 6.2.6, V would be a point against the assumptions. ⊓ ⊔

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Notice that, from Corollary 6.2.7, we already know that P2 and A2 cannot be isomorphic; the second part of Corollary 6.3.6 more generally shows that they are not even homeomorphic, as in A2 there actually exist curves with empty intersections. Similarly to Example 6.2.14, with the use of the Veronese isomorphism we can show that there exist quasi-projective varieties which cannot be either projective, or affine or quasi-affine. To do this, consider first the following preliminary result. Lemma 6.3.7 For any integer n > 2, let Λ ⊂ Pn be a linear subspace s.t. dim(Λ) 6 n − 2. Let W := Pn \ Λ. Then OW (W ) ∼ = K. Proof. Since W ⊂ Pn is an open dense subset of Pn , from Lemma 5.2.13 and Theorem 5.2.14 (h) we have (n) K(W ) ∼ = K(Pn ) = S((0)) .

Since OW (W ) ⊂ K(W ), then any regular function φ ∈ OW (W ) is of the (n) G1 , for some G1 , G2 ∈ Sd , with d > 0 an integer, G2 6= 0 and form φ = G 2 W ⊂ Up (G2 ) = Zp (G2 )c . Since dim(Λ) 6 n − 2, reasoning as in the proof of Claim 6.2.15 we get that G2 ∈ K∗ and so also G2 ∈ K. ⊓ ⊔ Example 6.3.8 We can now show that, for any point P ∈ P2 , W := P2 \ {P } is a quasi-projective variety which is neither projective, nor affine nor quasiaffine. With the use of projective transformations, it is not a restriction to assume P to be [1, 0, 0]. Now W is not projective: otherwise, for any line ℓ not passing through P , W \ ℓ would be affine, but this is W ∩ (P2 \ ℓ) ∼ = A2 \ {(0, 0)}, contradicting Example 6.2.14. By Lemma 6.3.7, W cannot be affine: otherwise A(W ) = OW (W ) ∼ = K, i.e. Ia (W ) woud be maximal i.e. W would be a point, a contradiction. Similarly, it cannot be quasi-affine: otherwise, if W denotes its affine closure, always by Lemma 6.3.7, we would have A(W ) ⊆ OW (W ) ∼ = K and we can conclude as before. We conclude this section by observing another main difference between affine and projective varieties. Corollary 6.2.4 enstablishes that the coordinate ring A(V ) of any affine variety V is invariant for the isomorphism class represented by V ; in other words, if V ⊂ An and W ⊂ Am are isomorphic affine varieties, then A(V ) ∼ = A(W ), no matter the embedding in different affine spaces. Projective varities behave differently; if V ⊂ Pn is any projective variety the homogeneous coordinate ring S(V ) in general is not invariant under isomorphism. To be more precise, if V, W ⊆ Pn are projective varieties which are isomorphic via a projectivity Φ ∈ PGL(n + 1, K) sending V to W (as it occurs

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159

st e.g. between any Veronese variety Vn,d and its standard model Im(νn,d )), by n+1 Definition 3.2.1, Φ gives rise to an affinity Φ ∈ GL(n + 1, K) of A for which

Ca (V ) ∼ = Ca (W ). Since these are affine varieties then, by Corollary 6.2.4, we have A(Ca (V )) ∼ = A(Ca (W )). Now A(n+1) ∼ = S(n) and from (3.8), we get that for any projective variety Z ⊆ Pn S(Z) =

S(n) ∼ A(n+1) = A(Ca (Z)). = Ip (Z) Ia (Ca (Z))

This implies that S(V ) ∼ = S(W ) as it occurs for coordinate rings of affine varieties. On the contrary if V ⊆ Pn and W ⊆ Pm are projective varieties, embedded in different ambient projective spaces, which are isomorphic as algebraic varieties, in general S(V ) is not isomorphic to S(W ) as the following easy example shows. Example 6.3.9 For any d > 1, any rational normal curve V1,d ⊂ Pd is isomorphic to P1 . On the one hand, we have S := S(P1 ) = S(1) ; on the other, since any V1,d is projectively equivalent inside Pd to the standard rational (d) normal curve, from above we have that S(V1,d ) = IpS(V1,d ) is isomorphic to the st st , the graded ). By the defintion of ν1,d homogeneous coordinate ring of Im(ν1,d st ring R := S(Im(ν1,d )) is the image of the (graded) K-algebra homomorphism md : S(d) −→ S(1) = S, defined by Yi −→ X0d−i X1i , 0 6 i 6 d, where Y0 , . . . , Yd the indeterminates in S(d) whereas X0 , X1 the ones in S(1) ; one can easily check that the kernel of the previous homomorphism is exactly the ideal generated by the maximal minors of the matrix of linear forms (6.18). Notice that md is not surjective; indeed focusing on the graded summands we get that S0 = R0 but R1 ∼ = Skd . = Sd and more generally, for any k > 1, Rk ∼ In other words, R is isomorphic to the graded subring of S = S(1) h i M h (1) i , S(1) = Sk d

k>0

d

where as in § 1.10.2 each graded summand is defined as i h (1) (1) := Skd . Sk d

6.3.3 Divisors Let Y be either an affine space An or a projective space P(V ), which we will always assumed to be endowed with a projective frame. We will denote by

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Div(Y ) the free abedlian group generated by the set Ω of all irreducible hypersurfaces in Y . Any element of Div(Y ) is of the form X rZ Z, D := Z∈Ω

where rZ ∈ Z are integers which are almost all zero except for finitely many of them. Such a D is called a divisor of Y , the integer rZ is called the multeplicity of Z in D and the hypersurfaces for which rZ 6= 0 are called the irreducible components of D. The hypersurface supp(D) := ∪rZ 6=0 Z is called the support of D. If rZ ∈ {−1, 0, 1} for any Z, then D is said to be reduced . One defines the degree of D as X rZ deg(Z). deg(D) := Z∈Ω

If fZ = P 0 is an equation of Z in Ω (in the chosen frame, when Y = P(V )), given D = Z∈Ω rZ Z and put Y fZrZ , fD := Z∈Ω

then fD = 0 is an equation of D. By the Hilbert nullstellensatz’s, two equations of a divisor D are essentially equal. From now on, let us focus on the case Y = P(V ) of dimension n, endowed with a system of homogeneous coordinates. Let D be a divisor and let Z ⊂ P(V ) be a linear subspace of dimension m. Let f = 0 be an equation of D and assume that Z has a parametric representation given by x = λ · A, where A a (m + 1) × (n + 1)-matrix of rank m + 1, x := (X0 , dots, Xn ) and λ = (λ0 , . . . , λm ) (cf. Esercise 3.4.8). The polynomial f (λ · A) is zero if an only if Z ⊆ supp(D). In such a case, one states that D contains Z and put Z ⊆ D. Otherwise, l’equazione f (λ · A) = 0 defines a divisor DZ in Z, which is called the intersection of Z with D. The previous definition is well-posed (cf. Exercise ??) and one has deg(DZ ) = deg(D). When Y = An one has similar definitions and considerations; the main difference is that in general deg(DZ ) 6 deg(D). Similarly, one can define in an obvious way the projective closure of a divisor D in An . Therefore, one has:

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161

Proposition 6.3.10 (Bezout’s theorem for linear sections) If D is a divisor of either an affine space or of a projective space and if Z is a subspace, then either Z is contained in D or D intersects Z along a divisor DZ of degree a most deg(D); moreover deg(DZ ) = deg(D) in the projective case. Example 6.3.11 If deg(D) = d and if Z is a line not contained in D, then DZ = m1 P1 + . . . + mh Ph where Pi are distinct points of Z and mi are positive integers, with 1 ≤ i ≤ h, such that d = m1 + . . . + mh . The positive integer mi is called the intersection multiplicity between Z and D at the point Pi and it is denoted by mPi (D, Z). One poses mP (D, Z) = 0 if P ∈ / supp(DZ ) and mP (D, Z) = ∞ for any P ∈ Z, when Z ⊆ D. Let us refer to the case Y = P(V ) of dimension n; fix a positive integer d and consider the set LP(V ),d LALA dei divisori effettivi di grado d in P(V ). Nel caso P(V ) = Pn , scriveremo Ln,d . Si ha LP(V ),d = P(Symd (V ∗ )). Un sottospazio di dimensione r di LP(V ),d si dice un sistema lineare di divisori (o anche, con abuso di lunguaggio, di ipersuperficie di grado d di P(V ). Se introduciamo in P(V ) un sistema di coordinate omogenee, mediante una proiettivt` a φ : Pn → P(V ), ne risulta indotto un isomorfismo omogeneo ∗ Sn → S(V ) che determina isomorfismi Sn,d → S(V ∗ )d per ogni d ≥ 0. Ci` o determina delle proiettivit`a biettive φd : Ln,d → LP(V ),d , che inducono delle coordinate omogenee in LP(V ),d : le coordinate [fi ]|i|=d (gli indici possono pensarsi ordinati in ordine P lessicografico) di un divisore D sono i coefficienti del polinomio f (x) = |i|=d fi xi , determinato a meno del prodotto per una costante moltiplicativa non nulla, tale che f = 0 sia un’equazione di D nel dato riferimento. P Se f (x) = |i|=d fi xi Sia L ⊆ LP(V ),d un sistema lineare di dimensione r. Se D0 , . . . , Dr ∈ L sono divisori linearmente indipendenti, che hanno equazioni fi = 0, per 0 ≤ i ≤ r, nel dato riferimento, allora un divisore D sta in L se e solo se ha equazione del tipo λ0 f0 + . . . + λr fr = 0, con ([λ0 , . . . , λr ] ∈ Pr . Un sistema lineare di dimensione 0 `e un divisore, un sistema di dimensione 1 si dice un fascio, un sistema di dimensione 2 si dice una rete. Il sistema vuoto ha dimensione −1. Notiamo che Z(f0 , . . . , fr ) = ∩Z∈L Z. Questo si dice il luogo base del sistema linear L e si indica con Bs(L). Se P ∈ P(V ) indichiamo con L(−P ) l’insieme dei divisori di L il cui supporto contiene P (si dice allora che contengono P ). Si ha L(−P ) = L se e solo se P ∈ Bs(L). Se P 6∈ Bs(L) allora L(−P ) `e un sottosistema lineare di L di dimensione r − 1

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6 Morphisms

Example 6.3.1. Un sistema lineare L di dimensione n di LP(V ),d , con P(V ) retta proiettiva, si dice una serie lineare di grado d e dimensione n e si indica con il simbolo gdn . la serie `e completa se L = LP(V ),d , cio`e se n = d. C’e’ un unica g11 , quella completa, costituita da tutti i divisori di grado 1 sula retta, cio`e dai suoi punti. Determiniamo le g21 . Introdotto un riferimento in P(V ), possiamo ridurci al caso di P1 . Esistono allora due polinomi omogenei di secondo grado non proporzionali f0 , f1 tali che gli elementi della g21 sono tutti e soli i divisori di equazione λ0 f0 + λ1 f1 = 0, con λ0 , λ1 non entrambi nulli. I due divisori Di di equazioni fi = 0, 0 ≤ i ≤ 1, non possono avere lo stesso supporto, altrimenti i polinomi f0 , f1 sarebbero proporzionali. Un primo caso `e quello in cui i supporti di D1 e D2 abbiano un punto P in comune. Allora P `e un punto base della g21 e i divisori di questa serie sono tutti e soli quelli del tipo P + Q al variare di Q sula retta. Supponiamo ora che i due divisori D1 , D2 abbiano supporti disgiunti, ossia che la g21 non abbia punti base. Allora la g21 `e costituita da divisori del tipo P1 + P2 , tali che per ogni punto P c’e’ un unico punto Q tale che P + Q ∈ g21 . La g21 determina allora l’applicazione σ : P(V ) → P(V ) che associa a P ∈ P(V ) il punto Q suddetto. La σ `e biettiva e involutoria ossia tale che σ −1 = σ. Per tal motivo le g21 senza punti base si dicono involuzioni. Lasciamo al Lettore di verificare che σ `e una proiettivit`a e che ogni proiettivit` a involutoria `e di questo tipo.

6.4 Morphisms and local properties We conclude this chapter by giving two fundamental results which will be frequently used later on (cf. Chapters 7 and 8). The first one gives a general criterion to verify when a continuous map between algebraic varieties is actually a morphism. Proposition 6.4.1 Let V and W be algebraic varieties and let ϕ : V → W be any continuous map. Then ϕ is a morphism if and only if there exists an open covering {Wj }j∈J of W such that, for any j ∈ J, there exists an open covering {Vij }i∈I(j) of ϕ−1 (Wj ) such that the continuous map ϕij : Vij → Vj induced by ϕ is a morphism, for any j ∈ J and any i ∈ I(j). The previous proposition states that being a morphism for a continuous map between algebraic varieties is a local property, i.e. it globally holds if and only if it is locally a morphism on open sets of suitable open coverings. Proof (Proof of Proposition 6.4.1). It is clear that if ϕ is a morphism, then each ϕij = ι−1 Wi ◦ϕ◦ιVij is a morphism, being a compositon of morphisms.

6.5 Exercises

163

Conversely, assume that any ϕij is a morphism and let U ⊆ W be any non empty open set and let f ∈ OW (U ) be any regular function on it. Let fij be the restriction of the continuous map f ◦ ϕ to the open set ϕ−1 (U ) ∩ Vij of V . Notice that fij coincides with the restriction to ϕ−1 (U ) ∩ Vij of f |U ∩Wj ◦ ϕij , i.e. fij ∈ OV (ϕ−1 (U ) ∩ Vij ). Since {ϕ−1 (U ) ∩ Vij }i∈I(j) is an open covering of ϕ−1 (U ), it follows that U ϕ (f ) is regular of ϕ−1 (U ), i.e. ϕ is a morpism. ⊓ ⊔ The second result is a generalization of Corollary 6.2.13 to any algebraic variety. Proposition 6.4.2 On any algebraic variety V there exists a basis for the topology ZarV consisting of open affine subsets. Proof. We must show that, for any point P ∈ V and for any open set U ⊂ V containinng P , there exists an open affine subset UP with P ∈ UP ⊆ U . Since U is also a variety, we may assume V = U . Moreover, since any variety can be covered by quasi-affine varieties, we can assume V to be quasi-affine in An , for some non negative integer n. a a Let Z := V \ V 6= ∅, where V the closure of V quasi-affine in An . Then a Z is a proper closed subset of V and let Ia (Z) ⊂ A(n) be its ideal. Then, since Z is closed and P ∈ / Z, there exists f ∈ Ia (Z) such that f (P ) 6= 0. Now Z ⊆ Za (f ) and P ∈ / Za (f ), so P ∈ UP := V \ (V ∩ Za (f )) which is an open subset of V , since V ∩ Za (f ) is a closed subset of V by induced topology; in particular UP is irreducible. On the other hand a

UP = V ∩ (An \ Za (f )) = (V ∪ Z) ∩ (An \ Za (f )) = V ∩ (An \ Za (f )), the latter equality following from Z ⊆ Za (f ); thus UP is also a closed subset of An \ Za (f ) which, by Lemma 6.2.12, is an affine open set of An . Hence UP is affine, as desired. ⊓ ⊔

6.5 Exercises Exercise 6.5.1. Let K be algebraically closed and let ϕ : A2 → A2 be the map defined by ϕ (x1 , x2 ) −→ (x1 , x1 x2 ). Prove that ϕ is a morphism, find its image and conclude that it is neither open nor closed in A2 .

Exercise 6.5.2. Let K be algebraically closed and let C = Za (x31 + x21 − x22 ) ⊂ A2 . Prove that C is an affine variety. Consider then the map A1 → C, t → (t2 − 1, t(t2 − 1)); prove tha it is a morphism. Is it an isomorphism?

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6 Morphisms

Exercise 6.5.3. Notation and assumption as in Exercise 5.1.4. Prove that the inverse of the rational map ψ therein is a morphism ϕ : A1 → C, which gives a parametrization of C. Prove finally that ϕ restricts to an isomorphism between A1 \ {3 points} and C \ {(0, 0)} (cf. [27, Ex. 4.7, p: 83]).

7 Products of algebraic varieties

In this chapter we will study products of algebraic varieties. Namely, we want to show how to give a structure of algebraic variety to the set-theoretic cartesian product V × W , when both V and W are algebraic varieties.

7.1 Products of affine varieties In Example 2.1.22 (i), we already introduced the product of two affine spaces Ar × As , where r and s non negative integers. As a set Ar × As is identified with Ar+s ; in particular, the cartesian product of two affine spaces has a natural structure of affine variety, the affine space Ar+s . The Zariski topology Zarar+s is finer than the product Zarra ×Zarsa of the two Zariski topology (recall Example 2.1.22 (iii)). The two projections Ar × As → Ar and Ar × As → As

(7.1)

are morphisms since they coincide with the projections πI and πJ as in (6.10), with multi-indices I = (1, 2, . . . , r) and J = (r + 1, . . . , r + s). More generally we have the following result. Proposition 7.1.1 Let r and s be non negative integer and let V ⊆ Ar and W ⊆ As be affine varieties. Then V ×W is an affine variety which is a Zariski closed irreducible subset of Ar+s . Proof. The fact that V × W is Zariski closed in Ar+s has been proved in Example 2.1.22 (i). More precisely, if V := Za (f1 , . . . , fn ) = Za (Ia (V )) and W := Za (g1 , . . . , gm ) = Za (Ia (W )), (2.8) enstablishes that V × W = Za (f1 , . . . , fn ) ∩ Za (g1 , . . . , gm ) = Za (Ia (V )) ∩ Za (Ia (W )), where Ia (V ), Ia (W ) are considered as ideals in A(r+s) , via the natural Kalgebra inclusions A(r) ֒→ A(r+s) and A(s) ֒→ A(r+s) , and where Za (Ia (V )) =

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7 Products

V × As is the closed s-dimensional cilynder over V in Ar+s and Za (Ia (W )) = Ar × W the closed r-dimensional cilynder over W in Ar+s . Moreover, Ia (V × W ) = (f1 , . . . , fn , g1 , . . . , gm ). To complete the proof, we are left to showing that V × W is irreducible. Assume that V ×W = Z1 ∪Z2 where Zi closed subsets of V ×W , for 1 6 i 6 2. For any point P ∈ V , WP := {P } × W

is a subset of V × W which is homeomorphic to W and so irreducible. Then WP ⊆ Zi , for i ∈ {1, 2}. Consider Vi := {P ∈ V : WP ⊆ Zi },

for

1 6 i 6 2.

For any point Q ∈ W , pose Vi (Q) := {P ∈ V | (P, Q) ∈ Zi },

for

1 6 i 6 2.

(V × {Q}) ∩ Zi = Vi (Q) × {Q},

for

1 6 i 6 2,

Then so Vi (Q) is closed, for any Q ∈ W and for any 1 6 i 6 2. Since one has \ Vi (Q) for 1 6 i 6 2 Vi = Q∈W

then V1 , V2 are closed in V . From V = V1 ∪ V2 and from the irreducibility of V , therefore one has either V = V1 (in which case V × W = Z1 ) or V = V2 (consequently, V × W = Z2 ). In all cases V × W is irreducible. ⊓ ⊔ Remark 7.1.2 As above, the projections π

π

V W V × W −→ V and V × W −→ W

(7.2)

are morphisms of affine varieties, since they are restrictions to V × W of the projections (7.1). 7.1.1 Coordinate ring of V × W affine and tensor products TO BE ADDED

7.2 Products of algebraic varieties Let us consider more generally V and W algebraic varieties. Thus, for some integers n and m, we have V ⊆ Pn , W ⊆ Pm , where the inclusions are locally closed morphisms. To endow the (set theoretical) cartesian product V × W with a structure of algebraic variety, it will suffice to define a set-theoretical injective map Ψ

V × W ֒→ PN ,

for some positive integer N , such that:

(7.3)

7.2 Products of algebraic varieties

167

(i) ψ(V × W ) ⊂ PN is a quasi-projective variety, (ii) for any affine open subsets UV ⊆ V and UW ⊆ W (recall Prop. 6.4.2), Ψ (UV × UW ) is an affine open subset of Ψ (V × W ); moreover such open sets determine an open covering of Ψ (V × W ), (iii) for any choice of affine open subsets UV ⊆ V and UW ⊆ W , the map Ψ |UV ×UW : UV × UW −→ Ψ (UV × UW ) is an isomorphism of affine varieties. Notice that (iii) is nothing but a compatibility condition between the structure of algebraic variety on Ψ (V × W ) as in (i) and the one given in § 7.1 on its affine open subsets as in (ii). More precisely, the subset Ψ (UV × UW ), which is an affine open subset w.r.t. the structure of algebraic variety of ψ(V × W ) ⊂ PN as in (i), has to be isomorphic (as affine variety) to the affine variety UV × UW , whose structure has been defined in § 7.1. Lemma 7.2.1 Given a set-theoretical injective map Ψ as in (7.3), with Ψ (V × W ) ⊂ PN a quasi-projective variety, then Ψ satisfies the compatibility condition (iii) above if and only if, for any points P ∈ V and Q ∈ W , there exist UP ⊆ V and UQ ⊆ W affine open neighborhoods of P ∈ V and Q ∈ W , respectively, s.t. Ψ |UP ×UQ defines an isomorphism of the affine variety UP × UQ onto its image Ψ (UP × UQ ) ⊆ PN and Ψ (UP × UQ ) is an affine open subset of ψ(V × W ). Proof. The implication (⇒) is trivial. For the converse, let UV ⊆ V and S U ⊆ W be any affine open subsets. Since U = V P ∈UV UP and UW = SW U , from § 7.1 we have that Q∈UW Q {UP × UQ }(P,Q)∈UV ×UW

is an open covering of UV × UW , formed by affine open sets by assumptions. The map Ψ |UV ×UW is therefore continuous and it is an isomorphism as it follows Propositions 1.11.3. ⊓ ⊔ A map Ψ for which conditions in Lemma 7.2.1 hold is said to satisfy the property of being local. Lemma 7.2.2 If a map Ψ as above exists, it is uniquely determined (up to isomorphism) by the property of being local. Proof. Assume there exist a map Ψ as above and another injective map Φ : V × W ֒→ PM , for some integer M , both of them satisfying the property of being local. Thus Φ ◦ Ψ −1 : Ψ (V × W ) → Φ(V × W )

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7 Products

is bijective. It suffices to show that Φ ◦ Ψ −1 is a morphism. The proof of this is identical to that of Lemma 7.2.1, using the fact that both Ψ and Φ satisfy the property of being local. ⊓ ⊔ The previous result states that, to give a structure of algebraic variety on V × W , one is reduced to show the existence of a map Ψ satisfying conditions (i), (ii) above and the property of being local. Once such a map has been constructed, the structure on V × W is uniquely determined up to isomorphisms. The aim of the next section is to construct such a map. 7.2.1 Segre morphism and the product of projective spaces Let us consider first V = Pn and W = Pm , for some non negative integers n and m. Definition 7.2.3 We define the Segre map of indices n and m by σn,m :

P n × Pm −→ PN ([x0 , . . . xn ], [y0 , . . . , ym ]) −→ [x0 y0 , x0 y1 , . . . , xi yj , . . . , xn ym ],

where N := (n + 1)(m + 1) − 1, with 0 6 i 6 n and 0 6 j 6 m. This map is well-defined and the image Σn,m = Im(σn,m ) is called the Segre variety of indices n and m. The use of the term ”variety” is justified by the following result. Lemma 7.2.4 The Segre map σn,m is bijective and its image Σn,m is a projective variety in PN . Proof. Take homogeneous coordinates [Wij ] in PN , with 0 6 i 6 n and 0 6 j 6 m. These coordinates are lexicographically ordered so that they are compatible with σn,m ; thus points in Σn,m satisfy homogeneous quadratic equations Wij Wkr − Wir Wkj = 0, 0 6 i, k 6 n, 0 6 j, r 6 m.

(7.4)

Let Z be the closed subset of PN cut out by quadric hypersurfaces as in (7.4); it is clear that Σn,m ⊆ Z. We claim that, for any point R ∈ Z, there is a unique pair (P, Q) ∈ Pn ×Pm such that R = σn,m (P, Q); this will imply that Σn,m = Z and that σn,m bijectively maps Pn × Pm onto Σn,m . 0 ] ∈ Z be any point. Without loss of To prove the claim, let R := [wij 0 generality, we can assume w00 6= 0 and so, by rescaling all the coordinates up 0 = 1 (the other cases can be to this multiplicative non zero factor, that w00 handled analogously); since R ∈ Z then

7.2 Products of algebraic varieties

169

0 0 0 0 0 wi0 w0j = wij w00 = wij .

Setting 0 0 0 0 P := [1, w10 , . . . , wn0 ] ∈ Pn and Q := [1, w01 , . . . , w0m ] ∈ Pm ,

this means that R = σn,m ((P, Q)) and that P and Q are uniquely determined. We are left to showing that Σn,m is irreducible. For any 0 6 i 6 n and 0 6 j 6 m, denote by N Uij := PN \ Zp (Wij ) the principal affine open set in PN isomorphic to AN and, similarly,

An ∼ = Zp (X0 )c := U0n ⊂ Pn , and Am ∼ = Zp (Y0 )c := U0m ⊂ Pm . Previous computations show that N Σn,m ∩ U00 = σn,m (U0n × U0m )

and that σn,m |U0n ×U0m is an isomorphism from the affine variety U0n × U0m ∼ = N An × Am ∼ , which is therefore an = An+m onto the affine closed set Σn,m ∩ U00 N , for any pair of indices affine variety. Similar conclusion holds for Σn,m ∩ Uij (i, j) with 0 6 i 6 n and 0 6 j 6 m. One has an open covering of Σn,m , [
N Σn,m ∩ Uij , Σn,m = i,j

where each Σn,m ∩

N Uij

is an affine open subset of Σn,m . Moreover \
N Σn,m ∩ Uij W := 6= ∅ i,j

is an open subset of Σn,m , which is also open in any affine open subset Σn,m ∩ N Uij , for any i, j as above; in particular, W is irreducible. Since W is contained in any open set of an affine open covering of Σn,m as a dense open subset, this forces Σn,m to be irreducible. ⊓ ⊔

The fact that Σn,m is closed in PN is also a consequence of a more general fact, which will be discussed later on (cf. Theorem 10.2.1 and Remark 10.2.2 (iii)). At last, we have:

Lemma 7.2.5 The bijection σn,m : Pn × Pm → Σn,m satisfies condition (ii) as in (7.3) and the property of being local. Proof. The proof of Lemma 7.2.4 shows that {σn,m (Uin × Ujm )}i,j is an open covering of the projective variety Σn,m , where each σn,m (Uin × Ujm ) is isomorphic to the affine variety Uin × Ujm , i.e. condition (ii) of (7.3) for the map σn,m holds. N Moreover, for any pair (i, j), Σn,m ∩ Uij is an affine open neighborhood of any of its points over which the Segre map is an isomorphism with Uin × Ujm . From Lemma 7.2.1, σn,m satisfy the property of being local. ⊓ ⊔

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7 Products

From Lemma 7.2.2, previous results determine (up to isomorphism) a structure of projective variety on Pn × Pm , which from now on, will be always identified with Σn,m by means of the Segre morphism σn,m . Remark 7.2.6 Notice e.g. that the Segre variety Σ1,1 is the quadric surface Zp (W00 W11 − W01 W10 ) ⊂ P3 which contains two families of lines, called rulings of the quadric surface. For this reason Σ1,1 is said to be doubly ruled. Lines of the two rulings arise from the isomorphism of Σ1,1 with P1 × P1 ; more precisely, for any P, Q ∈ P1 , Σ1,1 contains the lines ℓP := σ1,1 ({P } × P1 ) and rQ := σ1,1 (P1 × {Q}). Any two lines of the same ruling {ℓP }P ∈P1 are skew, the same holds for the lines of the other ruling; whereas, for any R ∈ Σ1,1 , there exist a line ℓP of the first ruling and a line rQ of the second one s.t. R = σ1,1 (P, Q) and ℓP ∩ rQ = R. 3 = Za (x3 − x1 x2 ) is the hyperbolyc paraboloid in A3 . Notice that Σ1,1 ∩ U00

Remark 7.2.7 (i) The projections Pn × Pm −→ Pn and Pn × Pm −→ Pm

(7.5)

are morphisms of projective varieties. This follows from Proposition 1.11.3 and the fact that their restrictions to the affine open subsets U0n × U0m are morphisms (cf. Rem 7.1.2). (ii) Notice that   [ N −1  Σn,m ∩ U0j U0n × Pm = σn,m j

n

m

is an open subset of P × P , so the map σn,m defines a structure of quasiprojective variety also on An × Pm . 7.2.2 Products of projective varieties We first consider the following preliminary result.

Proposition 7.2.8 (i) A subset T ⊆ Pn × Pm is closed if and only if it is defined by polynomials Gk (X0 , . . . , Xn , Y0 , . . . , Ym ), 1 6 k 6 s, where each polynomial Gk is bi-homogeneous, i.e. it is separately homogeneous with respect to the two set of indeterminates (X0 , . . . , Xn ) and (Y0 , . . . , Ym ).

7.2 Products of algebraic varieties

171

(ii) A subset T ⊆ An × Pm is closed if and only if it is defined by polynomials Fk (x1 , . . . , xn , Y0 , . . . , Ym ), 1 6 k 6 s, where each polynomial Fk is homogeneous with respect to the set of indeterminates (Y0 , . . . , Ym ). Proof. (i) Take any closed subset Z ⊂ PN , where N = (n + 1)(m + 1) − 1. Then Z := Zp (A1 (W00 , . . . , Wnm ), . . . , As (W00 , . . . , Wnm )), for some homogeneous polynomials Ak ∈ H(SN ), 1 6 k 6 s. −1 The closed subset σnm (Z) ⊂ Pn × Pm is defined by the equations Gk (X0 , . . . , Xn , Y0 , . . . , Ym ), 1 6 k 6 s, obtained by applying to the polynomials Ak ’s the indeterminate substitutions Wij = Xi Yj , 0 6 i 6 n, 0 6 j 6 m. Each of the polynomials G1 , . . . , Gs are separately homogeneous with respect to the two set of indeterminates (X0 , . . . , Xn ) and (Y0 , . . . , Ym ). Conversely, any polynomial G(X0 , . . . , Xn , Y0 , . . . , Ym ) which is homogeneous of degree d in the set of indeterminates (X0 , . . . , Xn ) and of degree r in the set of indeterminates (Y0 , . . . , Ym ) determines the same closed subset T ⊂ Pn × Pm defined by the vanishing locus of the set of polynomials Yjd−r G(X0 , . . . , Xn , Y0 , . . . , Ym ), 0 6 j 6 m,

(7.6)

where we assumed d > r. All polynomials in (7.6) are now homogeneous of the same degree d with respect to both set of indeterminates (X0 , . . . , Xn ) −1 and (Y0 , . . . , Ym ). Therefore T = σn,m (Z), where Z is the closed subset in PN defined by homogeneous polynomials in the indeterminates Wij obtained by polynomials in (7.6) via the indeterminate substitutions Wij = Xi Yj , 0 6 i 6 n, 0 6 j 6 m. (Notice that the indeterminate substitutions are not uniquely determined, so Z is not uniquely determined; on the other hand, two different choices differ by elements of the ideal generated by polynomials as in (7.4), so Z ∩ Σn,m is independent from the choices considered). (ii) An × Pm is open in Pn × Pm since it is Zp (X0 )c ; therefore, the assertion follows by dehomogenizing a system of equations for T ⊂ Pn × Pm , which is a closed subset as in (i). ⊓ ⊔ Let now V ⊂ Pn and W ⊂ Pm be projective varieties. We will show that V × W is a closed subvariety of Pn × Pm ; from § 7.2.1, this will endow V × W with a structure of projective variety. Suppose that Ip (V ) is generated by homogeneous polynomials Ph (X0 , . . . , Xn ) = 0, h = 1, . . . , s and that Ip (W ) is generated by homogeneous polynomials

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7 Products

Qk (Y0 , . . . , Ym ) = 0, k = 1, . . . , t. Then V × W is defined by the system of homogeneous polynomials Ph (X0 , . . . , Xn ) = 0 = Qk (Y0 , . . . , Ym ), h = 1, . . . , s, k = 1, . . . , t. From Proposition 7.2.8 (i), V × W is closed in Pn × Pm . We are left to showing that V ×W is irreducible. If either V or W is a point, then V × W is irreducible since homeomorphic to an irreducible topological space. Assume therefore that neither V nor W are points; whithout loss of generality, we can assume that V0 := V ∩ U0n 6= ∅ and W0 := V ∩ U0m 6= ∅. Now V0 × W0 is an affine variety (cf. § 7.1), so it is irreducible. Any polynomial G(X0 , . . . , Xn , Y0 , . . . , Ym ) which is separately homogeneous with respect to the two sets of indeterminates (X0 , . . . , Xn ) and (Y0 , . . . , Ym ) and which vanishes along V0 × W0 is such that, for any [q0 , . . . , qm ] ∈ W0 , G(X0 , . . . , Xn , q0 , . . . , qm ) ∈ Ip (V0 ) = Ip (V ) (recall (3.21)), i.e. G(X0 , . . . , Xn , Y0 , . . . , Ym ) vanishes also along V × W0 . Similarly, it vanishes along V0 × W , so it also vanishes along V × W . In other words, p V0 × W0 = V × W, which implies that V × W is irreducible. Remark 7.2.9 The projections πV : V × W −→ V and πW : V × W −→ W

(7.7)

are morphisms of projective varieties, since they are restrictions of morphisms as in (7.5). 7.2.3 Products of quasi-projective varieties Let V ⊂ Pn and W ⊂ Pm be quasi-projective varieties and denote by V and W their projective closures, respectively. Then C := V \ V is a closed subset of V and D := W \ W is a closed subset of W . Then  V × W = (V \ C) × (W \ D) = (V × W ) \ (V × D) ∪ (C × W ) ,

which shows that V × W is an open set of the projective variety V × W , so V × W is a quasi-projective variety. The fact that the projections πV and πW are morphisms follows as in Remark 7.2.9.

7.3 Diagonals, graphs and affine open sets

173

7.3 Diagonals, graphs and affine open sets As consequences of the previous analysis, we can deduce interesting properties of algebraic varieties. Let V be any algebraic variety; the subset ∆(V ) := {(P, P ) | P ∈ V } ⊂ V × V

(7.8)

is called the diagonal of V × V .

Proposition 7.3.1 For any algebraic variety V , ∆(V ) is closed in V × V . Proof. Since V ⊆ Pn , for some integer n, then

∆(V ) = ∆(Pn ) ∩ (V × V ),

thus it suffices to show that ∆(Pn ) is closed in Pn × Pn . One concludes by observing that ∆(Pn ) = Zp (X1 Y0 − X0 Y1 , . . . , Xn Y0 − X0 Yn ) and by using Proposition 7.2.8 (i).

⊓ ⊔

Recalling Proposition 6.4.2, with a similar strategy as above one can also prove the following result. Lemma 7.3.2 Let V be any algebraic variety and let U1 and U2 be non empty affine open subsets of V , then U1 ∩ U2 is an affine open subset of V .

Proof. One needs only to show that U1 ∩ U2 is isomorphic to a closed subset in some affine space. By assumption Ui is isomorphic to an affine variety Vi ⊆ Ani for some non negative integer ni , 1 6 i 6 2; thus U1 × U2 ⊂ An1 +n2

is an affine variety. One concludes by observing that U1 ∩ U2 = (U1 × U2 ) ∩ ∆(An1 +n2 )

and that ∆(An1 +n2 ) is closed in An1 +n2 .

Let now f : V → W be a morphism of algebraic varieties; the set Γf := {(P, f (P )) | P ∈ V } ⊆ V × W,

⊓ ⊔ (7.9)

is called the graph of the morphism. Proposition 7.3.3 For any algebraic varieties V and W and for any f ∈ Morph(V, W ), Γf is closed in V × W .

Proof. From Proposition 7.3.1, ∆(W ) is closed in W × W . Thus, one concludes by observing that Γf = (f × IdW )−1 (∆(W )) and that the map f × IdW : V × W → W × W

is a morphism, since it is a product of two morphisms.

⊓ ⊔

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7.4 Exercises

8 Rational maps of algebraic varieties

In this chapter we introduce the notion of rational maps and birational equivalence of algebraic varieties, which are important for classification of varieties. As we will see, a rational map is a morphism which is only defined on some non empty open subset of an algebraic variety. On the other hand, since any non empty open set is dense in a variety, this already carries a lot of information.

8.1 Rational and birational maps We start we the following preliminary result. Lemma 8.1.1 Let V and W be algebraic varieties, let ϕ, ψ ∈ Morph(V, W ) and assume there exists a non empty open subset U ⊆ V s.t. ϕ|U = ψ|U . Then ϕ = ψ. Proof. The morphisms ϕ and ψ determine a map ϕ × ψ : V → W × W, (ϕ × ψ)(P ) := (ϕ(P ), ψ(P )), ∀P ∈ V, which in fact is a morphism. From the assumption, one has (ϕ × ψ)(U ) ⊆ ∆(W ), where ∆(W ) the diagonal in W × W (cf. (7.8)). Since U is dense in V and ϕ × ψ is continuous, then (ϕ × ψ)(U ) = (ϕ × ψ)(V ) ⊆ (ϕ × ψ)(U ) ⊆ ∆(W ), the latter inclusion following from the fact that ∆(W ) is closed in W × W (cf. Proposition 7.3.1). This implies ϕ = ψ. ⊓ ⊔

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Definition 8.1.2 Let V and W be algebraic varieties. A rational map Φ between V and W , denoted by Φ : V 99K W , is an equivalence class of pairs (U, ϕU ), where U ⊆ V is a non empty open subset of V , ϕU ∈ Morph(U, W ) and where (U1 , ϕU1 ) and (U2 , ϕU2 ) are equivalent if ϕU1 and ϕU2 agree on U1 ∩ U2 . Thus, Φ = [U, ϕU ], where [U, ϕU ] the equivalence class of the pair (U, ϕU ). In such a case, ϕU is a representative morphism of Φ over the open set U . If Φ is a rational map and if (U1 , ϕU1 ), (U2 , ϕU2 ) are representative morphism of Φ, then (U1 ∪ U2 , ϕU1 ∪U2 ) is another representative of Φ, where ϕU1 ∪U2 |U := ϕUi , for i = 1, 2. i

The previous observation shows that there exists a non empty open subset of V , say UΦ , which is the biggest open subset of V where Φ|UΦ is a morphism. This open set is called the open set of definition (or even the domain) of the rational map Φ; sometimes UΦ is also denoted by Dom(Φ). From Examples 5.2.23 (i), (iii) and 6.2.5 (iv), the maps φ1 and φ3 are rational maps s.t. Uφ1 = Uφ3 = W = A1 \ {0}; similarly, from Example 6.3.2 (i), when B 6= ∅ the map νL as in (6.15) defines a rational Pn 99K Pr such that UνL = Pn \ B. In general the composition of rational maps is not well-defined. E.g. consider the map ϕ : A1 → A2 , x1 → (x1 , 0) (which is a morphism) and Φ : A2 99K A1 , (x1 , x2 ) 99K

x1 , x2

which is a rational map whose domain if UΦ = A2 \Za (x2 ); since ϕ(A1 )∩UΦ = ∅ the composition Φ ◦ ϕ is not defined. To avoid this kind of problems, one gives the following definition. Definition 8.1.3 A rational map Φ : V 99K W is said to be dominant if for some (and so every) representative (U, ϕU ) is a dominant morphism in the sense of Definition 6.1.5. To have some examples, notice that on any algebraic variety V , dominant rational maps Φ : V 99K A1 are nothing but rational functions Φ ∈ K(V ) \ K. Furthermore, if V is any (n + 1)-dimensional K-vector space and if we identify it with the affine space An+1 , the map π in (3.1) is a morphism (as it follows from Prop. 6.3.1) so it defines a rational surjective (so dominant) map An+1 99K Pn such that Dom(π) = An+1 \ {0}.

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Lemma 8.1.4 Let V, W, Z be algebraic varieties and let Φ : V 99K W and Ψ : W 99K Z be dominant rational maps. Then Ψ ◦ Φ : V 99K Z is a dominant rational map. Proof. By assumptions on Φ and Ψ , the representatives (UΦ , ϕUΦ ) and (UΨ , ψUΨ ) are dominant morphisms (which for simplicity will be simply denoted in the sequel by ϕ and ψ respectively). Then Im(ϕ) ∩ UΨ 6= ∅ and it contains a dense open subset U0 of UΨ , which is therefore also an open subset of W . Then ϕ−1 (U0 ) ⊆ V is an open subset where the composition ψ ◦ ϕ is defined. On this open set ψ ◦ ϕ is a dominant morphism. Indeed, by the dominance of Ψ , for any proper closed subset K ( Z one has that Im(ψ) is not contained in K and ψ −1 (K) is a proper closed subset of W . For the same reason, Im(ϕ) is not contained in ψ −1 (K), so Im(ψ ◦ ϕ) is not contained in K ( Z; since this holds for any proper closed subset K ( Z, this implies that Ψ ◦ Φ is a dominant rational map. ⊓ ⊔ Definition 8.1.5 A birational map Φ : V 99K W is a dominant rational map which admits an inverse, namely a rational map Ψ : W 99K V s.t. Ψ ◦ Φ = IdV and Φ ◦ Ψ = IdW , where the previous equalities are intended as rational maps. If there exists a birational map from V to W , the V and W are said to be birationally equivalent (or simply birational). By the very definition of birational map, V and W are birational varieties if and only if there exist non empty open subsets UV ⊆ V and UW ⊆ W which are isomorphic. For this reason, birational maps are sometimes called also birational isomorphisms. Theorem 8.1.6 Let V and W be algebraic varieties. There is a bijective correspondence between: (i) the set of dominant rational maps V 99K W , and (ii) the set of K-algebra monomorphisms K(W ) ֒→ K(V ). In this bijective correspondence, birational maps correspond to field isomorphisms. Proof. Let Φ : V 99K W be any dominant rational map; since (UΦ , ϕ) is a dominant morphism, from (6.5) we get a K-algebra monomorphism ϕ∗ : K(W ) ֒→ K(UΦ ). We then conclude by K(UΦ ) ∼ = K(V ). θ

Conversely, let K(W ) ֒→ K(V ) be any K-algebra monomorphism. From Proposition 6.4.2, W is covered by affine open subsets. Thus by Lemma 5.2.13 (ii) we can assume W to be affine. Let therefore y1 , . . . yn be generators of the affine coordinate ring A(W ) as a K-algebra of finite type. Since A(W ) ⊆ K(W ), then θ(y1 ), . . . , θ(yn ) ∈ K(V ). There exists a non empty open subset U ⊆ V s.t. θ(y1 ), . . . , θ(yn ) ∈ OV (U ) ⊂ K(V ) (it suffices to take the intersection of all the domains of definition of the rational functions θ(yi ), 1 6 i 6 n). This defines a K-algebra homomorphism

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8 Rational maps θ|

A(W ) −→ OV (U ), yi → θ(yi ) which is injective since θ is. Since W is affine, from Proposition 6.2.3, θ| corresponds to ϕθ ∈ Morph(U, W ). By the injectivity of θ| and by Corollary 6.2.8, we get that ϕθ is dominant. This correspondence defines therefore a dominant rational map Φ : V 99K W with representative ϕθ and is such that (ϕθ )∗ = θ, i.e. this correspondence is the inverse of that reminded at the beginning of the proof. ⊓ ⊔ Corollary 8.1.7 (i) V and W are birationally equivalent if and only if K(V ) ∼ = K(W ). (ii) Any algebraic variety V is birationally equivalent to any of its non empty open subset. (iii) Any algebraic variety V is birationally equivalent to an affine variety and to a projective variety. Proof. (i) This directly follows from Theorem 8.1.6. (ii) Since for any non empty open subset U ⊆ V one has K(U ) ∼ = K(V ), the statement follows from (i). (iii) From Proposition 6.4.2, any algebraic variety has an affine open set U . Moreover, since U is affine, its projective closure U is a projective variety for which U is an open dense subset. Then one concludes by (ii). ⊓ ⊔ Remark 8.1.8 (i) Birationality is an equivalence relation among algebraic varieties. For any algebraic variety V , the symbol [V ]bir will denote the equivalence class consisting of all algebraic varieties birationally equivalent to V . Any representative of [V ]bir will be called a model of the birational class; it is clear that [V ]bir contains all algebraic varieties which are isomorphic to V on the other hand it contains also several other models. (ii) For example, the rational map φ1 : A1 99K Y := Za (x1 x2 − 1) in Example 5.2.23 (i) and 6.2.5 (iv), is birational since Y is isomorphic to A1 \ {0} via φ1 ; on the other hand we already observed that Y is not isomorphic to A1 . Same occurs for e.g. the circle in Example 5.2.23 (iii), 6.2.5 (iv) and for the semi-cubic parabola in Examples 5.2.24, 6.2.5 (v), since we showed that both Za (x21 + x22 − 1) and Za (x31 − x22 ) \ {(0, 0)} are isomorphic to A1 \ {0}. More generally, recalling examples discussed in the previous chapters, we get e.g. that A1 , P1 , A1 \{0}, the circle, the hyperbola, the parabola, the semicubic parabola, the cubic Za (x22 −x21 (x1 −1)) (recall Example 5.2.25), the affine and the projective twisted cubic or more generally any affine and projective rational normal curve of degree d, etc. these are all models of [P1 ]bir : indeed,

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179

in all these cases, we proved that the field of rational function is isomorphic to K(x1 ) (these are all examples of rational curves, cf. § 8.2). (iii) On the contrary the affine curve Ya : Za (x22 − x1 (x1 − 1)(x1 − a), with 1, 0 6= a ∈ K, cannot be a model of [P1 ]bir : in example Example 5.2.25 we showed that K(Ya ) is an algebraic extension of degree 2 of the field K(x1 ) (Ya is a smooth plane cubic, as such it is called elliptic curve).

8.1.1 Some properties and examples of (bi)rational maps In this section, we shall discuss interesting examples of rational and birational maps. Let V be any algebraic variety. If one has a rational map Ψ : V 99K An , for some positive integer n, composition with the isomorphism φ0 : An → U0 and with the open inclusion ιU0 : U0 ֒→ Pn determines a rational map Ψe : V 99K Pn .

Conversely, take a rational map Ψe as above; let UΨe be its domain and let ψe e e ) ∩ H0 = ∅, then be its representative morphism over the open set UΨe . If ψ(U Ψ n Ψe corresponds to a rational map Ψ to A as above.

Corollary 8.1.9 Let V be any algebraic variety. Any non-constant morphism φ : V → P1 determines a rational map Φ : V 99K A1 , i.e. an element Φ ∈ K(V ). If moreover φ is not surjective, the map Φ is a morphism, i.e. Φ ∈ OV (V ). Proof. Only the last assertion needs some comment: if φ(V ) ( P1 , for any choice of P ∈ P1 \φ(V ) one has that P1 \{P } ∼ = A1 , from which one concludes. ⊓ ⊔

One basic question is: what about the converse of Corollary 8.1.9? In other words, given Φ ∈ K(V ) it induces a morphism φ : UΦ → A1 and so a rational map Φ : V 99K A1 ; is it true that it always extends to a morphism to P1 ? The answer in general is no, as the following example shows. X1 ∈ K(P2 ). This is regular on the principal Example 8.1.10 Let Φ := X 0 affine open set U02 ⊂ P2 , so it defines a rational map Φ : P2 99K A1 with a representative morphism

φ : A2 ∼ = U02 → A1 , which is the projection π1 : A2 → A1 , [1, x1 , x2 ] = (x1 , x2 ) → x1 = [1, x1 ]

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on the first coordinate (we identified A1 with U01 ⊂ P1 ). Notice that π1 is the restriction to the open set U02 of the projection from the linear subspace Λ := {[0, 0, 1]} ⊂ P2 πΛ : P2 99K P1 , [X0 , X1 , X2 ] 99K [X0 , X1 ]. Thus πΛ is a rational map whose domain is P2 \ Λ. In other words, Φ extends to a rational map to P1 but not to a morphism defined on the whole P2 . On the other hand, in some cases we have affirmative answer to the previous question, as the following examples show. Example 8.1.11 Any rational map Ψ : A1 99K An extends to a unique morphism ψe : P1 → Pn . It is enough to show it for n = 1, the general case being similar (using Proposition 6.2.2). The rational map Ψ : A1 99K A1 corresponds to a rational function Ψ ∈ K(A1 ) ∼ = Q(1) ∼ = K(t), where t an indeterminate over K. Thus, there exist (1) ∼ f, g ∈ A = K[t] s.t. Ψ = fg , where UΨ = Ua (g) = Za (g)c , i.e Ψ ∈ OA1 (Ua (g)). We can assume that g.c.d.(f, g) = 1. Ψ defines a morphism ψ : Ua (g) → A1 . We first show that the morphism ψ uniquely extends to a morphism A1 → P1 . To do this, identify the target A1 of ψ with U0 ⊂ P1 and consider the map ψ ′ : A1 → P1 , P → [g(P ), f (P )]. This map is well-defined as, for points P ∈ A1 \ Ua (g), one has [g(P ), f (P )] = [0, f (P )] = [0, 1] since f (P ) 6= 0 (otherwise (t − P ) would be a common divisor of f and g against the assumption on g.c.d.(f, g)). Observe moreover that ψ ′ is a morphism; indeed let f (t) = f0 + f1 t + . . . fn tn and g(t) = g0 + g1 t + . . . gm tm , with fi , gj ∈ K, fn , gm 6= 0, 0 6 i 6 n, 0 6 j 6 m, i.e. deg(f ) = n and deg(g) = m. Assume e.g. n > m (the other case being similar). Identify once 1 again A1 (the domain of ψ ′ ) with U0 ⊂ P1 ; thus t = X X0 and the homogeneous polynomials F (X0 , X1 ) = X0n f (

X1 X1 ) = h0 (f ) and G(X0 , X1 ) := X0n g( ) = X0n−m h0 (g) X0 X0

are such that [g(P ), f (P )] = [G(1, P ), F (1, P )]

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181

for any P ∈ A1 . This implies that ψ ′ is a morphism (cf. Prop. 6.3.1); moreover its uniqueness follows from Lemma 8.1.1. The next step is to show that ψ ′ uniquely extends to a morphism P1 → P1 . Since P1 = U0 ∪ {[0, 1]}, from the expression of F and G above, one has [G(0, 1), F (0, 1)] = [0, fn ] = [0, 1], since fn 6= 0. In particular, ψ ′ is the restriction to U0 of the map ψe : P1 → P1 , [X0 , X1 ] → [G(X0 , X1 ), F (X0 , X1 )],

which is a morphism from Proposition 6.3.1 (its uniqueness follows as above). ⊓ ⊔ Similarly, Example 8.1.12 Any rational map Φ : P1 99K Pn is always a morphism. We can assume Φ to be non-constant, otherwise there is nothing to prove. Let UΦ be the domain of Φ and let φ : UΦ → Pn be the representative morphism. Since being a morphism is a local property (cf. Prop. 6.4.1), from Proposition 6.3.1 we can directly assume that φ is globally defined on UΦ by a collection (1) of homogeneous polynomials F0 , . . . , Fn ∈ Sk , for some integer k > 1, with Zp (F0 , . . . , Fn ) = ∅. Then φ(P ) = [F0 (P ), . . . , Fn (P )], ∀ P ∈ UΦ , where with no loss of generality we can assume g.c.d(F0 , . . . , Fn ) = 1. Thus, for any point P = [p0 , p1 ] ∈ P1 \ UΦ , (F0 (P ), . . . , Fn (P )) 6= (0, . . . 0) otherwise the polynomials F0 , . . . , Fn would have the common divisor p1 X0 − p0 X1 , against the assumption. This means that Φ is wherever defined as a morphism. ⊓ ⊔ To have concrete applications of previous statements, consider: Example 8.1.13 Let K = C and let Ψ : A1 99K A2 be the rational map defined by   2 t − 1 −2t . , 2 t 99K 2 t +1 t +1 Its domain is UΨ = A1 \ {i, −i} and its representative morphism ψ over UΨ is such that ψ(UΨ ) ⊂ C = Za (x21 + x22 − 1) ⊂ A2 ,

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the first inclusion being strict since (1, 0) ∈ C is not contained in Im(ψ) as one can easily check. Ψ is more precisely birational; indeed, consider the pencil of lines through (1, 0) ∈ C, i.e. x2 = t(x1 − 1), where t ∈ K. This pencil defines a morphism C \ {(1, 0)} → A1 , (p1 , p2 ) →

p2 =t p1 − 1

which inverts the morphism ψ over the open set C \ {(1, 0)}, as for any t ∈ UΨ the intersection Za (x2 − t(x1 − 1), x21 + x22 − 1) (off (1, 0), the fixed point of the pencil of lines) determines the point  2  t − 1 −2t ∈ C \ {(1, 0)}. , t2 + 1 t 2 + 1 In particular Ψ is another rational parametrization of the unit circle, different from the parametrization considered in Examples 5.2.23 (iii), 6.2.5 (iv).

Fig. 8.1. Another birational parametrization of the circle (real part)

As in Example 8.1.11, ψ uniquely extends to a morphism ψ ′ : A1 → P1 , t → [t2 + 1, t2 − 1, −2t]. Notice indeed that ψ ′ is well-defined since ψ ′ (±i) = [0, 1, ±i] ∈ C ∩ H0 , where C the projective closure of C in P2 and where we identified A2 with the X1 affine chart U02 ⊂ P2 . Moreover, identifying A1 with U01 ⊂ P1 , one has t = X 0 ′ so ψ is [X0 , X1 ] → [X02 + X12 , X12 − X02 , −2X0 X1 ], which shows that the previous map is a morphism since (1)

Zp (X02 + X12 , X12 − X02 , −2X0 X1 ) = Zp (X02 , X12 , X0 X1 ) = Zp ((S+ )2 ) = ∅. Notice that Im(ψ ′ ) = C \ {[1, 1, 0]}. As in Example 8.1.11, finally ψ ′ extends to a morphism ψe : P1 → C as [0, 1] maps to [1, 1, 0] ∈ C (notice that this point coincides with the point where the inverse of ψ was not defined). ⊓ ⊔

8.2 Unirational and rational varieties

183

Consider now Λ ⊂ Pn a linear subspace, with Λ ∼ = Pk−1 , and let πΛ : Pn 99K Pk be the rational map given by the projection from Λ (recall Example 6.3.3). If V is any algebraic variety not contained in Λ, the restriction of πΛ to V defines a rational map πV : V 99K Pk (8.1) which is called projection of V on the subspace Pk ; πV is a morphism on V \ (V ∩ Λ); in particular, in V ∩ Λ = ∅, then πV ∈ Morph(V, Pk ). On the other hand, in some cases V \ (V ∩ Λ) does not fill up the whole domain of definition of πV , as the following example shows. Example 8.1.14 Let C ⊂ P2 be the conic as in Example 8.1.14. The pencil of lines therein defines a projection of A2 onto the x2 -axis and so a linear projection πP : P2 99K P1 , where P1 identified with the line H1 = Zp (X1 ) ⊂ P2 and where the center of projection is the base point of the pencil P = [1, 1, 0] (recall Figure 8.1). Since P is on C, πP defines a rational map πC which is a projection of C to the linear subspace P1 . This is a morphism certainly over C \ {P } and it is an example of stereographic projection of an irreducible conic onto P1 . For any point Q = (0, y0 ) on the x2 -axis in A2 = U0 , with y0 6= ±i, the unique line of the pencil P ∨ Q intersects C outside P at a unique further  2  y0 −1 2y0 point, say PQ ∈ C, which is PQ = y2 +1 , y2 +1 . From Example 8.1.13, this 0 0 is a morphism. Identifying the line H1 with P1 , πC is therefore a birational map C 99K P1 whose domain certainly contains C \ {P }. On the other hand, C ∼ = P1 via a Veronese morphism ν1,2 . Up to the composition with ν1,2 , from Example 8.1.12 πC extends to a unique morphism defined on the whole C. The geometric interpretation of such an extension is given by considering the tangent line X0 − X1 = 0 to C at P and then intersecting it with the line H1 , which gives the point [0, 0, 1].

8.2 Unirational and rational varieties We start with some important definitions. Definition 8.2.1 An algebraic variety V is said to be unirational if there exist a positive integer r, a dominant rational map Φ : Pn 99K V and a dense open subset U ⊆ Im(Φ) s.t., for any P ∈ U , Φ−1 (P ) is a finite set. Such a map Φ is said to be generically finite dominant rational map.

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From Theorem 8.1.6 the map Φ is associated with a field extension K(V ) ⊆ K(x1 , . . . , xr ) = Q(n) (after Chapter 9, one more precisely will realize that the previous extension has to be necessarily algebraic). Definition 8.2.2 An algebraic variety V is said to be rational if it is birational to Pr (or Ar ), for some non negative integer r. In particular, if V is rational then it is unirational. Moreover, from Corollary 8.1.7 (i), we get: Corollary 8.2.3 V is rational if and only if K(V ) ∼ = Q(r) , i.e. if and only if K(V ) is a purely trascendental extension of K of trascendental degree r, for some non negative integer r. All the curves listed in Remark 8.1.8 (ii) are rational curves, whereas the curve Ya in (ii) therein is not rational (recall indeed it has been called elliptic curve). The so called L¨ uroth problem is a fundamental problem in algebraic geometry: does unirationality imply rationality? In 1876, L¨ uroth proved that in the curve case the answer is yes, i.e. any unirational curve is also rational (cf. e.g. [7, p. 148]). In 1894 Castelnuovo showed that also unirational surfaces are rational (cf. e.g. [2] and [3]). Only after almost one century Clemens and Griffiths (1972) showed that a cubic threefold (i.e. an algebraic variety of dimension three) is unirational but in general not a rational variety, providing an example for three dimensions that unirationality does not imply rationality. In what follows we shall give several examples of rational varieties which are birational but not isomorphic to Pn , for n > 2. 8.2.1 Stereographic projection of a rank-four quadric surface A nice example of a surface which is birational (but not isomorphic) to P2 is given by an irreducible, doubly ruled quadric Q ⊂ P3 . This is a rank-four quadric in P3 , i.e. the homogeneous quadratic polynomial defining Q is a quadratic form in the indeterminates X0 , . . . , X3 whose associated symmetric matrix has non-zero determinant. Since K is algebraically closed, all these quadrics are projectively equivalent. Therefore with no loss of generality, we can assume that Q = Zp (X0 X3 − X1 X2 ), i.e. it is P1 × P1 embedded via the Segre morphism σ2,2 as an irreducible, doubly ruled quadric in P3 . Take P0 = [1, 0, 0, 0] ∈ Q. From (8.1), the rational map πP0 : P3 99K P2 , [X0 , X1 , X2 , X3 ] 99K [X1 , X2 , X3 ] induces the projection

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185

πQ : Q 99K P2 which is called stereographic projection of the rank-four quadric surface to a plane. This rational map is a morphism over the open subset Q \ {P0 }; differently from the stereographic projection in Example 8.1.14, by its expression πQ cannot be extended (as a morphism) also at the point P0 and its target is the plane H0 = Zp (X0 ) ⊂ P3 . Our aim is to show that πQ is a birational map to such a plane. . To do this consider H3 = Zp (X3 ) ⊂ P3 and the open set U := Q \ (Q ∩ H3 ) ⊂ Q. Notice that Q ∩ H3 = Zp (X1 , X3 ) ∪ Zp (X2 , X3 ) = ℓ13 ∪ ℓ23 , where ℓ13 and ℓ23 are the two lines passing through P0 of the two different rulings of Q. Thus, ϕ := πQ |U : U → U ′ ⊂ H0 ∼ = P2 is a morphism (since P0 ∈ ℓ13 ∪ ℓ23 ) whose image is the open set U ′ := H0 \ (H0 ∩ H3 ), which is isomorphic to P2 minus a line (such a line is given by ℓ03 = Zp (X0 , X3 ) ⊂ H0 ). Claim 8.2.4 ϕ : U → U ′ is an isomorphism. Proof. For any point Q = [0, q1 , q2 , q3 ] ∈ U ′ , consider the line P0 ∨ Q; this line has parametric equations in P3 given by X0 = s, X1 = q1 t, X2 = q2 t, X3 = q3 t, [s, t] ∈ P1 and intersects U at the point [ q1q3q2 t, q1 t, q2 t, q3 t], which is [ q1q3q2 , q1 , q2 , q3 ], as t 6= 0 since P0 ∈ / U . In other words ϕ−1 (Q) = [q1 q2 , q1 q3 , q2 q3 , q32 ] which is a morphism as it follows from its expression and the fact that q3 = 6 0 since Q ∈ U ′ . ⊓ ⊔ In particular, πQ is birational (but not a morphism) so Q is a rational surface. Identifying the plane H0 with P2 , with homogeneous coordinates [X1 , X2 , X3 ], from Claim 8.2.4, the inverse of ϕ is induced by the rational map ν : P2 99K P3 , [X1 , X2 , X3 ] 99K [X1 X2 , X1 X3 , X2 X3 , X32 ],

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where ν = νL as in (6.15) with (2)

L = Span{X1 X2 , X1 X3 , X2 X3 , X32 } ⊂ S2 . In particular νL is not defined at B := {P1 = [1, 0, 0], P2 = [0, 1, 0]} ⊂ P2 , which is the base locus of the linear system of plane conics P(L), and Im(νL ) ⊆ Q. To sum up, πQ and νL are two birational maps, one is the inverse of the other (as rational maps), none of them is a morphism. Notice moreover that ℓ03 ⊂ H0 is the line joining P1 , P2 , the two points of indeterminacy of νL , that νL contracts ℓ03 \ {P1 , P2 } to P0 ∈ Q and, conversely, that πQ contracts ℓ13 \ P0 = {[s, 0, t, 0], s, t ∈ K, t 6= 0} to P1 ∈ H0 and ℓ23 \ P0 = {[s, t, 0, 0], s, t ∈ K, t 6= 0} to P2 ∈ H0 ; this explains the choice of the open set U and U ′ to construct the isomorphism ϕ. 8.2.2 Monoids Other examples of rational varieties in any dimension n > 1 can be easily constructed as follows. Let Fd (x1 , . . . , xn ), Fd−1 (x1 , . . . , xn ) ∈ K[x1 , . . . , xn ] be reduced homogeneous polynomials of degree d and d − 1, respectively. Consider the hypersurface Z := Za (Fd + Fd−1 ) ⊂ An , (8.2) which is called (affine) monoid of degree d with vertex the origin O ∈ An . Notice that Fd (x1 , . . . , xn ) + Fd−1 (x1 , . . . , xn ) = 0 is the reduced equation of Z. The projective closure of Z is the hypersurface Z ⊂ Pn given by Fd (X1 , . . . , Xn ) + X0 Fd−1 (X1 , . . . , Xn ) = 0,

(8.3)

which is called (projective) monoid of degree d with vertex P0 = [1, 0, 0, . . . , 0] ∈ Pn . More generally, any hypersurface V ⊂ An which is the transform of Z via a linear transformation of An as well as its projective closure V will be called monoid of degree d. Example 8.2.5 (i) The parabola Z = Za (x2 − x21 ) ⊂ A2 is a (affine) monoid of degree two with vertex O = (0, 0).

8.2 Unirational and rational varieties

187

(ii) The semi-cubic parabola Z = Za (x22 − x31 ) ⊂ A2 is a (affine) monoid of degree three with vertex O. Identifying A2 with U0 , Z is given by Zp (X0 X22 − X13 ), which is a (projective) monoid with vertex the fundamental point P0 = [1, 0, 0] ∈ P2 . (iii) Similarly, the plane nodal cubic Z := Za (x31 + x21 − x22 ) ⊂ A2 is a (affine) monoidal curve, whose vertex is the origin and whose projective closure is the (projective) monoid Zp (X13 + X12 X0 − X22 X0 ) ⊂ P2 with vertex P0 . (iv) The rank-four quadric surface Q = Zp (X0 X3 − X1 X2 ) ⊂ P3 is a (projective monoid) of degree two with vertex P0 , being Q0 = Q∩U0 = Za (x3 −x1 x2 ) an affine monoid of vertex O. Similarly, the quadric cone Zp (X0 X3 −X12 ) ⊂ P3 (i.e. a rank-three quadric) is a monoidal surface too, always with vertex the point P0 . As in § 8.2.1, we can prove the following result. Proposition 8.2.6 For any n > 2, any monoid of degree d > 2 is rational. Proof. With no loss of generality, we can focus on the case of Z ⊂ Pn a monoid with vertex P0 , i.e. whose reduced equation is given by (8.3). In such a case, we will more precisely show that the projection πZ : Z 99K H0 = Zp (X0 ) ⊂ Pn from the point P0 ∈ Z is birational onto H0 ∼ = Pn−1 . The restriction of πZ at Z \ {P0 } is a surjective morphism ϕ : Z \ {P0 } → H0 ∼ = Pn−1 , [X0 , . . . , Xn ] → [X1 , . . . , Xn ]. Similarly to the quadric case, the dominant rational map πZ is called stereographic projection of the monoid from its vertex P0 . . It suffices to find a dominant rational map inverting πZ . To do this observe that for any [a1 , . . . , an ] ∈ H0 \ (Zp (Fd−1 ) ∩ H0 ), i.e. s.t. Fd−1 (a1 , . . . , an ) 6= 0, one has   Fd (a1 , . . . , an ) , a1 , . . . , an ; ϕ−1 ([a1 , . . . , an ]) = Fd−1 (a1 , . . . , an ) i.e. the rational map

Ψ : Pn−1 ∼ = H0 99K Pn ,

defined by [X1 , . . . , Xn ] 99K [Fd (X1 , . . . , Xn ), X1 Fd−1 (X1 , . . . , Xn ), . . . , Xn Fd−1 (X1 , . . . , Xn )]

is the desired rational inverse.

⊓ ⊔

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8 Rational maps

8.2.3 Blow-up of Pn at a point Here we use products of projective varieties to introduce an important birational transformation. Consider Pn with homogeneous coordinates [X] := [X0 , . . . , Xn ] and Pn−1 with homogeneous coordinates [Y ] := [Y1 , . . . , Yn ]. In Pn × Pn−1 consider the closed subset defined by (cf. Prop. 7.2.8 (i))

Let

fn := Zp (Xi Yj − Xj Yi ) , for 1 6 i, j 6 n. P fn → Pn σ:P

(8.4) (8.5)

fn of the first projection π1 : be the morphism defined by the restriction to P n n−1 n n P ×P → P ; σ is called the blow-up of P at the point P0 , where P0 = [1, 0, . . . , 0] the fundamental point of Pn . Similar approach holds for an arbitrary point P ∈ Pn (cf. e.g. [15]); for simplicity in what follows we will focus only on the case of P0 . fn \σ −1 (P0 ) is a quasi-projective variety isomorphic (via Proposition 8.2.7 P fn is rational, being birational to Pn . σ) to Pn \ {P0 }. In particular, P

Proof. For any Q = [q0 , . . . , qn ] ∈ Pn \ {P0 } there exists i ∈ {1, . . . , n} for q which qi 6= 0. Thus ([q0 , . . . , qn ], [y1 , . . . , yn ]) ∈ σ −1 (Q) if and only if yj = yi qji , for any 1 6 j 6 n; in other words fn ⊂ Pn × Pn−1 σ −1 (Q) = ([q0 , . . . , qn ], [q1 , . . . , qn ]) ∈ P

is a unique point. Consider the map τ

Pn \ {P0 } −→ Pn × Pn−1 −1 Q = [q0 , . . . , qn ] −→ σ (Q) = ([q0 , . . . , qn ], [q1 , . . . , qn ]).

(8.6)

Composing τ with the Segre isomorphism ∼ =

σn,n−1 : Pn × Pn−1 −→ Σn,n−1 ⊂ Pn(n+1)−1 gives a map γ = σn,n−1 ◦ τ defined by

γ([q0 , . . . , qn ]) = [q0 q1 , q0 q2 , . . . , qn2 ]

which is therefore a morphism by Prop. 6.3.1. Since σn,n−1 is an isomorphism, one deduces that τ is a morphism. fn \ σ −1 (P0 ) is a quasi-projective variety. In particular, one gets that P Indeed fn \ σ −1 (P0 ); τ (Pn \ {P0 }) = P fn is closed and σ−1(P0 ) is closed the latter is locally closed in Pn × Pn−1 (as P being σ a morphism) and it is irreducible (since Pn \ {P0 } is irreducible and τ is a morphism (cf. Cor. 4.1.6)). At last, by its expression, we have τ = (σ|Pn \{P0 } )−1 i.e. τ is an isomorphism. ⊓ ⊔

8.2 Unirational and rational varieties

189

On the contrary, by (8.4) one gets σ −1 (P0 ) = π1−1 (P0 ) ∼ = Pn−1 ;

(8.7)

fn . σ −1 (P0 ) is called the exceptional divisor of σ in P

Remark 8.2.8 A geometric interpretation of the isomorphism in (8.7) can be given as follows. Let H be any hyperplane in Pn not passing through P0 ; up to a projectivity of Pn fixing P0 , we can assume with no loss of generality that this hyperplane coincides with H0 = Zp (X0 ). Any point Q ∈ H uniquely determines the line rQ := P0 ∨ Q in Pn ; conversely, any line r ⊂ Pn passing through P0 intersects H at a unique point Qr = r ∩ H such that P0 ∨ Qr = r. In other words, the set LP0 := {lines in Pn through P0 } ∼ Pn−1 . bijectively corresponds to the hyperplane H = For any Q = [0, q1 , . . . , qn ] ∈ H, the line rQ = P0 ∨ Q has parametric equations X0 = λ, Xi = µ qi , 1 6 i 6 n, with [λ, µ] ∈ P1 . (8.8) By (8.6), at the points of rQ \ {P0 } the map τ restricts to the morphism: τ |rQ \{P0 } :

fn \ σ −1 (P0 ) ⊂ Pn × Pn−1 rQ \ {P0 } −→ P [λ, µq1 , . . . , µqn ] −→ ([λ, µq1 , . . . , µqn ], [q1 , . . . , qn ]),

(8.9)

where the previous expression follows from the fact that

([λ, µq1 , . . . , µqn ], [µq1 , . . . , µqn ]) = ([λ, µq1 , . . . , µqn ], [q1 , . . . , qn ]), as µ 6= 0 on rQ \ {P0 }. Composing τ |rQ \{P0 } with the Segre morphism σn,n−1 , one gets a rational map τ |rQ : rQ 99K Pn(n+1)−1 .

Since rQ ∼ = P1 , from Example 8.1.12, τ |rQ is a morphism extending τ |rQ \{P0 } at the point P0 ; by (8.8), this extension is given by τ |rQ (P0 ) := ([1, 0, . . . , 0], [q1 , . . . , qn ]).

(8.10)

fn rf Q := τ (rQ ) ⊂ P .

(8.11)

Let

1 From Proposition 8.2.7 and (8.10), rf Q is isomorphic to P in particular it is an irreducible rational curve. Moreover, by its definition −1 rf (P0 ) = τ |rQ (P0 ) ∩ σ −1 (P0 ) = [q1 , . . . , qn ] ∈ σ −1 (P0 ). Q∩σ

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In particular, the map ∼ H σ−1(P0 ) = LP0 −→ −1 ∼ Q (P0 ) = rQ −→ rf Q∩σ [0, q1 , . . . , qn ] ←→ [q1 , . . . , qn ]

can be interpreted as the isomorphism (8.7). fn is irreducible. Proposition 8.2.9 P Proof. Notice that

fn = (P fn \ σ −1 (P0 )) ∪ σ −1 (P0 ). P

fn \ σ −1 (P0 )) is irreducible and σ −1 (P0 ) is closed From Proposition 8.2.7, (P n f fn \ σ −1 (P0 ) is dense and irreducible in P ; it therefore suffices to show that P fn . in P To prove this, it is enough to show that any point of σ −1 (P0 ) is in the fn of some algebraic subset contained in P fn \ σ −1 (P0 ). closure in P −1 In Remark 8.2.8 we showed that any point of σ (P0 ) is given by −1 τ |rQ (P0 ) = rf (P0 ), Q∩σ

for some point Q ∈ H, and moreover for any Q ∈ H the map τ |rQ is an ∼ 1 fn fn −1 (P0 ), isomorphism. Thus, rf Q = P is the closure in P of τ (rQ \{P0 }) ⊂ P \σ 1 where τ (rQ \ {P0 }) ∼ A . ⊓ ⊔ =

f1 = ∼ P1 . Example 8.2.10 (i) By definition of blow-up, it is clear that P f2 is the rational surface in P2 × P1 given by the equation (ii) For n = 2, P Zp (X1 Y2 − X2 Y1 ). Repeating verbatim the proof of Proposition 8.2.7, one easf2 ∩ (U 2 × U 1 ) is isomorphic to the hyperbolic paraboloid ily realizes that e.g. P 0 0 Σ := Za (x2 − yx1 ) ⊂ A3 , where A3 is U02 × U01 ∼ = A2 × A1 and where affine X2 Y2 X1 , y=X . coordinates (x1 , x2 , y) are given by x1 = X0 , x2 = X 0 1 −1 2 1 The exceptional divisor σ (P0 ) ∩ (U0 × U0 ) is the line Za (x1 , x2 ) contained in Σ. Its points (0, 0, y), with y varying in A1 , bijectively correspond to direction coefficients y of lines through the origin of A2 (i.e. P0 in the chart U02 ) in the pencil {x2 − yx1 = 0} ⊂ A2 . 8.2.4 Blow-ups and resolution of singularities Here we discuss some examples which show how blow-ups can be used to resolve singularities of algebraic varieties. As in the previous section, things more generally hold for a blow-up at any arbitrary point P of Pn . On the other hand, up to projectivities, one can always reduce to the basic case of the blow-up at the fundamental point P0 .

8.2 Unirational and rational varieties

191

Let V ⊆ Pn be any algebraic variety passing through P0 . Consider the fn → Pn as above and let blow-up σ : P WP0 := σ −1 (V \ {P0 }),

(8.12)

fn . where the closure is taken in P

Claim 8.2.11 WP0 is a projective variety which is birational to V . Proof. V \ {P0 } is an algebraic variety, being an open dense subset of V ; furthermore, since σ|V \{P0 } is an isomorphism between V \{P0 } and its image, fn , i.e. it is a quasi-projective σ(V \ {P0 }) is irreducible and locally closed in P variety. This implies that WP0 is irreducible and birational to V . Moreover, fn is closed in Pn × Pn−1 , WP is also projective. since P ⊓ ⊔ 0 σ −1 (V ) is called the total transform of V via σ . We let −1 g (V ); V P0 := WP0 ∩ σ

(8.13)

g since V is quasi-projective and σ is a morphism, V P0 is locally closed in WP0 g (so irreducible) and birational to V . VP0 is called the proper transform of g V via σ. The restriction of σ at V P0 is a birational morphism which will be denoted by g σV : V P0 → V and called blow-up of V at P0 .

Example 8.2.12 (i) Let V := Zp (X0 X12 − X23 ) ⊂ P2 ; notice that V is the projective closure of the semi-cubic parabola V0 = V ∩ U0 = Za (x21 − x32 ) ∈ A2 ∼ = U0 so V is a monoid in P2 with vertex the fundamental point P0 = [1, 0, 0] (cf. Example 8.2.5 (ii)). Since V is projective, then T := V × P1 is a projective subvariety of P2 × P1 , whose defining equation is simply f2 → P2 be the blow-up of P2 at P0 . As in Zp (X0 X12 − X23 ). Let now σ : P f2 is defined by Zp (X1 Y2 − X2 Y1 ). Example 8.2.12 (ii), P The total transform of V is therefore: f2 = Zp (X0 X 2 − X 3 , X1 Y2 − X2 Y1 ). σ −1 (V ) = T ∩ P 1 2

Take affine charts U02 ⊂ P2 and U21 ⊂ P1 , with affine coordinates x1 =

X2 Y1 X1 , x2 = , t= . X0 X0 Y2

(8.14)

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From (8.14), the corresponding open subset of the total transform is given by: σ −1 (V ) ∩ (U02 × U21 ) = Za (x21 − x32 , x1 − x2 t) = Za (x22 (x2 − t2 ), x1 − x2 t). The latter is equivalent to Za (x1 − tx2 , x2 − t2 ) ∪ Za (x22 , x1 ) which reads also as Za (x1 − t3 , x2 − t2 ) ∪ Za (x2 , x1 ).

(8.15)

In U02 × U21 ∼ = A2 × A 1 ∼ = A3 , with affine coordinates (x1 , x2 , t), the first algebraic subset in (8.15) is an affine rational curve in A3 with polynomial parametrization, as in (3.31). More precisely Za (x1 − t3 , x2 − t2 ) is the image of the morphism A1 −→ A3 , t −→ (t3 , t2 , t).

The second algebraic subset in (8.15) is simply the t-axis in A3 ; this line coincides with the affine part of the exceptional divisor σ −1 (P0 ) ∩ (U02 × U21 ) (cf. Example 8.2.10 (ii)). g f2 ∩ (U 2 × U 1 ), the proper transform (V To sum up, in the affine chart P 0 )P0 0 2 2 of the semi-cubic parabola V0 ⊂ A is isomorphic to the affine twisted cubic C ⊂ A3 as in § 3.3.12. g f2 of (V g The projective closure in P 0 )P0 is the proper transform VP0 (which in this case coincides with WP0 since V is already projective). From Claim 3.3.16, VeP0 is therefore isomorphic to the projective twisted cubic given by the image of the Veronese morphism ν1,2 : P1 −→ P3 , [Y1 , Y2 ] −→ [Y13 , Y12 Y2 , Y1 Y22 , Y23 ].

−1 g The total transform σ −1 (V ) coincides with V (P0 ), where σ −1 (P0 ) ∼ = P0 ∪ σ 1 P . g As for the blow-up morphism σV : V P0 → V , previous computations show that σV |VeP ∩(U 2 ×U 1 ) is nothing but the restriction to C of the projection 0

0

2

πI : A3 → A2 onto the first two coordinates, i.e. with multi-index I = (1, 2). Geometrically speaking πI is the projection onto the plane Za (t) ⊂ A3 from the point at infinity of the t-axis; under this projection one therefore has πI |C : C −→ V0 , (t3 , t2 , t) −→ (t3 , t2 )

i.e. C ⊂ A3 maps onto the semi-cubic parabola V0 ⊂ A2 . We will see that V0 is singular at the origin (cf. Example 11.1.3 (ii)), whereas the affine twisted cubic C ⊂ A3 is a smooth curve, i.e. it is nonsingular at any of its points (cf. Example 11.1.3 (iv)). In other words, the semi-cubic parabola acquires its (cuspidal) singularity at O because of the projection πI |C ; indeed, the intersection between C ⊂ A3 (i.e. the proper

8.2 Unirational and rational varieties

193

transform of V0 ) and the t-axis (i.e. the affine part of the exceptional divisor) is given by Za (x1 − t3 , x2 − t2 , x1 , x2 ). (8.16) As a set, this intersection is supported at the unique point O = {(0, 0, 0)} ∈ C, which corresponds to the coefficient direction m = 0 of the line Za (x2 ) ⊂ A2 . On the other hand, the intersection multiplicity at O of the system of equations (8.16) is two (cf. Def. 11.1.4) as the t-axis is the tangent line to C ⊂ A3 at O; thus, the cuspidal singularity of V0 at O is due to the fact that πI |C is a tangential projection and the singular point is where the parametrization is not regular (i.e. where the Jacobian matrix vanishes). Conversely, the affine twisted cubic C ⊂ A3 can be viewed as a nonsingular birational model of V0 , since C is the proper transform of V0 under the blow-up πI |C and one says that the morphism πI |C resolves the cuspidal singularity of V0 at O. (ii) Let V := Zp (X13 + X0 (X12 − X22 )) ⊂ P2 ; this is a (projective) monoid of vertex P0 being the projective closure of the (affine) monoid V0 = V ∩ U02 = Za (x31 + x21 − x22 ) ⊂ A2 (cf. Example 8.2.5 (iii)). Computations as in (i) show that the total transform of V is given by σ −1 (V ) = Zp (X13 + X0 (X12 − X22 ), X1 Y2 − X2 Y1 ).

(8.17)

Take affine charts U02 ⊂ P2 and U11 ⊂ P1 , with affine coordinates x1 =

X2 Y2 X1 , x2 = , s= . X0 X0 Y1

From (8.17), we get that σ −1 (V ) ∩ (U02 × U11 ) is given by Za (x31 + x21 − x22 , x2 − x1 s) = Za (x21 (x1 + 1 − s2 ), x2 − x1 s), i.e. Za (x1 − s2 + 1, x2 − s3 + s) ∪ Za (x1 , x2 ).

(8.18)

The first algebraic subset in (8.18) is the image of the morphism A1 −→ A3 ,

s −→ (s2 − 1, s3 − s, s)

3 so the proper transform Vg 0P0 is an affine twisted cubic C ⊂ A (cf. Exercise 3.4.11); the second algebraic subset is the s-axis, i.e. as above the affine part of the exceptional divisor. g As in (i) above, the proper transform V P0 of the monoid V0 coincides with WP0 and it is isomorphic to the projective closure of C. g The blow-up morphism σV : V P0 → V in the above affine charts reads as the restriction to C of the projection onto the plane Za (s) ⊂ A3 from the

194

8 Rational maps

point at infinity of the s-axis. In this case, the intersection between C and the s-axis is given by Za (x1 − s2 + 1, x2 − s3 + s, x1 , x2 );

(8.19)

this system of equations gives {Q1 = (0, 0, 1), Q−1 = (0, 0, −1)} ∈ C, where the points of intersection correspond to the coefficient directions m = ±1 of the two lines Za (x2 − x1 ) and Za (x2 + x1 ) in the plane Za (s). In other words, the nodal singularity of V0 at O is due to the fact that σV is a secant projection of C to the s-plane. More precisely, the projection is given by C −→ V0 , (s2 − 1, s3 − s, s) −→ (s2 − 1, s3 − s); the s-axis is a secant line to C at the points Q1 and Q2 and these two points are identified under the projection, i.e. they both map to O ∈ V0 ; this creates the nodal singularity at O (i.e. a point where the parametrization is not injective). In the other direction, the affine twisted cubic C ⊂ A3 is a non-singular birational model of the plane nodal cubic V0 and σV resolves its (nodal) singularity at O.

8.3 Birational transformations of an algebraic variety Any rational map Φ : V 99K V is called a rational transformation of V . If Φ is birational, then it is called a birational transformation of V . The set of birational transformations of V is denoted by Bir(V ); from Lemma 8.1.4 this is a group with respect to the composition ◦ which contains Aut(V ) as a subgroup. For any integer n > 1, Bir(Pn ) is called the Cremona group of Pn . Here we consider a basic example of Cremona tranformation. Consider q : Pn 99K Pn , [X0 , . . . , Xn ] 99K [

1 1 ,..., ], X0 Xn

which reads also as q([X0 , . . . , Xn ]) = [X1 · · · Xn , . . . , X0 · · · Xn−1 ]. This map is an involution on the open set U := U0 ∩ U1 ∩ . . . ∩ Un ⊂ Pn , n i.e. q|U is an isomorphism with q|−1 U = q|U , in particular q ∈ Bir(P ).

8.3 Birational transformations of an algebraic variety

195

For n = 1, U = P1 \ {[0, 1], [1, 0]} and q|U is noting but the automorphism of U given by [X0 , X1 ] → [X1 , X0 ].

From Example 8.1.12, q is an automorphism of P1 , i.e. q ∈ Aut(P1 ) = PGL(2, K). If otherwise n = 2, the map q is given by q([X0 , X1 , X2 ]) = [X1 X2 , X0 X2 , X0 X1 ]

and, for this expression, it is called elementary quadratic transformation of P2 . The map q coincides with the rational map νL as in (6.15) given by the linear (2) system associated to L = Span{X1 X2 , X0 X2 , X0 X1 } ⊂ S2 . This means that the indeterminacy locus of q coincides with the base locus B of the linear system of conics P(L) which is B := {P0 = [1, 0, 0], P1 = [0, 1, 0], P2 = [0, 0, 1]}. These points are the fundamental points of P2 , which are also called fundamental points of q. From above, q is an involutory isomorphism over the open set U = P2 \ {H0 , H1 , H2 }, where Hi = Zp (Xi ) the coordinate line in P2 , 0 6 i 6 2. On the contrary, q contracts Hi \ (Hi ∩ B) to the fundamental point Pi , for any i ∈ {0, 1, 2}; for this reason, the lines Hi , 0 6 i 6 2, are called exceptional lines of q. Moreover, since q is an involution as a birational map, one states also that q dilate the fundamental points to P2the exceptional lines. Any line ℓ = Zp ( i=0 ai Xi ) not intersecting B maps to the conic Zp (a0 X1 X2 + a1 X0 X2 + a2 X0 X1 ).

Any line through one of the fundamental points is instead mapped to another line; more precisely, take e.g. ℓ = Zp (λX1 + µX2 ) a line through P0 , with [λ, µ] ∈ P1 . Consider ℓ \ {P0 }, whose points are given by [µ, µ, −λ] with µ 6= 0; then q([µ, µ, −λ]) = [−µλ, −µλ, µ2 ] = [λ, λ, −µ], i.e. the projective closure of q(ℓ \ {P0 }) is the line rℓ = Zp (µX1 + λX2 ), which cuts out on H0 the point [0, λ, µ]. In other words, the points on the exceptiona line H0 are in one-to-one correspondence with the directions of lines through the fundamental point P0 ; the same occurs for the other fundamental points (exceptional lines, respectively).

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8 Rational maps

8.4 Exercises

9 Dimension

Here we define the dimension of an algebraic variety V with the use of its field of rational functions K(V ). For other equivalent definitions see e.g. [15, page 5, Prop. I.1.7, Thm.I.1.8.A, Ex. I.2.6].

9.1 Dimension of an algebraic variety Let V be any algebraic variety. In Corollary 5.2.20 we proved that K(V ) is a finitely generated field extension of the base field K. In particular trdegK (K(V )) < +∞, as this number equals the maximal number of algebraically independent elements among the generators of K(V ) over K (recall the proof of Corollary 5.2.20). One poses dim(V ) := trdegK (K(V )) (9.1) Immediate consequences of the previous definition are the following. Proposition 9.1.1 (i) dim(An ) = dim(Pn ) = n. (ii) If U ⊆ V is a non empty open subset, then dim(U ) = dim(V ). (iii) Let n > 1 and let Λ be a linear (resp., affine) subspace of Pn (resp., of An ) s.t. Λ ∼ = Ph (resp., Λ ∼ = Ah ), with h > 0. Then dim(Λ) = h. (iv) dim(V ) = 0 if and only if V is a point. (v) If Φ : V 99K W is a dominant rational map, then dim(W ) 6 dim(V ). Proof. (i) It follows from Corollary 5.2.17 and from Remark 1.8.4 (iii). (ii) It follows from Lemma 5.2.13 (ii). (iii) If Λ ⊂ An is an affine subspace s.t. Λ ∼ = Ah , identifying An with the affine chart U0 , the projective closure Λ of Λ is Pn is a linear subspace isomorphic

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9 Dimension

to Ph and Λ is a dense open subset of Λ (cf. § 3.3.5). From (ii) above, we can therefore focus on the case Λ ⊂ An an affine subspace. Since Λ ∼ = Ah , then A(Λ) ∼ = A(h) = K[x1 , . . . , xh ], where the xi ’s are indeterminates over K. Since Λ is affine, from Theorem 5.2.14 (d), K(Λ) = Q(A(Λ)) ∼ = Q(A(h) ) = Q(h) and we are done. (iv) If V is a point, then V ∼ = A0 and we conclude by (iii). Conversely, assume dim(V ) = 0. By (ii) above, we can assume V to be affine. In particular, we can reduce to the case that V is an irreducible, closed subset of An , for some positive integer n. From Theorem 5.2.14, we have K ⊆ A(V ) ⊆ K(V ) = Q(A(V )). From the assumption dim(V ) = 0, it follows that K ⊆ K(V ) is an algebraic extension. On the other hand, since K is algebraically closed, we must have K = K(V ). This forces A(V ) = K and so Ia (V ) ⊂ A(n) to be a maximal ideal. From the Hilbert ”Nullstellensatz”-weak form (cf. Theorem 2.2.1), V is a point. (v) If Φ : V 99K W is dominant, from Theorem 8.1.6, we have a field extension K(W ) ⊂ K(V ). Since trdegK (K(V )) = dim(V ) < +∞, from the field extensions K ⊆ K(W ) ⊆ K(V ) we get also trdegK(W ) (K(V )) < +∞. We conclude by Proposition 1.8.5. ⊓ ⊔ As for terminology, when dim(V ) = 1, V is called (irreducible) curve; if otherwise dim(V ) = 2 then V is called (irreducible) surface, if dim(V ) = 3, V is an (irreducible) threefold and so on. More generally, any algebraic set Y is said to be of of pure dimension if all its irreducible components have the same dimension. In particular, a curve is an algebraic set of pure dimension 1, a surface is an algebraic set of pure dimension 2 and so on. When otherwise Y is not pure, since Y is an algebraic set, then it has finitely many irreducible components; therefore dim(Y ) is defined to be the maximum among all the dimensions of its irreducible components. Some important comments to Proposition 9.1.1 are the following. Remark 9.1.2 (i) From Proposition 9.1.1 (v), the dimension is a birational invariant of algebraic varieties. (ii) In particular if V is a rational variety of dim(V ) = n, then it is birational to Pn (and even to An ). Moreover K(V ) is a purely trascendental extension of K. Indeed, in such a case, K(V ) is isomorphic to Qn so trdegK (K(V )) = n. (iii) Revisiting Examples 5.2.23, 5.2.24, 6.2.5 (ii)-(v), 8.1.13 and 8.2.12 (i)-(ii), these are all irreducible rational curves (in the sense above) since they all are indeed birational to P1 .

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199

The same conclusion holds for affine (resp., projective) rational normal curves of degree d > 1, since they all are isomorphic to A1 (resp., to P1 ). (iv) The elliptic curve Ya in Example 5.2.25, with a 6= 0, 1, is an example of an irreducible curve which is not rational. Indeed, therein we proved that K(Ya ) is an algebraic extension of degree 2 of K(x1 ) ∼ = Q(1) , as x2 ∈ K(Ya ) has minimal polynomial Px1 (t) = t2 − x1 (x1 − 1)(x1 − a) ∈ Q(1) [t], where t an indeterminate over K. In other words, the field extensions K ⊂ K(x1 ) ⊂ K(Ya ) are such that the first one is purely trascendental (of trascendence degree 1) whereas the second extension is algebraic of degree two, so trdegK (K(Ya )) = trdegK (Q(1) ) = 1 = dim(Ya ). For a = 0, 1 we instead obtain plane (nodal) cubic curves which are monoids, so they are irreducible rational curves. Proposition 9.1.3 For any algebraic varieties V and W , dim(V × W ) = dim(V ) + dim(W ). Proof. From Propositions 6.4.2 and 9.1.1 (ii), we can reduce to the case that V and W are both affine. In particular, we can consider them as irreducible, closed subsets V ⊂ An and W ⊂ Am , for some integers n and m. By Proposition 7.1.1, V × W ⊂ An × Am ∼ = An+m is an affine variety. Let v := dim(V ) and w := dim(W ). By assumptions on V and W , we have A(V ) =

K[y1 , . . . , ym ] K[x1 , . . . , xn ] and A(W ) = . Ia (V ) Ia (W )

From (1.18), we have trdegK (A(V )) = v and trdegK (A(W )) = w. Since A(V ) is generated by (the images of) x1 , . . . xn as an integral K-algebra of finite type, the previous equality implies that {x1 , . . . , xn } contains a trascendence basis over K determined by v 6 n elements. For simplicity, assume this basis to be {x1 , . . . , xv }, i.e. Q(v) ∼ = K(x1 , . . . , xv ) ⊂ Q(A(V )), where the field extension is algebraic. As for A(W ), assume that {y1 , . . . , yw } is a trascendence basis over K. From Corollary 6.2.8, the two projection morphisms V ևV ×W ։W give rise to injective K-algebra homomorphisms

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9 Dimension

A(V ) ֒→ A(V × W ) ←֓ A(W ). Since {x1 , . . . , xv } is a set of elements of A(V ) which are algebraically independent over K, these elements remain algebraically independent over K when viewed as elements of A(V × W ). Same occurs for the set {y1 , . . . , yw }. We want to show that {x1 , . . . , xv , y1 , . . . , yw } is a set of algebraically independent elements over K in A(V × W ). Reasoning recursively, it suffices to show that {x1 , . . . , xv , y1 } is formed by algebraically independent elements over K. Assume there exists a non-zero polynomial f (t1 , . . . , tv+1 ) ∈ K[t1 , . . . , tv+1 ] such that f (x1 , . . . , xv , y1 ) = 0. As K[t1 , . . . , tv+1 ] = (K[t1 , . . . , tv ]) [tv+1 ], the polynomial f (x1 , . . . , xv , tv+1 ) has to be the zero polynomial, since all its coefficients are polynomial expressions with coefficients in K in the x1 , . . . , xv , which are by assumption algebraically independent. Therefore f (x1 , . . . , xv , y1 ) is identically zero. From (2.8), K[x1 , . . . , xn , y1 , . . . , ym ] A(V × W ) ∼ = Ia (V ) + Ia (W ) so A(V ×W ) is generated as an integral K-algebra of finite type by (the images of) x1 , . . . , xn , y1 , . . . , ym . Thus, K(x1 , . . . , xv , y1 , . . . , yw ) ⊂ Q(A(V ×W ) is an algebraic extension and so one concludes. ⊓ ⊔ Proposition 9.1.4 Let V be any algebraic variety. If W ⊂ V is a closed subvariety strictly contained in V then dim(W ) < dim(V ). Proof. Up to replacing V with a non empty, affine open subset intersecting W , we can assume that V is affine. Then W is also affine, being closed in V . Since W is properly contained in V , one has that Ia,V (W ) 6= (0) is a prime ideal and A(V ) A(W ) ∼ = Ia,V (W ) (recall notation as in (5.7)). In particular, one has a surjective K-algebra homomorphism Φ

A(V ) ։ A(W ). From (1.19), we get dim(V ) > dim(W ). Assume by contradiction that equality holds. If v := dim(V ), this means there exist x1 , . . . , xv ∈ A(V ) algebraically independent over K such that Φ(x1 ), . . . , Φ(xv ) ∈ A(W ) are algebraically independent over K too. Take any non-zero z ∈ Ia,V (W ) ⊂ A(V ). By maximality condition, x1 , . . . , xv , z ∈ A(V ) are algebraically dependent over K, i.e. there exists a non zero polynomial f (t1 , . . . , tv+1 ) ∈ K[t1 , . . . , tv+1 ] s.t.

9.1 Dimension of an algebraic variety

201

f (x1 , . . . , xv , z) = 0. Since A(V ) is an integral domain, we can assume f to be irreducible; for this reason the indeterminate tv+1 does not divide f (t1 , . . . , tv+1 ) (otherwise, by irreducibility, f would be f = αtv+1 for some α ∈ K∗ which implies z = 0 against the assumption). In this case, using the fact that Φ is a K-algebra homomoprhism and that z ∈ Ia,V (W ), we would get that f (t1 , . . . , tv , 0) ∈ K[t1 , . . . , tv ] is a non zero polynomial for which 0 = Φ(f (x1 , . . . , xv , z)) = f (Φ(x1 ), . . . , Φ(xv ), Φ(z)) = f (Φ(x1 ), . . . , Φ(xv ), 0) contradicting that Φ(x1 ), . . . , Φ(xv ) ∈ A(W ) are algebraically independent over K. ⊓ ⊔ For any subvariety V of an algebraic variety V , one can define its codimension in V , which is denoted by codimV (W ) := dim(V ) − dim(W ).

(9.2)

From Propositions 9.1.1 (ii) and 9.1.4, this integer is always non-negative; it is zero when W is a non empty open subset of V whereas it is strictly positive when W is a closed and strictly contained in V (or more generally a locally closed subset of a proper, closed subvariety of V ). Proposition 9.1.5 Let V be any closed subvariety of An (equivalently, Pn ). Then dim(V ) = n − 1 if and only if V is an irreducible hypersurface. In particular any hypersurface of An (equivalently, Pn ) is of pure dimension n − 1. Proof. As usual, it suffices to consider V to be affine. (⇒) If dim(V ) = n − 1 < n = dim(An ), V is a proper, closed subvariety of An so Ia (V ) 6= (0). Take any non zero f ∈ Ia (V ); since Ia (V ) is prime, we can moreover assume f to be irreducible. Then V ⊆ Za (f ) ⊂ An , where the second inclusion is strict. Therefore dim(Za (f )) < n so necessarily n − 1 = dim(V ) = dim(Za (f )). Since V is closed in An , then V is also closed in Za (f ). By Proposition 9.1.4, we must have V = Za (f ). (⇐) Let V = Za (f ), for some non zero irreducible polynomial f ∈ K[x1 , . . . , xn ]. Up to re-labeling the indeterminates, we can assume f to be non constant with respect to the indeterminate x1 ; then any g ∈ (f ) is non constant with respect to x1 , i.e. (f ) does not contain any non zero polynomial h(x2 , . . . , xn ). Since A(V ) = K[x1 , . . . , xn ]/(f ) is generated by (the images of) x1 , . . . , xn , in particular (the images of) x2 , . . . , xn ∈ A(V ) are algebraically independent over K; thus dim(V ) > n − 1. By Proposition 9.1.4 equality must hold. For the last part of the statement, any (possibly reducible) affine hyperSℓ surface is given by Za (f ) = i=1 Za (fi ), where f = f1 · · · fℓ is its reduced equation (recall (2.6) and (2.7)). Any Za (fi ) is an irreducible component of Za (f ) to which one applies the first part of the proof. ⊓ ⊔

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9.2 Exercises

10 Completeness of projective varieties

Here we discuss an important topological property of projective varieties.

10.1 Complete algebraic varieties Complete varieties are the analogues in the category of algebraic varieties of compact topological spaces in the category of Hausdorff topological spaces. Recall that the image of a compact topological space under a continuous map is compact and hence is closed if the image space is Hausdorff. Moreover, a Hausdorff topological space X is compact if and only if, for all topological space Y , the projection πY : X × Y → Y (with X × Y endowed with the product topology) is a closed map, i.e. it maps closed subsets of X × Y to closed subsets of Y (cf. [4, 10.2]). For algebraic varieties, some of the previous requests are empty some others make no sense. Indeed: • compactness (in the Zariski topology) is a property which is satisfied by all algebraic varieties, since they all are noetherian topological spaces (cf. § 4.2); • any algebraic variety V , which is not reduced to a single point, is never a Hausdorff space; • the Zariski topology ZV ×W of the product V × W of two algebraic varieties V and W never coincides with the product topology ZarV × ZarW , unless at least one of them is reduced to a single point. Thus, in the category of algebraic varieties we will consider the following: Definition 10.1.1 An algebraic variety V is said to be complete if, for all algebrac varieties W , the projection morphism πW : V × W → W of algebraic varieties is a closed map. Notice that, in the previous definition, V × W is an algebraic variety as in Chapter 7, i.e. endowed with ZarV ×W .

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Corollary 10.1.2 Let V be a complete algebraic variety. Let W be an algebraic variety and let ϕ : V → W be a morphism. Then ϕ(V ) ⊆ W is a closed subvariety. In particular, if ϕ is dominant then it must be surjective, i.e. ϕ(V ) = W . Proof. For the first part of the statement, notice first that ϕ(V ) is irreducible since V is irreducible and ϕ is a morphism (cf. Cor.4.1.6); thus, we are left to prove that ϕ(V ) is closed in W . From Proposition 7.3.3, the graph Γϕ is closed in V × W ; then one concludes by using that V is complete and that πW (Γϕ ) = ϕ(V ). The second assertion follows from the fact that ϕ(V ) is a closed subset of W containing an open (so dense) subset of W . ⊓ ⊔ The previous corollary gives motivation for the terminology complete: if V and W are algebraic varieties, with V a subvariety of W , and V is complete, then if dim(V ) = dim(W ) necessarily V = W . This behavior is in contrast with e.g. the open inclusion ιU0 : An ∼ = U0 ֒→ Pn , for n > 1; in other words, for any n > 1, An cannot be a complete variety. Next examples more generally show affine varieties which are not complete. Example 10.1.3 (i) Take V = W = A1 and let V × W ∼ = A2 with affine 2 coordinates (x1 , x2 ) and Zariski topology Zara . Consider the closed subset Z = Za (x1 x2 − 1) ⊂ A2 and the projection onto the second coordinate π2 ((x1 , x2 )) = x2 . Then π2 (Z) = A1 \ {0} is not closed in W = A1 . X1 b of Z in , the closure Z On the other hand, if we take V = P1 with x = X 0 1 1 P × A is given by X1 x2 − X0 = 0 and the projection π2 extends to πA1 : P1 × A1 → A1 , ([X0 , X1 ], x2 ) → x2 .

Notice that b πA−1 1 (0) ∩ Z = {([X0 , X1 ], 0) | 0X1 − X0 = 0} = {([0, 1], 0)}

b = A1 is closed. b surjectively dominates A1 , i.e. πA1 (Z) so Z (ii) Let V := Za (x1 x2 − x3 ) be the hyperbolic paraboloid in A3 . From Remark 7.2.6, V is the image of A1 ×A1 ∼ = U01 ×U01 in A3 via the restriction to U01 ×U01 of the Segre morphism σ1,1 : P1 × P1 → P3 . Take the multi-index I = (1, 2) and consider the projection morphism πI : A3 → A2 . The map φ := (πI )|V : V → A2 is the morphism φ in Example 6.2.5 (vi), where we showed that φ(V ) is neither closed nor open in A2 (it is a constructible set). In particular, Corollary 10.1.2 does not hold so V cannot be complete.

10.2 The main theorem of elimination theory

205

10.2 The main theorem of elimination theory The core of this section is to show that projective varieties are complete. Theorem 10.2.1 If V is a projective variety, then it is complete. Proof. By the definitions we have to show that, for any algebraic variety W , the projection πW : V × W → W is a closed morphism. Since V is projective, we can assume that V is a closed, irreducible subset of Pr , for some integer r. From the fact that V × W is therefore closed in Pr × W , it suffices to consider V = Pr . From Proposition 6.4.2, ZarW has a basis consisting of affine open sets. Since the property of being closed is local (recall e.g. the proof of Proposition 5.1.4 (i)), we can verify the property of being closed in any open set of an affine open covering of W ; with no loss of generality, we can therefore reduce to the case W to be affine. In such a case, W can be considered as an irreducible, closed subset of An , for some integer n. By the induced topology ZarW , we can reduce to the case W = An . To sum up, we are reduced to show that the second projection π2 : Pr × An → An is closed. To prove this, let Z ⊆ Pr ×An be any closed subset. From Proposition 7.2.8, Z is defined by polynomial equations gj (X, y) = 0, 1 6 j 6 s, for some integer s, where X = (X0 , . . . , Xr ), y = (y1 , . . . , yn ) and where the polynomials gj are homogeneous of degree dj with respect to the set of indeterminates X, 1 6 j 6 s. A point P = (p1 , . . . pn ) ∈ An is such that P ∈ π2 (Z) if and only if −1 π2 (P ) ∩ Z 6= ∅, i.e. if and only if Zp (g1 (X, P ), . . . , gs (X, P )) 6= ∅,

(10.1)

where (r)

gj (X, P ) := gj (X0 , . . . , Xr , p1 , . . . , pn ) ∈ Sdj , 1 6 j 6 s. Since K is algebraically closed, by the Homogeneous Hilbert ”Nullstellensatz”weak form (cf. Theorem 3.2.4), (10.1) is equivalent to (r)

(S+ )d * (g1 (X, P ), . . . , gs (X, P )), ∀ d > 1. Let

o n (r) Zd := P ∈ An | (g1 (X, P ), . . . , gs (X, P )) + (S+ )d ;

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T since π2 (Z) = d>1 Zd , it suffices to show that Zd is closed for any d > 1. For any P ∈ An and for any d > 1 consider the vector space homomorphism (r) (r) (r) ρd (P ) : Sd−d1 ⊕ · · · Sd−ds −→ Sd defined by ρd (P ) (F1 (X), . . . , Fs (X)) =

s X

Fj (X) gj (X, P ) ,

j=1

(r)

where Sd−dt = (0) for those dt > d. With this set-up, P ∈ Zd if and only if   r+d (r) = dim(Sd ). (10.2) rk(ρd (P )) < d
Condition (10.2) is equivalent to the vanishing of all the minors of order r+d d of the matrix associated to the linear map ρd (P ) in the canonical bases of these vector spaces. These minors are polynomial expressions in the pi ’s, for 1 6 i 6 n. Replacing the coordinate pi with the indeterminate yi , 1 6 i 6 n, one gets that Zd is the closed subset of An defined as the vanishing locus of such polynomial minors, as desired. ⊓ ⊔ The previous result is called the Main Theorem of Elimination Theory. Motivation for the terminology is clearly described by the strategy of the proof; we started with some equations gj (X, y) = 0, 1 6 j 6 s, and we asked for the image of the projection map (X, y) → y, which can be written as  y ∈ An | ∃ X ∈ Pr s.t. gj (X, y) = 0, ∀ 1 6 j 6 s ;

in other words, we eliminated the indeterminates X from the problem. The statement of the theorem is that the set of all such y ∈ An can itself be written as the solution set of some polynomial equations. Elimination theory is concerned with providing algorithms for passing from the equations defining Z ⊆ Pr × An to equations defining π2 (Z) ⊂ An . For example, let Z ⊆ P1 × An be the closed subset defined by two polynomials g1 (X0 , X1 , y) := s0 (y)X0m + s1 (y)X0m−1 X1 + · · · + sm X1m and

g2 (X0 , X1 , y) := t0 (y)X0n + t1 (y)X0n−1 X1 + · · · + tn X1n ,

with si (y), tk (y) ∈ K[y] = A(n) , 0 6 i 6 m, 0 6 k 6 n. For any P ∈ An , the two polynomials P (1) g1P (X0 , X1 ) := g1 (X0 , X1 , P ) ∈ S(1) m , g2 (X0 , X1 ) := g1 (X0 , X1 , P ) ∈ Sn

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207

have a common zero in P1 if and only if the resultant is s.t. R(δ0 (g1P ), δ0 (g2P )) = 0. Indeed, if this common zero is [0, 1] ∈ P1 then g1P (0, 1) = g2P (0, 1) = 0 gives sm (P ) = tn (P ) = 0, i.e. the last column of the Sylvester matrix (1.8) of δ0 (g1P ), δ0 (g2P ) is zero; if otherwise the common zero is [1, a] ∈ P1 , for some a ∈ K, the assertion follows from Theorem 1.3.17 (cf. also the proof of Proposition 1.10.17)). From this observation, since
δ0 (g1 (X0 , X1 , y)), δ0 (g2 (X0 , X1 , y)) ∈ K[y] [X1 ],

π2 (Z) ⊂ An is given by

Za (RX1 (δ0 (g2 (X0 , X1 , y)), δ0 (g2 (X0 , X1 , y))) ⊆ An , as RX1 (δ0 (g2 (X0 , X1 , y)), δ0 (g2 (X0 , X1 , y))) ∈ K[y]. Elimination theory does this in general, with the use of elimination ideals (for details, see e.g. [8, Ch. 8, § 5]). 10.2.1 Some consequences In the previous section, we have just seen that every projective variety is complete; the converse is not true. On the other hand, it is quite hard to exhibit explicit examples of complete varieties which are not projective; there are examples in dimension 2 due to Nagata and in dimension 3 to Hironaka (cf. e.g. [15], Exercise 7.13, p. 171, and Example 3.4.1, p. 443). We will certainly do not treat such examples, so for practical purposes the terms ”projective variety” and ”complete variety” can be considered almost synonymous. Remark 10.2.2 Theorem 10.2.1 gives different (and shorter) proofs of some results already encountered in the previous chapters, as well as some new consequences. For example: (i) If V is a projective variety and W is an affine variety, any morphism ϕ : V → W is constant; in particular OV (V ) = K. This has been already proved in Corollary 6.2.7 and in Theorem 5.2.14 (e). We can give an alternative proof, using the completeness of V . Indeed, since W is affine, we can assume that W is a closed subvariety of An , for some n > 0 . If we identify An with the affine chart U0 ⊂ Pn and if we consider the morphism ϕ′ := ιU0 ◦ ιW ◦ ϕ : V → Pn ,

by the completeness of V we get that ϕ′ (V ) is closed in Pn . On the other hand, ϕ′ (V ) = ϕ(V ) ⊆ W ⊆ An is closed in W which is closed in An ; then one concludes by Corollary 6.2.6. ⊓ ⊔

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(ii) If V is a projective variety and ϕ : V → Pn is a morphism, then ϕ(V ) ⊂ Pn is closed in particular it is a projective variety. Recall that this does not occur for arbitrary algebraic varieties (cf. Example 10.1.3). (iii) From (ii), e.g. any Veronese variety Vn,d , any Segre variety Σn,m , etcetera, are always projective varieties. (iv) Let Λ ⊂ Pn be a linear space inducing a projection πΛ : Pn 99K Pk , for suitable k < n. If V ⊂ Pn is a projective variety s.t. V ∩ Λ = ∅, the projection πV : V → Pk is a morphism. Thus, πV (V ) ⊂ Pk is again a projective variety which is called projection of V on the given Pk . (v) Let V ⊂ Pn be a projective variety which contains more than one point and let F ∈ H(S(n) ) be a non-constant homogeneous polynomial. Then V ∩ Zp (F ) 6= ∅ This has already been proved in Corollary 6.3.6. We can give here an alternative proof which uses completeness of V . Assume indeed by contradiction (n) there exists a homogeneous polynomial F ∈ Sd , for some integer d > 1, such that F (P ) 6= 0 for any P ∈ V . Let P, Q ∈ V be two distinct points and let G ∈ Ip (P ) be homogeneous, of degree d, such that G(Q) 6= 0. Consider the morphism ϕ : V → P1 , R → [F (R), G(R)], ∀ R ∈ V (this is well defined by the assumption on F ). Then, ϕ(V ) is closed and irreducible in P1 ; therefore ϕ is either constant or it is surjective. Since [0, 1] ∈ / ϕ(V ), this would imply ϕ is constant. This is a contradiction as ϕ(P ) = [F (P ), G(P )] = [1, 0] and ϕ(Q) = [F (Q), G(Q)] 6= [1, 0] by the choice of G. ⊓ ⊔

10.3 Exercises

11 Tangent spaces and smoothness

In this chapter we consider the local study of an algebraic variety V , i.e. we examine its structure near a point P ∈ V . Throughout this chapter the field K is taken to be algebraically closed and of characteristic 0.

11.1 Tangent space at a point of an affine variety. Smoothness We start as usual with the affine case. Let V ⊂ Ar be an affine variety. Let Ia (V ) := (f1 , . . . , fs ) ⊆ A(r) be its radical ideal and let P = (p1 , . . . , pr ) ∈ V be a point. Definition 11.1.1 The affine tangent space to V at the point P is the affine subspace of Ar defined by the system of linear equations r X ∂fj i=1

∂xi

(P )(xi − pi ) = 0, 1 6 j 6 s.

It will be denoted by TV /Ar ,P . It is easy to observe that the previous definition depends only on the point P ∈ V and on the ideal Ia (V ), i.e. it does not depend on the choice of generators of Ia (V ). When in particular V is an affine subspace of Ar , at any P ∈ V we have TV /Ar ,P = V . Let   ∂fj , 1 6 i 6 r, 1 6 j 6 s J= ∂xi be the Jacobian matrix, which is a s × r matrix with polynomial entries. For any integer h ∈ {0, . . . , r}, let

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Jh ⊂ A(r) be the ideal generated by the minors of order r − h + 1 of J. One can consider the set V ∩ Za (Jh ) := {P ∈ V | rk(J(P )) 6 r − h} i.e.  V ∩ Za (Jh ) = P ∈ V | dim(TV /Ar ,P ) > h , (11.1)

where dim(TV /Ar ,P ) denotes the dimension as a (classical) affine subspace, i.e. the dimension of the K-vector space determined by the direction of TV /Ar ,P . From above, the right-side-member of (11.1) is in particular a closed subset of V . Thus dim(TV /Ar ,P ) reaches its minimal value on a non empty open subset U of V (possibly U = V , as we will see below). U is formed by points P in V where the Jacobian matrix J(P ) has maximal rank.

Definition 11.1.2 A point P ∈ U is said to be a smooth (or also simple or non-singular) point for V . Otherwise, P is said to be a singular point for V . The (possibily empty) proper, closed subset V \ U is denoted by Sing(V ). If Sing(V ) = ∅, V is said to be a smooth variety (or also a non-singular variety), otherwise V is said to be singular. Example 11.1.3 (i) Let V = Za (x2 −x21 ) ⊂ A2 be the parabola. Its Jacobian matrix is J = (−2x1 1); notice that Za (J0 ) = Za (J1 ) = A2 . This implies that dim(TV /A2 ,P ) > 1 at any P ∈ V ; since moreover J2 = (2x1 , 1), then Za (J2 ) = ∅ so the previous inequality is an equality, for any P ∈ V . From the previous definitions, V is a smooth irreducible curve. Notice that at any point P ∈ V one has dim(TV /A2 ,P ) = 1 = dim(V ). One easily checks that, similarly, the affine twisted (t, t2 , t3 ) ⊂ A3 (more generally, any affine rational normal curve of degree d in Ad , given by (t, t2 , t3 , . . . , td ), for t ∈ A1 ) is a smooth, irreducible curve. (ii) Consider now the semi-cubic parabola, i.e. the cuspidal plane cubic V = Za (x22 − x31 ) ⊂ A2 (cf. e.g. Example 8.2.12 (i)). We have J = (−3x21 2x2 )

11.1 Tangent space at a point of an affine variety. Smoothness

211

and, as above, Za (J0 ) = Za (J1 ) = A2 so any P ∈ V is s.t. dim(TV /A2 ,P ) > 1. For h = 2, J2 = (−3x21 , 2x2 ) = (x21 , x2 ) so V ∩ Za (J2 ) = {O := (0, 0)}. Thus, for any P ∈ V \ {O} one has dim(TV /A2 ,P ) = 1 = dim(V ). On the contrary, at the origin one has dim(TV /A2 ,O ) > 2; since TV /A2 ,O has to be an affine subspace of A2 , we must have TV /A2 ,O = A2 and the previous inequality is an equality. From above, V is singular; more precisely Sing(V ) = {O = (0, 0)} and the dense open subset of smooth points is U := V \ {0}. (iii) Identical behavior for the plane nodal cubic V = Za (x31 + x21 − x22 ) ⊂ A2 which as in (ii) is monoid of degree 3 in A2 (cf. Example 8.2.12 (ii)). In this case J = (3x21 + 2x1 − 2x2 ) and once again V ∩ Za (J2 ) = {O := (0, 0)} which is the singular locus of V . (iv) When more generally V is an (irreducible) hypersurface V = Za (f ) ⊂ Ar , the Jacobian matrix is simply   ∂f ∂f ∂f . ... J= ∂x1 ∂x2 ∂xr For any 0 6 h 6 r − 1, one has Za (Jh ) = Ar therefore any P ∈ V is such that dim(TV /Ar ,P ) > r − 1 = dim(V ). The only non-trivial information is contained in V ∩ Za (Jr ) = Sing(V ). If this locus is empty, then V is smooth; in particular V is smooth if and only if J(P ) 6= (0 0 . . . 0) for any P ∈ V . Otherwise any P ∈ V ∩ Za (Jr ) is a singular point. Notice that Sing(V) = V ∩ Za (Jr ) is defined by the ideal   ∂f ∂f ∂f f, ... ∂x1 ∂x2 ∂xr confirming that Sing(V ) is a proper, closed subset in V , since Ia (V ) = (f ) is principal. Thus, at any smooth point Q ∈ V one has dim(TV /Ar ,Q ) = r − 1 = dim(V ).

(11.2)

212

11 Tangent spaces and smoothness

If otherwise P ∈ V is singular, then dim(TV /Ar ,P ) = r = dim(Ar ). (v) One word of warning concerning the singular locus of an affine hypersurface. Recall that Sing(V ) = V ∩Za (Jr ). Indeed, it can happen that Za (Jr ) 6= ∅ even if V ∩ Za (Jr ) = ∅, i.e. V is a smooth variety. Consider e.g. V to be the hypersphere in Ar determined by the vanishing locus of the polynomial f = x21 + x22 + . . . + x2r − 1. In this case Za (Jr ) = {O = (0, 0, . . . , 0)} but Sing(V ) = ∅ as O ∈ / V. We can give a useful geometric interpretation of the affine tangent space to an affine variety. With notation as above, let P ∈ V and let b := (b1 . . . , br ) ∈ Kr be a non-zero vector. Consider parametric equations of the line ℓP,b passing through P and with direction b, i.e. ℓP,b : xi = bi t + pi , 1 6 i 6 r, t ∈ K. Points of V ∩ ℓP,b corresponds to the solutions of the system of equations φ1 (t) = · · · = φs (t) = 0 where φi (t) := fi (b1 t + p1 , . . . , br t + pr ), 1 6 i 6 s, and where Ia (V ) = (f1 , . . . fs ) as above. If one takes φ(t) = g.c.d.(φ1 (t), . . . , φs (t)),

(11.3)

the points of ℓP,b ∩ V bijectively corresponds to the roots of the polynomial φ(t) ∈ K[t]. The root t = 0 of φ(t), corresponding to P ∈ ℓP,b ∩ V , will occur with a certain multiplicity µ > 1. Definition 11.1.4 The intersection multiplicity between ℓ := ℓP,b and V at the point P is the multiplicity µ of the root t = 0 of the polynomial φ(t). It is denoted by µ(V, ℓ; P ). It is easy to check that the previous definition is independent from the chosen parametric equations of ℓP,b and from the chosen generators of Ia (V ). Remark 11.1.5 (i) φ(t) is identically zero if and only if ℓP,b is entirely contained in V ; in such a case we will set µ(V, ℓ; P ) = +∞. (ii) If a line ℓ does not pass through P , we will put µ(V, ℓ; P ) = 0. (iii) It t = 0 is a simple root of φ(t), i.e. it has algebraic multiplicity 1, then µ(V, ℓ; P ) = 1 and we will say that the intersection at P is simple or even transverse. When otherwise µ(V, ℓ; P ) > 2, we will say that the line touches V at P or even that they have contact of order greater than 1 at P

11.2 Tangent space at a point of a projective variety. Smoothness

213

Proposition 11.1.6 Let V ⊂ Ar be an affine variety and let P ∈ V be a point. The affine tangent space TV /Ar ,P is the union of all the lines in Ar which touch V at P . Proof. As above, let Ia (V ) = (f1 , . . . , fs ). To simplify notation, let x := (x1 , . . . xr ) be the vector of indeterminates and let p = (p1 , . . . , pr ) be the coordinate vector of the point P ∈ V . For any j ∈ {1, . . . , s}, by Taylor expansion we set fj (x) = fj ((x − p) + p) = hj (x − p) + gj (x − p),

(11.4)

where hj is a linear form and where gj is a polynomial whose homogeneous factors have at least degree two. For any 1 6 j 6 s, one has fj (b t + p) = hj (b t) + gj (b t) = t hj (b) + t2 qj (t), for suitable qj ∈ K[t]. The line ℓP,b touches V at P if and only if h1 (b) = h2 (b) = · · · = hs (b) = 0.

(11.5)

On the other hand, from Taylor’s formula applied to (11.4), one has hj (x − p) =

r X ∂fj i=1

∂xi

(p)(xi − pi )

so (11.5) enstablishes that b belongs to the direction of TV /Ar ,P , which is the sub-vector space of Kr defined by a homogeneous linear system whose coefficient matrix is precisely J(P ). ⊓ ⊔

11.2 Tangent space at a point of a projective variety. Smoothness The case of a projective variety is similar to the affine case; we shall briefly mention to it. Let V ⊂ Pr be a projective variety and let P = [p0 , p1 , . . . , pr ] ∈ V be a point. Let Ip (V ) = (F1 , . . . Fs ) be its homogeneous radical ideal. Definition 11.2.1 The projective tangent space to V at the point P is the linear subspace of Pr defined by the homogeneous linear system r X ∂Fj (P )Xi = 0, 1 6 j 6 s. ∂Xi i=0

It will be denoted by TV /Pr ,P .

214

11 Tangent spaces and smoothness

As in the affine case, the definition is independent from the choice of the generators of Ip (V ). If p0 6= 0, i.e. if P ∈ V ∩ U0 := V0 , the hyperplanes in the linear system defininig TV /Pr ,P are the projective closures of the affine hyperplanes in Definition 11.1.1 defining TV0 /Ar ,P . Thus, from § 3.3.5, the linear space TV /Pr ,P is the projective closure of the affine space TV0 /Ar ,P . Same conclusion holds for any other non empty affine chart Vi = V ∩ Ui , i = 1, . . . , r. Let LP ⊂ Pr be any line passing through the point P ∈ V ; let moreover Q = [q0 , q1 , . . . , qr ] 6= P be any other point of LP . To simplify notation, we pose p := (p0 , p1 , . . . , pr ) and q = (q0 , q1 , . . . , qr ) whereas x = (X0 , . . . , Xr ) denotes the row-matrix of homogeneous coordinates in Pr . Thus, parametric equations of LP are given by x = λp + µq, [λ, µ] ∈ P1 . Points of V ∩ LP corresponds to the solutions [λ, µ] ∈ P1 of the system of equations F1 (λ p + µ q) = F2 (λ p + µ q) = · · · = Fs (λ p + µ q) = 0. The previous system of equations is equivalent to a unique homogeneous equation F (λ, µ) = 0, where F (λ, µ) := g.c.d.(F1 (λ p + µ q), · · · , Fs (λ p + µ q)). The multiplicity of the root [1, 0] (corresponding to the point P ) will be called intersection multiplicity between LP and V at the point P and will be denoted by µ(V, LP ; P ). Similarly to the affine case, µ(V, L; P ) = 0 if the line L does not pass through P , whereas µ(V, L; P ) = +∞ when the line L passes through P and it is entirely contained in V . It is easy to verify that the definition of µ(V, L; P ) is compatible with that given in Definition 11.1.4, i.e. if P ∈ V0 = V ∩ U0 6= ∅ and if ℓ := L ∩ U0 , then µ(V, L; P ) = µ(V0 , ℓ; P ). Similarly to the affine case, one can show that TV /Pr ,P is the union of all projective lines L ⊂ Pr touching V at P , i.e. such that µ(V, L; P ) > 2. Example 11.2.2 (i) When V = Zp (F ) is an (irreducible) hypersurface in Pr , (r) with F ∈ Sd for some positive integer d, its Jacobian matrix is

11.3 Zariski tangent space of an algebraic variety. Smoothness

J=



∂F ∂F ∂F ... ∂X0 ∂X1 ∂Xr



215

.

As in the affine case, for any 0 6 h 6 r − 1, one has Zp (Jh ) = Pr therefore any P ∈ V is such that dim(TV /Pr ,P ) > r − 1 = dim(V ). When h = r, Jr is a homogeneous ideal generated by homogeneous polynomials of degree (d − 1) and Sing(V ) is simply given by Zp (Jh ), as it follows from Euler’s identity (cf. Proposition 1.10.13 (i)). For any smooth point Q ∈ V one has therefore dim(TV /Pr ,Q ) = r − 1 = dim(V ); on the contrary, P ∈ Za (Jr ) belongs to Sing(V ) and

dim(TV /Pr ,P ) = r = dim(Pr ).

√ (r) In particular, V is smooth if and only if Zp (Jh ) = ∅, i.e. Jh = S+ . (ii) For V = Zp (F ) an irreducible hypersurface in Pr as above, if L ⊂ Pr is a line not contained in V one has X µ(V, L; P ) = d P ∈L

since the left-side-member above equals the sum of the (algebraic) multiplicities of the roots of the homogeneous polynomial F (λ p + µ q) ∈ K[λ, µ]d , for any pair of points P, Q ∈ L.

11.3 Zariski tangent space of an algebraic variety. Smoothness The previous definitions of tangent spaces take into account the embedding of the variety V either in an affine or in a projective space; therefore these definitions are not intrinsic. It is possible to intrinsically associate a K-vector space to any point P of any algebraic variety V in such a way that, under this association, one reobtains the affine tangent space in Definition 11.1.1 (viewed as a vector space whose origin is P ) when V is affine. Take V any algebraic variety and let P ∈ V be any point. From Theorem 5.2.10 and Example 5.2.22, we know that (OV,P , mV,P ) is a local ring, with residue field K. To simplify notation, in what follows we will simply denote by m the maximal ideal mV,P . O The OV,P -module mm2 is annihilated by m, therefore it is a V,P m -module, i.e. it is a K-vector space.

216

11 Tangent spaces and smoothness

Lemma 11.3.1 For any P ∈ V ,

m m2

is a finite dimensional K-vector space.

Proof. From Corollary 5.2.19, for any open set U ⊆ V containing P one has an isomorphism of local rings OU,P ∼ = OV,P . Thus, by Proposition 6.4.2, we can reduce to V an affine variety. In particular we can consider V ⊆ Ar , for some integer r, as an irreducible, closed subset. In such a case, from Claim 5.2.15, we have OV,P ∼ = A(V )mV (P ) and m ∼ = mV (P )A(V )mV (P ) . Thus

m ∼ mV (P )A(V )mV (P ) . = m2 mV (P )2 A(V )mV (P )

On the other hand, from Theorem 5.2.14 (b), mP mV (P ) ∼ , = Ia,V (P ) = Ia (V ) where mP = (x1 − p1 , . . . , xr − pr ) the maximal ideal in A(r) corresponding to P ∈ Ar and Ia (V ) ⊂ A(r) the radical ideal of V ⊆ Ar . Therefore   m mP =r 6 dim dim m2 m2P ⊓ ⊔

as desired.

Definition 11.3.2 For any point P ∈ V , the K-vector space  m ∨ m2

is called the Zarisky tangent space of V at P and it is denoted by TV,P . Notice that the definition of TV,P depends only on the local ring OV,P , i.e. TV,P is of local nature. In particular, for any open neighborhood U of P in V one has TV,P ∼ = TU,P . Definition 11.3.3 A K-linear map D : OV,P → K is called a K-derivation from OV,P to K if, for any Φ, Ψ ∈ OV,P and for any λ ∈ K, one has D(λ) = 0 and D(Φ Ψ ) = Φ(P ) D(Ψ ) + Ψ (P ) D(Φ). The set of all K-derivations from OV,P to K determines a sub-vector space of HomK (OV,P , K), which will be denoted by DerK (OV,P , K).

11.3 Zariski tangent space of an algebraic variety. Smoothness

217

Proposition 11.3.4 (i) For any algebraic variety V and any point P ∈ V , there exists a canonical isomorphism of K-vector spaces TV,P ∼ = DerK (OV,P , K). (ii) If V ⊆ Ar is an affine variety and P ∈ V , there exists a canonical isomorphism of K-vector spaces TV,P ∼ = TV /Ar ,P , where TV /Ar ,P is considered as a K-vector space whose origin is the point P . Proof. (i) If D ∈ DerK (OV,P , K), for any Φ, Ψ ∈ m we have D(Φ Ψ ) = 0, i.e. D|m2 = 0. This implies that D induces a K-linear homomorphism m → K, LD : m2
∨ i.e. LD ∈ mm2 = TV,P . Conversely, given L ∈ TV,P , define DL : OV,P → K, Φ → L(Φ − Φ(P )), where Φ − Φ(P ) ∈ mm2 . For any λ ∈ K, DL (λ) = L(λ − λ) = L(0) = 0. Now for any Φ, Ψ ∈ OV,P , one has DL (Φ Ψ ) = DL ((Φ − Φ(P ) + Φ(P )) ((Ψ − Ψ (P ) + Ψ (P )) = DL ((Φ − Φ(P )) (Ψ − Ψ (P )) + (Φ − Φ(P )) Ψ (P ) + (Ψ − Ψ (P )) Φ(P ) + Φ(P ) Ψ (P )) .

Since L is linear, by its definition DL is also linear. Using that Φ(P ) Ψ (P ) ∈ K and (Φ − Φ(P )) (Ψ − Ψ (P )) ∈ m2 , the latter expression equals DL ((Φ − Φ(P )) Ψ (P ) + (Ψ − Ψ (P )) Φ(P ))

which, by definition of DL , is       L (Φ − Φ(P )) Ψ (P ) + (Ψ − Ψ (P )) Φ(P ) = Ψ (P ) L Φ − Φ(P ) +Φ(P ) L Ψ − Ψ (P )

which is Ψ (P ) DL (Φ) + Φ(P ) DL (Ψ ). In other words, DL ∈ DerK (OV,P , K). Since the maps D → LD and L → DL are one the inverse of the other, we conclude. (ii) Let V ⊂ Ar be affine; thus, for any P ∈ V , OV,P ∼ = A(V )mV (P ) and let x1 , . . . xr ∈ OV,P be the images of the indeterminates of A(r) . For any D ∈ DerK (OV,P , K), let λi = D(xi ), 1 6 i 6 r.

(11.6)

218

11 Tangent spaces and smoothness

For any f ∈ Ia (V ), by the linearity of D one has: 0 = D(f (x1 , . . . , xr )) =

r X ∂fj i=1

∂xi

(P )λi ,

i.e. (λ1 , . . . , λr ) ∈ Kr lies in the direction of TV /Ar ,P , i.e. (p1 , . . . , pr ) + (λ1 , . . . , λr ) ∈ TV /Ar ,P . Conversely, for (p1 , . . . , pr ) + (λ1 , . . . , λr ) ∈ TV /Ar ,P , posing as in (11.6) D(xi ) := λi , we can define a derivation D ∈ DerK (OV,P , K) as follows: D(g) :=

r X ∂fj i=1

and D

g h

=

∂xi

(P )λi , ∀ g ∈ A(V )

D(g) h(P ) − D(h) g(P ) g , ∀ ∈ OV,P . h(P )2 h

The map TV /Ar ,P → DerK (OV,P , K) just constructed is well-defined, K-linear and it is the inverse of the previous one; so TV /Ar ,P ∼ = DerK (OV,P , K). We then conclude by part (i). ⊓ ⊔ As in the affine case, a point P ∈ V is said to be smooth (or simple, or non-singular) if dim (TV,P ) = minQ∈V dim (TV,Q ) ; otherwise P is said to be a singular point. Applying Proposition 11.3.4 (ii) to any affine open set of an affine open covering of V and using (11.1) for any such affine open set, one deduces that the set of simple points in V is a non-empty, dense open set in V . The set Sing(V ) of singular points is a proper, closed subset of V ; the algebraic variety will be said to be either singular or smooth according to either Sing(V ) 6= ∅ or Sing(V ) = ∅, respectively. Definition 11.3.5 Let V and W be algebraic varieties, P ∈ V be a point and ϕ : V → W be a morphism. The local, K-algebra homomorphism ϕP : OW,ϕ(P ) → OV,P induces a homomorphism of K-vector spaces dϕP : TV,P −→ TW,ϕ(P ) which is called the differential of ϕ at the point P . has

If ϕ : V → W and ψ : W → Z are morphisms, for any point P ∈ V one d(ψ ◦ ϕ)P = (dψϕ(P ) ) ◦ (dϕP ) and (dIdV )P = IdTV,P .

(11.7)

11.3 Zariski tangent space of an algebraic variety. Smoothness

219

Corollary 11.3.6 If ϕ : V → W is an isomorphism of algebraic varieties then, for any point P ∈ V , dϕP is an isomorphism of K-vector spaces, Proof. It immediately follows from (11.7).

⊓ ⊔

Theorem 11.3.8 below gives an alternative definition for the dimension of an algebraic variety; we first need an auxiliary result. Proposition 11.3.7 Any algebraic variety V is birationally equivalent to a hypersurface of a projective space and to a hypersurface of an affine space. Proof. It suffices to prove e.g. the second statement. Let dim(V ) = v and let {s1 , . . . , sv } be a trascendence basis of K(V ) over K. Since char(K) = 0, by the Primite Element Theorem (cf. e.g. [21, Thm. 4.6, p.243]), there exists θ ∈ K(V ) such that K(V ) ∼ = K(s1 , . . . , sv , θ). By the assumption on {s1 , . . . , sv }, the elements s1 , . . . , sv , θ ∈ K(V ) are algebraically dependent over K. There exists therefore a non-zero polynomial f ∈ K[x1 , . . . , xv , xv+1 ] (we may assume to be irreducible) s.t. f (s1 , . . . , sv , θ) = 0. Since {s1 , . . . , sv } is a trascendence basis of K(V ) over K, f is not constant with respect to the indeterminates xv+1 . Let W := Za (f ) ⊂ Av+1 ; then the affine coordinate ring A(W ) =

K[x1 , x2 , . . . , xv+1 ] A(v+i) = , Ia (W ) Ia (W )

where the xi ’s are indeterminates, is an integral K-algebra. Since Ia (W ) = (f ) is principal, as in the proof of Prop. 9.1.5, the elements x1 , . . . , xv ∈ A(W ) ⊂ Q(A(W )) ∼ = K(W ) are algebraically independent, where xi denotes the image of the indeterminate xi , 1 6 i 6 v + 1. Therefore K(W ) ∼ = K(x1 , . . . , xv )[xv+1 ], = K(x1 , . . . , xv , xv+1 ) ∼ where the second isomorphism follows from the fact that xv+1 is algebraic over K(x1 , . . . , xv ). If we let y be an indeterminate, then K(x1 , . . . , xv )[y] K(x1 , . . . , xv )[xv+1 ] ∼ . = (f (x1 , . . . , xv , y)) Since {s1 , . . . , sv } is a trascendence basis over K, then

220

11 Tangent spaces and smoothness

K(x1 , . . . , xv )[y] ∼ K(s1 , . . . , sv )[y] . = (f (x1 , . . . , xv , y)) (f (s1 , . . . , sv , y)) By the definition of θ, K(s1 , . . . , sv )[y] ∼ = K(s1 , . . . , sv )[θ] ∼ = K(s1 , . . . , sv , θ) ∼ = K(V ). (f (s1 , . . . , sv , y)) To sum up, K(V ) ∼ = K(W ). One then concludes by using Corollary 8.1.7 (i). ⊓ ⊔ Theorem 11.3.8 Let V be an algebraic variety and let P ∈ V be any smooth point of V . Then dim(V ) = dim (TV,P ) . (11.8) Proof. The set U of smooth points of V is a non empty open set, therefore we can asusme V to be affine (cf. Prop. 6.4.2). From Definition 11.1.1, (11.8) holds true if V = Ar . Similarly, from (11.2), (11.8) holds true if V is a hypersurface in Ar . If V and W are birational algebraic varieties, then (11.8) holds true for V if and only if it holds true for W : indeed there exist non empty open sets UV ⊆ V and UW ⊆ W which contain smooth points of V and W , respectively, and which are isomorphic so we can we apply Corollary 11.3.6. Since, from Proposition 11.3.7, any algebraic variety is biratioanlly equivalent to an affine hypersurface, the proof is complete. ⊓ ⊔

11.4 Exercises

References

1. Atiyah M. F. and Macdonald I. G., Introduction to Commutative Algebra, Reading, Massachusetts - Menlo Park, California, Addison - Wesley Publishing Co., 1969. 2. Barth W.P., Hulek K., Peters, C.A.M. and Van de Ven A. Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 4, SpringerVerlag, Berlin, 2004, ISBN 978-3-540-00832-3. 3. Beauville A., Complex algebraic surfaces, London Mathematical Society Student Texts 34 (2nd ed.), Cambridge University Press, 1996, ISBN 978-0-521-49510-3. 4. Bourbaki N., Elements of Mathematics. General Topology. Chapters 1-4, Springer, 1989. 5. Ciliberto C., Lecture Notes on Algebraic Geometry, University of Rome ”Tor Vergata”, private notes. 6. Clemens C. H. and Griffiths P. A., ”The intermediate Jacobian of the cubic threefold”, Annals of Mathematics, 95 (2), 1972, 281356. 7. Cohn P. M., Algebraic Numbers and Algebraic Functions, Chapman Hall/CRC Mathematics Series 4, 1991, CRC Press. 8. Cohn D., Little J., O’Shea D., Ideals, varieties and algorithms, Springer, 1992. 9. Debarre O., Introduction a ` la g´eom´etrie alg´ebrique, cours de DEA, 1999/2000, ´ et M2, 2007/2008, Ecole normale sup´erieure, available at http : //www.math.ens.f r/ ∼ debarre/DEA99.pdf 10. Dolgachev I., Introduction to Algebraic Geometry, Lecture Notes 2013, University of Michigan at Ann Arbor, available at http : //www.math.lsa.umich.edu/ ∼ idolga/631.pdf 11. Eisenbud D., Commutative Algebra, with a view toward Algebraic Geometry, Springer, Berlin, 1995. 12. Fulton W., Algebraic Curves. An introduction to Algebraic Geometry, 2008 (slightly modified version of the 1969 text), available at http : //www.math.lsa.umich.edu/ ∼ wf ulton/ 13. Gathmann A., Algebraic Geometry, Lecture Notes 202/2003, University of Kaiserslautern, available at http : //www.mathematik.uni − kl.de/ ∼ gathmann/class/alggeom − 2002/main.pdf 14. Harris J., Algebraic Geometry. A first course, Graduate Texts in Mathematics, 133, Springer - Verlag, 1992.

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References

15. Hartshorne J., Algebraic Geometry, Springer - Verlag, GTM 52, 1977. 16. Hassett B., Introduction to Algebraic Geometry, Cambridge University Press, 2007. 17. Hulek K., Elementary Algebraic Geometry, Student Mathematical Library, 20, American Mathematical Society, 2003. 18. Kendig K., Elementary Algebraic Geometry, Graduate Texts in Mathematics, 44, Springer - Verlag, 1977. 19. Kosniowski C., A First Course in Algebraic Topology, Cambridge University Press, 1980. 20. Kunz E., Introduction to Commutative Algebra and Algebraic Geometry, Birkhauser, 1985. 21. Lang S., Algebra, Third Ed., Reading-Massachusetts, Addison-Wesley, 1995. 22. Matsumura H., Commutative Algebra, Readings Massachusetts, B. Cummings Publishing Program, 1980. 23. Milne J.S., (Basic First Course in) Algebraic Geometry, available at http : //www.jmilne.org/math/CourseN otes/ag.html 24. Miranda R., Algebraic curves and Riemann surfaces, Graduate Studies in Mathematics, 5, American Mathematical Society, 1995. 25. Mumford D., The Red Book of Varieties and Schemes, Lecture Notes in Mathematics, 1358, Springer-Verlag, 1988. 26. Mumford D., Algebraic Geometry I. Complex Projective Varieties, Classics in Mathematics, Springer-Verlag, 1976. 27. Reid M., Undergraduate Algebraic Geometry, London Mathematical Society Student Texts, 12, Cambridge University Press, Cambridge, 1988. 28. Sernesi E., Appunti del corso di Geometria Algebrica, University of Rome ”La Sapienza”, private notes. 29. Shafarevich I. R., Basic Algebraic Geometry 1. Varieties in Projective space, Springer-Verlag, 1994. 30. Verra A., Geometria ed Algebra delle curve piane, University ”Roma Tre”, 2009 slides, available at http : //www.mat.uniroma3.it/users/verra/SOLON GHELLO.pdf

Index

trdegK (F), 24 (I : J), 46 (OY,W , mY,W ), 122 G–graded module, 39 G-graded ring, 30 JAS extended ideal in localization, 44 R-algebra, 16 R-algebra of finite type, 17 R-bilinear map of R-modules, 26 R-module, 15 R-module generated by T , 15 R-module homomorphism, 16 R-submodule, 15 Rf , 46 R([f ]) , 46 R(p) , 46 Rp , 46 UΦ , 176 OY , 116 OY,W , 120 ∆(V ), 173 Γf , 173 K-algebra homomorphisms, 145 K-algebra of rational functions defined in a subvariety, 120 K-derivation from OV,P to K, 217 K0 -rational points of an AAS, 70 O(P(V ), P(W )), 92 Σ1,1 , 170 Σn,m , 168 Q(n) , 116 (n) Q0 , 116 dimK (K[X0 , . . . , Xn ]d ), 35

IY (W ), 122 Q0 (S(Y )), 128 Q0 (S), 34 Zarn p , 78 ZarP(V ),Y , 79 ZarP(V ) , 78 p(n, d), 35 σn,m , 168 x ⊗ y, 26 Aff(An ), 94 Aut(V ), 141 Bir(V ), 194 Div(Y ), 160 HomK (R, S), 145 Isom(V, W ), 141 Morph(V, W ), 141 PGL(V ), 92 Specm(A), 128 (Zariski) closed subsets of An , 54 (a.c.c.) ascending chain condition on open sets, 111 (affine) monoid of degree d, 186 (d.c.c.) descending chain condition on closed sets, 110 (non-negatively) graded ring, 31 (projective) Grassmann formula, 88 (projective) monoid of degree d, 186 a parametric representation of a projective twisted cubic, 104 a parametric representation of an affine twisted cubic, 103 AAS, 50 affine charts of Pn , 84

224

Index

cone over Y ⊂ Pn , 79 coordinate ring, 107 group of An , 94 hypersurface, 58, 59 open set of an algebraic variety, 149 affine rational curve with a polynomial parametrization, 97 affine rational curve with polynomial parametrization, 148 affine space An K , 49 affine subspace of An , 58 affine tangent space, 209 affine twisted cubic, 98 affine variety, 110, 149 affinity, 94 algebra of regular functions on U , 116 algebraic affine aet, 50 algebraic closed set, 110 algebraic closure of a field, 1 algebraic locally-closed set, 110 algebraic projective set, 78 algebraic subvariety, 120 algebraic variety, 110 algebraically independent elements over a field, 22 APS, 78 automorphism of an algebraic variety, 141 affine affine affine affine affine

base locus of a linear system of hypersurfaces in Pn , 155 Bezout’s theorem in A2 -weak form, 69 birational class of birationally equivalent algebraic varieties, 179 birational isomorphism, 177 birational maps, 177 birational transfromation of an algebraic variety, 194 birational varieties, 177 birationally equivalent algebraic varieties, 177 blow-up of Pn at P0 , 188 blow-up of a subvariety at a point, 191 closed immersion, 143 closure of a subset in An , 61 codimension of a subvariety, 201 complete algebraic variety, 203

complete linear system of hypersurfaces of degree d is Pn , 155 constructible set, 148 contact of order greater than 1 at a point, 213 contracted ideal w.r.t. a ring homomorphism, 3 coordinate affine subspace, 57 coordinate axes of Pn , 83 coordinate axis of an affine space, 57 coordinate field of an AAS, 70 coordinate linear subspace of Pn , 83 coordinate vector of a point in An K , 49 coordinates of a point in An K , 49 Cremona group, 194 cross ratio, 103 cuspidal plane cubic, 135 cyclic points of a circle, 135 cylinders, 60 decomposition of a polynomial in homogenous part, 35 degree of a divisor, 160 degree of a field extension, 16 degree of a projective hypersurface, 95 degree of an affine hypersurface, 59 degree-g graded part of a G-graded ring, 30 degree-zero rational functions in X0 , . . . , Xn , 116 dehomogenized polynomial w.r.t. Xi , 37 dehomogenizing operator w.r.t. Xi , 37 dense open subset, 106 derivatives in D[x], 6 Diagonal of V × V , 173 differential of a morphism at a point, 218 dimension of P(V ), 75 dimension of an algebraic variety, 197 direction of an affine subspace, 91 directrix of a cylinder, 60 discriminant in D[x], 11 divisor, 160 Dom(Φ), 176 domain of a linear system of hypersurfaces in Pn , 155 domain of a rational function, 120 domain of a rational map Φ, 176 dominant morphism, 142

Index dominant rational map, 176 doubly ruled quadric surface, 170 dual projective space, 83 elementary quadratic transformation of P2 , 195 Ellipsis, hyperbola and parabola, 135 elliptic curve, 179 Euler’s identity, 36 Euler’s Lemma, 9 evaluation at P , 50 exceptional divisor of a blow-up at a point, 189 exceptional lines of elementary quadratic transformation of P2 , 195 extended ideal w.r.t. a ring homomorphism, 3 extension of scalars, 28 field, 1 field extension, 16 field of definition of an AAS, 70 field of rational functions K(x1 , . . . , xn ), 17 field of rational functions on an algebraic variety, 119 finite R-algebra, 16 finite trascendence degree over a field, 24 finitely generated R-module, 15 finitely generated field extension, 17 finitely generated ideal, 2 fundamental affine open set of Y , 85 fundamental hyperplanes of Pn , 83 fundamental points of Pn , 76 fundamental points of elementary quadratic transformation of P2 , 195 generically finite rational map, 184 global sections of a pre-sheaf, 137 graded ring, 30 graph of a morphism of algebraic varieties, 173 Hilbert Nullstellenstaz-strong form, 63 Hilbert Nullstellenstaz-weak form, 62 Hilbert’s basis theorem, 13

225

homogeneized polynomial w.r.t. Xi , 37 homogeneous component of an element, 31 homogeneous coordinate ring, 107 homogeneous differential operators, 40 homogeneous elements of degree g, 30 homogeneous free resolution of the ideal of projective twisted-cubic, 101 homogeneous Hilbert Nullstellenstazstrong form, 82 homogeneous Hilbert Nullstellenstazweak form, 80 homogeneous homomorphism of graded rings, 39 homogeneous homomorphism of rings of degree d, 39 homogeneous ideal of a subset of Pn , 79 homogeneous localization of a graded ring w.r.t. a multiplicative system, 43 homogeneous localization w.r.t. a homogeneous, non-nilpotent element, 46 homogeneous localization with respect to a homogeneous, prime ideal, 46 homogeneous morphism of degree 0, 39 homogeneous polynomial, 34 homogeneous Study’s principle, 95 homogenizing operator w.r.t. Xi , 37 homography, 91 homomorphism of R-algebras, 16 hyperbolic paraboloid, 148 hyperplane at infinity of an affine chart, 85 ideal, 1 ideal of a subset of An , 61 ideal of a subvariety of an algebraic variety, 122 ideal of germs of regular functions vanishing at a point, 121 improper point of An , 85 improper points of Y ⊂ An , 85 indeterminacy locus of a morphism, 155 integral domain, 1 integral element over a subring, 18 intersection multiplicity at a point P with a line, 212 intersection of a family of ideals, 2

226

Index

intersection of projective subspaces, 88 irreducible affine hypersurface, 66 irreducible components, 112 irreducible components of a divisor, 160 irreducible components of an affine hypersurface, 66 irreducible topological space, 105 irredundant decomposition, 112 irrelevant ideal of a non-negatively graded ring, 31 isomorphic algebraic varieties, 141 isomorphism of algebraic varieties, 141 itegral closure of a subring in a ring, 19 Jacobian matrix, 209 L¨ uroth problem, 184 leading coefficient of f (x), 6 linear envelope of a subset in P(V ), 88 linear envelope of affine subspaces, 102 linear system of dimension r of hypersurfaces of degree d in Pn , 155 linearly independent points in P(V ), 89 local isomorphism of local rings, 128 local ring, 45 local ring of a subvariety W in a variety Y , 122 localization homomorphism, 42 localization of a ring w.r.t. a multiplicative system, 42 localization w.r.t. a non-nilpotent element, 46 localization with respect to a prime ideal, 46 locally closed subset of Pn , 85 locally-closed immersion, 143 Main theorem of Elimination Theory, 206 map satisfying the property of being local, 167 matrix equation of projective twisted cubic, 98 maximal ideal, 2 model of a birational class of algebraic varieties, 179 monic polynomial, 6 morphism and local property, 162

morphism of algebraic varieties, 141 multi-tensor product, 27 multiple factor of a polynomial, 5 multiple factor of a polynomial in D[x], 10 multiple root of f ∈ D[x], 6 multiplicative system in a ring, 42 multiplicity of a root of f ∈ D[x], 6 multiplicity of an irreducible hypersuraces in a divisor, 160 Noetherian ring, 12 Noetherian topological space, 110 non degenerate subset of An , 91 non degenerate subset of P(V ), 88 non-singular point, 210, 218 non-singular variety, 210 numerical projective space, 75 open immersion, 143 open Riemann surface of a complex conic, 56 open set of definition of a linear system of hypersurfaces in Pn , 155 open set of definition of a rational function, 120 open set of definition of a rational map Φ, 176 open subset in ZarAn , 54 orthogonal subspace to a subspace of P(V ), 89 parametric representation of a subspace in Pn , 89 parametric representation of an affine subspace, 58, 91 partial derivatives in D[x1 , . . . , xn ], 7 plane nodal cubic, 187 pluri–homogeneous polynomials, 41 point in An K , 49 points at infinity of An , 85 points at infinity of Y ⊂ An , 85 points in general linear position in P(V ), 89 polynomial map of affine varieties, 148 power of an ideal, 2 pre-sheaf on a topological space, 137 prime ideal, 2 principal ideal, 2 principal ideal domain (PID), 4

Index principal open affine set, 59 principal open affine sets of Pn , 84 principal open projective sets, 95 principal open set of P(V ), 78 principal open subset associated to a regular function, 118 principal ring, 4 product of 2 ideals, 2 product of a family of ideals, 2 product of two AAS’s, 59 projection of An onto one of its axis, 144 projection of An onto the coordinates I = {i1 , i2 , . . . , im }, 148 projection of Pn to Pr with center a linear space, 155 projection of a projective variety on a given subspace, 208 projection of an algebraic variety, 183 projective closure, 85 projective cone over Z ⊂ Pn , 94 projective cone with vertex a linear space, 102 projective hypersurface, 95 projective space P(V ), 75 projective subspace generated by a subset of P(V ), 88 projective subspace of P(V ), 87 projective tangent space, 214 projective twisted cubic, 98 projective twisted cubic is determinantal, 98 projective variety, 110 projectively isomorphic projective spaces, 92 projectivities of Pn , 155 projectivity, 92 proper transform of a subvariety via blow-up at a point, 191 proportionality relation, 75 pure dimension, 198 purely trascendental extension of a field, 24 Pythagorean triples, 70 quadric cylinders in A3 , 60 quasi-affine variety, 110 quasi-projective variety, 110 quotient field, 1

227

quotient ideal, 46 Rabinowithch’s trick, 64 radical ideal, 3 radical of an ideal, 2 rational functions in x1 , . . . , xn , 116 rational functions on An , 116 rational functions on Y , 118 rational map of algebraic varieties, 176 rational normal curve of degree d in Pd , 156 rational parametrization of a conic, 56 rational transformation of an algebraic variety, 194 rational variety, 184 real trace of a complex conic, 57 reduced divisor, 160 reduced equation of a projective hypersurface, 95 reduced equation of an affine hypersurface, 59 reduced ring, 3 reducible topological space, 105 regular function at a point, 116 regular function of a locally-closed algebraic set, 116 relation of algebraic dependence over a field, 22 representative morphism of a rational map over an open set, 176 restricted sheaves to open subsets, 138 restriction of scalars, 28 resultant of two polynomials in D[x], 9 resultant polynomial w.r.t. an indeterminate, 11 ring homomorphism, 1 ring of differential operators, 40 ring of germs of regula function at a point, 121 ring of polynomials R[x], 1 ring of polynomials R[x1 , . . . , xn ], 1 ring of total fractions of a ring, 43 root of f ∈ D[x], 6 roots of a homogeneous polynomial F (X0 , X1 ), 39 sections of a pre-sheaf on an open set, 137 Segre map of indices n and m, 168

228

Index

Segre variety of indices n and m, 168 Semi-cubic parabola, 135, 137 set of APS’s of P(V ), 78 set of generators for an ideal, 2 set of generators of an R-module, 15 set of homogeneous elements of a subset, 30 sheaf on a topological space, 138 simple field extension, 17 simple intersection at a point with a line, 213 simple point, 210, 218 simple ring, 2 simple root of f ∈ D[x], 6 singular point, 210, 218 singular variety, 210, 218 smooth plane cubic, 179 smooth point, 210, 218 smooth quadric surface in P3 , 170 smooth variety, 210, 218 stereographic projection of a monoid from its vertex, 187 stereographic projection of a rank-four quadric surface to a plane, 185 stereographic projection of an irreducible conic on P1 , 183 structural sheaf of an algebraic variety, 119 Study’s principle, 67 sub-field of degree-zero, homogeneous fractions of a graded ring, 34 sum of 2 ideals, 2 sum of a family of ideals, 2 support of a divisor, 160 Sylvester matrix of two polynomials, 9 symmetric algebra over V , 40 system of equations for an AAS, 50

system of homogeneous equations for an APS, 77 syzygies for the projective twisted cubic, 99 syzygy module of the projective twisted cubic, 99 tensor product of R-modules, 27 total transform of a subvariety via blow-up at a point, 191 transposed homography, 102 transverse intersection, 213 trascendence basis of a field extension, 23 trascendence degree of a field extension, 24 trascendence degree of an integral K-algebra over K, 25 unique factorization domain (UFD), 4 unirational variety, 184 unit, 1 vector of homogeneous coordinates, 75 Veronese morphism νn,d , 155 Veronese variety Vn,d , 155 vertices of the pyramid of Pn , 76 Zariski tangent space, 216 Zariski topology of An , 54 Zariski topology of a subset of An , 54 Zariski topology of a subset of P(V ), 79 Zariski topology on P(V ), 78 Zariski topology on Pn , 78 zero–set, 50 zero–set in Pn , 77 zero–set of a homogenous polynomial in P(V ), 77 zero-locus of a regular function, 117