CHAPTER 13 Coherent Sheaves; Invertible Sheaves In this chapter, k is an arbitrary field. a Coherent sheaves Let V SpmA be an affine variety over k, and let M be a finitely generated A-module. D There is a unique sheaf of OV -modules M on V such that, for all f A, 2 .D.f /;M/ M . A A M /: D f D f ˝ Such an OV -module M is said to be coherent. A homomorphism M N of A-modules ! defines a homomorphism M N of OV -modules, and M M is a fully faithful functor ! 7! from the category of finitely generated A-modules to the category of coherent OV -modules, with quasi-inverse M .V;M/. 7! Now consider a variety V . An OV -module M is said to be coherent if, for every open affine subset U of V , M U is coherent. It suffices to check this condition for the sets in an j open affine covering of V . n For example, OV is a coherent OV -module. An OV -module M is said to be locally n free of rank n if it is locally isomorphic to OV , i.e., if every point P V has an open n 2 neighbourhood such that M U O . A locally free OV -module of rank n is coherent. j V Let v V , and let M be a coherent OV -module. We define a .v/-module M.v/ as 2 follows: after replacing V with an open neighbourhood of v, we can assume that it is affine; hence we may suppose that V Spm.A/, that v corresponds to a maximal ideal m in A (so D that .v/ A=m/, and M corresponds to the A-module M ; we then define D M.v/ M A .v/ M=mM: D ˝ D It is a finitely generated vector space over .v/. Don’t confuse M.v/ with the stalk Mv of M which, with the above notations, is Mm M A Am. Thus D ˝ M.v/ Mv=mMv .v/ A Mm: D D ˝ m This is Chapter 13 of Algebraic Geometry by J.S. Milne. July 8, 2015 Copyright c 2015 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permission from the copyright holder. 1 2 13. COHERENT SHEAVES;INVERTIBLE SHEAVES Nakayama’s lemma (AG 1.3) shows that M.v/ 0 Mv 0: D ) D The support of a coherent sheaf M is Supp.M/ v V M.v/ 0 v V Mv 0 : D f 2 j ¤ g D f 2 j ¤ g Suppose V is affine, and that M corresponds to the A-module M . Let a be the annihilator of M : a f A f M 0 : D f 2 j D g Then M=mM 0 m a (for otherwise A=mA contains a nonzero element annihilating ¤ ” M=mM ), and so Supp.M/ V.a/: D Thus the support of a coherent module is a closed subset of V . Note that if M is locally free of rank n, then M.v/ is a vector space of dimension n for all v. There is a converse of this. PROPOSITION 13.1. If M is a coherent OV -module such that M.v/ has constant dimen- sion n for all v V , then M is a locally free of rank n. 2 PROOF. We may assume that V is affine, and that M corresponds to the finitely generated A-module M . Fix a maximal ideal m of A, and let x1;:::;xn be elements of M whose images in M=mM form a basis for it over .v/. Consider the map n X A M; .a1;:::;an/ ai xi : W ! 7! Its cokernel is a finitely generated A-module whose support does not contain v. Therefore there is an element f A, f m, such that defines a surjection An M . After 2 … f ! f replacing A with Af we may assume that itself is surjective. For every maximal ideal n of A, the map .A=n/n M=nM is surjective, and hence (because of the condition on ! the dimension of M.v/) bijective. Therefore, the kernel of is contained in nn (meaning n n) for all maximal ideals n in A, and the next lemma shows that this implies that the kernel is zero. 2 LEMMA 13.2. Let A be an affine k-algebra. Then \ m 0 (intersection of all maximal ideals in A). D PROOF. When k is algebraically closed, we showed (AG, 2.18) that this follows from the strong Nullstellensatz. In the general case, consider a maximal ideal m of A kal. Then ˝k A=.m A/ , .A kal/=m kal; \ ! ˝k D and so A=m A is an integral domain. Since it is finite-dimensional over k, it is a field, and \ so m A is a maximal ideal in A. Thus if f A is in all maximal ideals of A, then its image \ al 2 in A k is in all maximal ideals of A, and so is zero. 2 ˝ b. Invertible sheaves. 3 For two coherent OV -modules M and N , there is a unique coherent OV -module M O N such that ˝ V .U;M O N / .U;M/ .U;N / ˝ V D ˝ .U;OV / for all open affines U V . The reader should be careful not to assume that this formula holds for nonaffine open subsets U (see example 13.4 below). For a such a U , one writes S U Ui with the Ui open affines, and defines .U;M O N / to be the kernel of D ˝ V Y Y .Ui ;M O N / Ã .Uij ;M O N /: ˝ V ˝ V i i;j Define Hom.M;N / to be the sheaf on V such that .U;Hom.M;N // HomO .M;N / D U (homomorphisms of OU -modules) for all open U in V . It is easy to see that this is a sheaf. If the restrictions of M and N to some open affine U correspond to A-modules M and N , then .U;Hom.M;N // HomA.M;N /; D and so Hom.M;N / is again a coherent OV -module. b Invertible sheaves. An invertible sheaf on V is a locally free OV -module L of rank 1. The tensor product of two invertible sheaves is again an invertible sheaf. In this way, we get a product structure on the set of isomorphism classes of invertible sheaves: ŒL ŒL def ŒL L : 0 D ˝ 0 The product structure is associative and commutative (because tensor products are associative and commutative, up to isomorphism), and ŒOV is an identity element. Define L Hom.L;OV /: _ D Clearly, L_ is free of rank 1 over any open set where L is free of rank 1, and so L_ is again an invertible sheaf. Moreover, the canonical map L L OV ; .f;x/ f .x/ _ ˝ ! 7! is an isomorphism (because it is an isomorphism over any open subset where L is free). Thus ŒL ŒL ŒOV : _ D 1 For this reason, we often write L for L_. From these remarks, we see that the set of isomorphism classes of invertible sheaves on V is a group — it is called the Picard group, Pic.V /, of V . We say that an invertible sheaf L is trivial if it is isomorphic to OV — then L represents the zero element in Pic.V /. 4 13. COHERENT SHEAVES;INVERTIBLE SHEAVES PROPOSITION 13.3. An invertible sheaf L on a complete variety V is trivial if and only if both it and its dual have nonzero global sections, i.e., .V;L/ 0 .V;L /: ¤ ¤ _ PROOF. We may assume that V is irreducible. Note first that, for any OV -module M on any variety V , the map Hom.OV ;M/ .V;M/; ˛ ˛.1/ ! 7! is an isomorphism. Next recall that the only regular functions on a complete variety are the constant functions (see AG, 7.9, in the case that k is algebraically closed), i.e., .V;OV / k where k is D 0 0 the algebraic closure of k in k.V /. Hence Hom.OV ;OV / k , and so a homomorphism D 0 OV OV is either 0 or an isomorphism. ! We now prove the proposition. The sections define nonzero homomorphisms s1 OV L; s2 OV L : W ! W ! _ We can take the dual of the second homomorphism, and so obtain nonzero homomorphisms s1 s2_ OV L OV : ! ! The composite is nonzero, and hence an isomorphism, which shows that s2_ is surjective, and this implies that it is an isomorphism (for any ring A, a surjective homomorphism of A-modules A A is bijective because 1 must map to a unit). 2 ! c Invertible sheaves and divisors. Now assume that V is nonsingular and irreducible. For a divisor D on V , the vector space L.D/ is defined to be L.D/ f k.V / div.f / D 0 : D f 2 j C g We make this definition local: define L.D/ to be the sheaf on V such that, for any open set U , .U;L.D// f k.V / div.f / D 0 on U 0 : D f 2 j C g [ f g P The condition “div.f / D 0 on U ” means that, if D nZZ, then ordZ.f / nZ 0 C D C for all Z with Z U . Thus, .U;L.D// is a .U;OV /-module, and if U U , then \ ¤ ; 0 .U ;L.D// .U;L.D//: We define the restriction map to be this inclusion.