The lemma below, called the projection formula, is very useful to compute a pushforward. Lemma 51 (Projection formula). Let f :(X, ) (Y, ) be a morphism of ringed OX ! OY spaces. Let be an -module, be a locally free -module of finite rank. Then there F OX E OY is a natural isomorphism f ( f ⇤ ) f . ⇤ F⌦OX E ' ⇤F⌦OY E Proof. Since f ⇤(f ) f ⇤f f ⇤ , there is a natural morphism f ⇤f f ⇤ ⇤F⌦E ' ⇤F⌦ E ⇤F⌦ E! f ⇤ .Sincef ⇤ and f are adjoint to each other, there is a corresponding natural F⌦ E ⇤ morphism ' : f f ( f ⇤ ). Note that being ' an isomorphism is a local ⇤F⌦E! ⇤ F⌦ E property, hence, we may replace Y by a sufficiently small open subset. Since f and the ⇤ tensor product commutes with the direct sum, we may assume furthermore that = Y . 1 E O The conclusion follows from the fact that f ⇤ Y = f − Y f 1 X = X . O O ⌦ − OY O O Note that the category of -modules is an abelian category: we can talk about kernel, OX cokernel, and so on. In particular, we are able to consider (cochain) complexes of - OX modules, and do homological algebra. Fiberwisely, vector bundles on give vector OX spaces; and similarly, vector spaces also form an abelian category. However, when we focus on their global nature (= vector bundles themselves, or equivalently, locally free sheaves), they do not form an abelian category. Example 52. Let X = P1 =ProjC[x, y], and let Z = V (x) X be a closed subset of ⇢ X. The ideal sheaf IZ is indeed locally free: On an affine open subset Ux Spec C[y/x], which corresponds to the set of 1 ' points [x : y] P where x = 0, we have IZ (Ux)=C[y/x]sincethereisno 2 6 condition. On the other hand, on an affine open subset Uy Spec C[x/y], ' which corresponds to the set of points [x : y] P1 where y = 0, an element of 1 2 6 IZ (Uy) is a regular function on P [1 : 0] which has zero at Z = [0 : 1]. In x \{ } particular, IZ (Uy) C[x/y]. We conclude that IZ is locally free, since ' y · it can be covered by open (affine) subsets such that on IZ is free on each open subset. We have a natural inclusion I , whose cokernel is isomorphic to which is Z !OX OZ not a locally free sheaf on . OX Hence, it is natural to extend the category of locally free sheaves by adding more objects so that the cokernels of maps of locally free sheaves appear as objects in the enlarged category. In most places, we are in the (locally) Noetherian setting, so that the finite rank case will behave much better. Therefore, it is also reasonable to extend the category of locally free sheaves of finite rank in this manner. In general, there are three finiteness conditions for an A-module M: (1) finitely generated, there is a surjection A n M 0 for some integer ⊕ ! ! n; 21 Topic 1 : Sheaf (2) finitely presented, so that the finite number of generators have only finitely many relations, i.e., there is a right exact sequence A m ⊕ ! A n M 0 for some integer m, n; ⊕ ! ! (3) coherent, M is finitely generated, and any homomorphism A n M ⊕ ! (not necessarily surjective) has the finitely generated kernel. When A is a Noetherian ring, all these conditions are equivalent. Since we will only work over Noetherian schemes (= schemes which can be covered by a finite number of open affine subschemes Spec Ai where Ai are Noetherian), or even better: affine/projective varieties over a field (of finite dimension), we do not need to distinguish them. Definition 53. Let (X, ) be a ringed space. An -module is quasi-coherent it OX OX F has a local presentation, that is, for every point x X, there is an open neighborhood 2 U X and a right exact sequence ✓ I J ⊕ ⊕ 0 OX |U !OX |U !F|U ! for some (possibly infinite) set I and J. In other words, a quasi-coherent sheaf is locally a cokernel of (locally) free sheaves. An -module is coherent if it is quasi-coherent, and locally coherent, that is, it OX F satisfies the following properties: (i) is of finite type, i.e., for every point x X, there is an open neigh- F n2 borhood U X and a surjection ⊕ 0 for some integer ✓ OX |U !F|U ! n; (ii) for any open subset U X, any integer n, and any morphism ' : n ✓ ⊕ ,thekernelker' is of finite type. OX |U !F|U As we discussed above, the notion becomes much simpler for noetherian schemes. We start by defining the sheaf of modules M on Spec A associated to an A-module M. Definition 54. Let A be a ring and letfM be an A-module. The sheaf associated to M, denoted by M,isdefinedby D(s) M 7! s f for each s A and the distinguished open subset D(s) Spec A, together with the 2 ✓ natural restriction maps. In particular, the stalk at p Spec A is isomorphic to the localized module (M) = M , 2 p p and Γ(Spec A, M)=M. We first see that the map M M is functorial: f f 7! Proposition 55. Let A, B be rings, ' : B A be a ring homomorphism, and let f ! f :SpecA Spec B be the corresponding morphism of spectra. Then, ! (1) the map M M gives an exact, fully faithful functor from the category of A-modules 7! to the category of -modules; OSpec A f 22 (2) if M and N are two A-modules, then (M A N)⇠ M N; ⌦ ' ⌦OSpec A (3) if M afamilyofA-modules, then ( M ) M ;f e { i} ⊕ i ⇠ ' i (4) for any A-module M, we have f (M) (BM)⇠, wheref BM means M considered as ⇤ ' B-modules; f (5) for any B-module N, we have f (N) (N A) . ⇤ ' ⌦B ⇠ Proof. Note that the map M M givese a functor since a homomorphism of A-modules 7! M N commutes with the localization M N for each s A. It is exact, since ! s ! s 2 it commutes with every localizationf at a prime ideal p A, and the exactness can be ⇢ measured at the stalks. It is also fully faithful: the functor ⇠ gives a natural map HomA(M,N) Hom (M,N); • ! OSpec A the global section functor Γ(Spec A, ) gives the inverse map Hom (M,N) • − fOSpece A ! HomA(M,N). f e In particular, we embed the category of A-modules into the category of -modules. OSpec A All the other statement follow from the definitions and statements in terms of modules. Definition 56. Let (X, ) be a scheme. An -module is quasi-coherent if X OX OX F can be covered by affine open subsets U =SpecA such that M for some A - i i F|Ui ' i i module M for each i. is coherent if furthermore each M can be taken to be a finitely i F i generated Ai-module. f Example 57. (1) Let X be any scheme. The structure sheaf is quasi-coherent and OX coherent. (2) Let X =SpecA be an affine scheme, Y =SpecA/I be the closed subscheme of X defined by an ideal I A, together with a closed immersion i : Y X. The sheaf ✓ ! i Y is a quasi-coherent (and coherent) X -module, which is isomorphic to (^A/I). ⇤O O (3) Let Y be a closed subscheme of a scheme X with the inclusion i : Y, X.The ! sheaf = i 1 is not a quasi-coherent sheaf on Y in general. In fact, it needs OX |Y − OX not to be an -module. OY n (4) Let X = A =Speck[x1, ,xn]. Let be the constant sheaf associated to the ··· K group K = k(x , ,x ) which is the function field (= the field of rational functions) 1 ··· n of X.Then is a quasi-coherent -module, but not coherent unless n = 0. K OX Maybe the following proposition seems to be too strong, and in fact, we have to check carefully that their global property comes from their localizations. The idea of proof is almost the same as using the “partition of unity” when we define the structure sheaf of a spectrum, so we will skip its proof. OSpec A 23 Topic 1 : Sheaf Proposition 58. Let X be a scheme. An -module is quasi-coherent if and only OX F if for every affine open subset U =SpecA of X, there is an A-module M such that M.IfX is noetherian, then is coherent if and only if furthermore we can take F|U ' F M as a finitely generated A-module. f We mentioned several times that the global section functor X Γ(X, ) is left exact. 7! − The sheaf cohomology we will see later measures how Γ(X, ) is di↵erent from the exact − functor. One distinguished example we seen is a flasque sheaf, that is, a sheaf such that every restriction map is surjective. When is flasque, and we have a short exact F 0 sequence 0 0, then we check that the induced maps on global !F0 !F!F00 ! sections again form a short exact sequence 0 Γ(X, 0) Γ(X, ) Γ(X, 00) 0. ! F ! F ! F ! Naively speaking, a flasque sheaf forces that the global section functor becomes “exact”. Hence, it is the “simplest” sheaf in this viewpoint.