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Coherent Sheaves on X Form an Abelian Category.” LECTURES ON ALGEBRAIC GEOMETRY MATH 202A 61 7. Coherent sheaves The purpose of this section is to make sense of the statement that “Coherent sheaves on X form an abelian category.” 7.1. Sheaves of modules. For any ring R, the category Mod-R of all R-modules and subcategory mod-R of finitely generated R-modules are abelian categories. This means in particular that a homomorphism f : M N have kernel ker f, cokernel coker f and image (using the basic property that the cokernel! of ker f M is isomorphic to the kernel of N coker f): ! ! im(f)=M/ker f =ker(N coker f) ! The basic example of a quasi-coherent sheaf over a variety or scheme X isaΓ(X)-module. It is coherent if it is finitely generated. One very important example is for a grade ring R. Then the quasi-coherent sheaf over X = Proj R corresponding to Rn is denoted oX (n). For more general varieties and schemes X, a quasi-coherent sheaf is a sheaf which is locally a module over Γ(U, oX ): Definition 7.1.1. Suppose that (X, oX ) is any ringed space. Then a sheaf of oX -modules over X is a sheaf F of additive groups so that (1) For every open subset U X, F (U)isaΓ(U, o )-module. ✓ X (2) If V U, the restriction map F (U) F (V ) is morphism of Γ(U, oX )-modules. (F (V ⇢) becomes a module over Γ(U) by! the ring homomorphism Γ(U) Γ(V ).) ! This looks like a perfectly good definition. We need an explicit example to see what is wrong with it and why a “coherence” condition is needed. The basic good example is the affine case X = Spec(R). Let M be any R-module. Then the corresponding sheaf is denoted M and is given on basic open sets by M(X )=M R f f ⌦R f So, at each point x =[p], the stalk if M R . f ⌦R p Proposition 7.1.2. Every element of M R has the form ⌦R p x 1 := x f ⌦ f where x M,f / p. 2 2 Proof. By definition, elements of M R are sums x r /f .Butr /f = s / f for ⌦R p i ⌦ i i i i i j some si R. So, 2 P Q si xi ri/fi = ⌦ fj X P Q ⇤ Corollary 7.1.3. For any open U in Spec(R), a section of M over U is given by a section σ[p]=x/f of the stalk bundle so that x, f are locally constant. I.e., for every point [p], there is a nbh V U so that x, f can be chosen to be constantf on V . ✓ Exercise 7.1.4. Show that, if Γ(U, M)=0 then Γ(V,M) = 0 for all open subsets V of U. f f 62 KIYOSHI IGUSA BRANDEIS UNIVERSITY 7.2. Definition. Definition 7.2.1. A quasi-coherent sheaf over a scheme X is an oX -module F so that there is an open covering of X by affine open sets U = Spec(R ) so that F U = M for ↵ ↵ | ↵ ↵ some R↵-module M↵. f The following theorem show that the category of quasi-coherent sheaves on X is a gen- eralization of the category of sheaves of the form M on X = Spec(R). Theorem 7.2.2. Every quasi-coherent sheaf F over X = Spec(R) is isomorphic to M f where M = F (X). f This follows from the following lemma. Lemma 7.2.3. For f R not nilpotent consider Xf = Spec(Rf ). 2 n (a) If s F (X), s Xf =0then n s.t. f s =0 (b) If t 2F (X ) then| n, s F9(X) so that f nt = s X . 2 f 9 9 2 | f Proof that Lemma implies Theorem. Let M = F (X). By definition of oX -module, F (Xf ) is an Rf -module. So, we have: φ : M = F (X) R F (X ) f ⌦R f ! f given by φ(x a)=res(x)a. It suffices to show that this is an isomorphism for all f. (a) implies⌦ that φ is a monomorphism: m m m Take any s/f Mf in the kernel of φ.Thenφ(s/f )=φ(s)/f =0implies k so k 2 k k n k 9 that f φ(s)=0=φ(f s). So, f s Xf = 0. By (a) n so that f (f s)=0inM.Then m | 9 s/f =0inMf . So, φ is a monomorphism. (b) implies that φ is onto: Take any t F (X ). By (b) there are n, s M so that t = s/f n X . So, φ is onto. 2 f 2 | f ⇤ Proof of Lemma. (a) We are given s F (X), X = Spec(R) so that s X = 0. 2 | f Since X is quasi-compact, by definition of quasi-coherent, we have X = Xi, Xi = Spec(R ) so that F X = M . We may assume that X = X . Recall that X X = X . i | i i i gi f \S gi fgi So, Xf = Xfgi and f Γ(X , M )=(M ) S fgi i i f So, s Xf =0i↵s Xfgi =0 (Mi)f for each i. Let yi = s Xi.Thenyi (Xi)f =0implies n | | 2 n | n | f i yi = 0. Let n = max ni.Thenf s restrictedf to Xi is f yi = 0. Since X = Xi this implies that f ns =0inF (X). This proves (a). ni S (b) Let t F (Xf ). Since F (Xfgi )=(Mi)f , t Xfgi = xi/f . Let n = max ni.Then t X = y /f2n. In other words, f nt X = y M|. The restrictions of y ,y to X X may | fgi i | fgi i 2 i i j i\ j not agree. But their restriction to Xi Xj Xf are equal since yi,yj are both restrictions n \ \ mij mij of f t F (Xf ). By (a) this implies that, for some mij, f yi = f yj in F (Xi Xj). 2 m \ Let m = max mij.Thenf yi Mi = F (Xi) agree on Xi Xj. By definition of a sheaf m2 m+n \ !s F (X) so that s Xi = f yi for each i.Sincef t F (Xf ) restricts to the same elements9 2 in X X ,wemusthave| s X = f m+nt. 2 f \ i | f ⇤ Corollary 7.2.4. An oX -module F is quasi-coherent if and only if, for every affine open subset U of X, F U = M where M =Γ(U, o ). | ⇠ X f LECTURES ON ALGEBRAIC GEOMETRY MATH 202A 63 7.3. Coherent sheaves over projective space. We take the simplest case X = P (⌦) = [x, y]:x, y not both zero 1 { } Recall that this is a union of two affine open subsets: X = X X x [ y where 1 Xx = [x, y]:x =0 = A˚ Spec(k[s]) { 6 } ⇡ Xy = [x, y]:y =0 Spec(k[t]) { 6 }⇡ with intersection 1 Xx Xy = [x, y]:xy =0 Spec(k[t, t− ]) \ { 6 }⇡ We ignore the fact that the variety has more points than the scheme. (Recall that the scheme is the quotient space of the variety given by identifying x, y if x = y and taking the push-forward of the structure sheaf.) Proposition 7.3.1. A quasi-coherent sheaf on P1 is given uniquely be the following data. (1) A k[s]-module M0 (2) A k[t]-module M1 1 1 (3) An isomorphism M0 k[s, s− ] ⇠= M1 k[t− ,t] where the rings are identified with st =1. ⌦ ⌦ Proof. By the theorem, any quasi-coherent sheaf F restricted to the affine open sets Xx,Xy is given by F X = M and F X = M . | x 0 | y 1 On Xx Xy the restrictions of M0 and M1 agree i↵(3). ⇤ \ f f For Pn, there are more affine sets and the isomorphisms need to be compatible. As an example, take n = 2. (The wordingf suggestsf how to generalize to any n.) Proposition 7.3.2. A quasi-coherent sheaf on P2 is given uniquely be the following data. (1) An Ri-module Mi for i =0, 1, 2 where R0 = k[T1,T2], R1 = k[T0,T2], R2 = k[T0,T1].) (2) An Rij-module Mij for i = j 0, 1, 2 where Rij = k[T0,T1,T2]/(1 TiTj). 6 2{ } − (3) One R012-module M012 where R012 = k[T0,T1,T2]/(1 T0T1T2). − so that for any inclusion I J of two nonempty subsets of 0, 1, 2 we have isomorphisms ⇢ { } (7.1) M R = M I ⌦RI J ⇠ J Proof. P2 is a union of three affine open sets X0,X1,X2 and all intersections are also affine Xi Xj = Xij, XI XJ = XI J . Furthermore, XJ = Spec(RJ ) for all nonempty J 0, 1, . \ \ [ ✓{ } By the theorem, any quasi-coherent sheaf on XJ has the form MJ for some RJ -module MJ . The isomorphisms (7.1) are exactly what is needed to paste these together. ⇤ f Definition 7.3.3. An oX -module F is called a coherent sheaf if X has a covering by affine open sets U = Spec(R ) so that F U = M for some R -module M . ↵ ↵ | ↵ ⇠ ↵ ↵ ↵ It can be shown that, for projective space, these affine open sets can be taken to be the f open set Xi. So, the quasi-coherent sheaf over Pn given above is coherent if and only if each Mi is a finitely generated Ri module.
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