2 Sheaves and Cohomology 2.1 Sheaves and Presheaves We fix a topological space X. Later we will include assumptions that are satisfied by smooth manifolds. 2.1.1 Definitions and Examples Definition 2.1. A presheaf of abelian groups F on X assigns to each open U ⊆ X an abelian group F (U) = Γ(U; F ) and for every inclusion of open sets V ⊆ U a homomorphism of abelian groups F ρUV : F (U) ! F (V ), often called the restriction map, satisfying F 1 [P1] ρUU = F(U) F F F [P2] for W ⊆ V ⊆ U, we have ρVW ◦ ρUV = ρUW . If F and G are two presheaves (of abelian groups) on X, then a morphism ' : F ! G consists of the data of a morphism 'U : F (U) ! G (U) for each open set U ⊆ X such that if V ⊆ U is an inclusion, then we have commutative diagrams 'U F (U) / G (U) F G ρUV ρUV (V ) / (V ): F 'V G Remark 2.2. We may form a category TopX whose objects are open sets in X and whose mor- phisms are simply inclusions of open sets. Then the above definition says that a presheaf is a ◦ contravariant functor TopX ! Ab, and that a morphism of presheaves is a natural transforma- tion of the associated functors. Definition 2.3. A sheaf F of abelian groups on X is a presheaf which, for any open set U ⊆ X and any open covering fUigi2I of U, satisfies the two additional properties: [S1] if s 2 F (U) is such that sjUi = 0 for all i 2 I, then s = 0; [S2] if si 2 F (Ui) such that sijUi\Uj = sjjUi\Uj for all i; j 2 I, then there exists s 2 F (U) such that sjUi = si for each i. Note that such an element is unique by [S1]. A morphism of sheaves is a morphism of the underlying presheaves. Let A be a sheaf of rings on X, i.e. one for which A (U) is a ring for each open U ⊆ X and for which the restriction maps are ring homomorphisms. If F is a sheaf of abelian groups such that for every open U ⊆ X, F (U) is an A (U)-module, then F is called a sheaf of A -modules. Examples 2.4. (a) The constant presheaf Let X be a topological space and let A be an abelian p group. We define the constant presheaf with values in A to be the presheaf AX to be the presheaf such that p AX (U) = A Ap X 1 for all non-empty U ⊆ X, and ρUV = A for all V ⊆ U. (b) The constant sheaf With X and A as above, we define the constant sheaf with values in A to be the sheaf AX whose sections AX (U) over U are locally constant functions U ! A; this is the same as the set of continuous functions U ! A, where A is given the discrete topology. 1 0 0 0 (c) Functions Let C = CX be the sheaf for which C (U) is the set of continuous R-valued functions on U. Then C0 is a sheaf. Similarly, if X is a smooth (respectively, complex) mani- 1 1 fold, then CX (respectively, OX ) is a sheaf, where CX (U) (respectively, OX (U)) is the ring of smooth (respectively, holomorphic) functions on U. These are all sheaves of rings. (d) Skyscraper sheaves Let X and A be as above, and fix x 2 X. Then we can define a sheaf Ax by ( A if x 2 U Ax(U) := 0 otherwise. 0 In the case that that A = C, then Cx is a sheaf of CX -modules, where a function acts on a section simply by multiplying by its value at x. (e) Vector bundles Let X be as above and let π : E ! X be a vector bundle over X. Then for any U ⊆ X, U 7! Γ(U; E) defines a sheaf. For a vector bundle E, we will typically also denote its sheaf of sections by E, as no confusion is likely to occur. By Remark , E is a sheaf of C0-modules; it is a sheaf of 1 CX -modules or OX -modules if E is smooth or holomorphic, respectively. 2.1.2 Stalks We first recall the notion of a colimit. Definition 2.5. Let I be a small category (i.e., Ob I is a set) and C any category. Let α : I ! C be a functor; we thus, think of this data as a set of objects of C indexed by I , with morphisms between these objects indexed by morphisms in I . Any object X 2 Ob C determines such a functor ∆X : I ! C , namely, the constant functor, i.e., ∆X takes all objects of I to X and all morphisms to 1X . If Y 2 Ob C is another object and f : X ! Y a morphism, then there is an induced natural transformation Φ = Φf : ∆X ! ∆Y which is simply f at each object of I . We say that α has a colimit X 2 Ob C if there is a natural transformation λ : α ! ∆X such that if Y 2 Ob C and τ : α ! ∆Y is a natural transformation, then there is a unique f : X ! Y such that α λ τ τ = Φf ◦ λ. Ó ∆X / ∆Y Φf commutes. [The colimit is often called the “direct limit,” but for the sake of standardizing lan- guage, “colimit” is preferable.] It is not hard to see that if a colimit exists, then it is unique up to a unique isomorphism. Lemma 2.6. If I is a small category and α : I ! Ab is an functor, then colim α exists. L Proof. For i 2 Ob I , we will write Xi for α(i). Then let Xe := i2Ob I Xi be the direct sum and let N ⊆ Xe be the subgroup generated by elements of the form (··· ; 0; xi; ··· ; −α(f)(xi); ··· ) for morphisms f : i ! j. It is left as an exercise to show that X := X=Ne is the colimit. L Remark 2.7. From the construction above, any element of colim α is a tuple in Xi with finitely many non-zero components, corresponding to i1; : : : ; ir 2 Ob I . Suppose that for any such i1; : : : ; ir, there is j 2 Ob I with arrows ip ! j for 1 ≤ p ≤ r. Then by using the relation in the proof above, one can represent any element of colim α by one in the image of Xj ! X. 2 Let X be a topological space and fix a point x 2 X. Recall that TopX was the category whose objects are open sets in X and whose morphisms are inclusions. Consider the (full) subcategory ◦ TopX;x whose objects are open neighbourhoods of x. It is easy to see that TopX;x has the property mentioned in Remark 2.7, since the intersection of any finite set of open neighbourhoods of x is also one. ◦ Now, if F is a presheaf, which we may think of as a functor F : TopX ! Ab, then we can ◦ restrict it to F : TopX;x ! Ab. By Lemma 2.6, Fx := colimTopX;x F = colimx2U F (U) exists; we call it the stalk of F at x. By properties of the colimit, for each open neighbourhood U of x, we have a natural map F (U) ! Fx and these “are compatible” with the restriction maps. Remark 2.7 says that any element of Fx is the image of some s 2 F (U) for some neighbourhood U of x. We may repeat the above definition for an arbitrary set S ⊆ X, instead of fxg, by defining the subcategory TopX;S of TopX with objects open sets U ⊆ X with S ⊆ U. In this way, we can extend the presheaf F to all subsets of X by F (S) := colimS⊆U F (U): 2.2 Sheafification Proposition 2.8. Let F be a presheaf on X. Then there exists a sheaf F + together with a mor- phism θ : F ! F + such that if ' : F ! G is any morphism and G is a sheaf, then there is a unique morphism Φ: F + ! G such that F + θ 8 F Φ ' = Φ ◦ θ ' & G commutes. Definition 2.9. The sheaf F + is called the sheafification of F . Remark 2.10. If F is a sheaf, then θ : F ! F + is an isomorphism. Proof. We define F + as follows. Let U ⊆ X be open. We let F +(U) be the set of elements Y σ 2 Fx x2U such that for each x 2 U there is an open neighbourhood V of x contained in U and a section s 2 F (V ) such that pry(σ) = sy for all y 2 V , where pry is the obvious projection map. We leave it as an exercise to show that this is a sheaf. Q Given s 2 F (U), there is a unique σ 2 x2U Fx, where prx(σ) = sx; clearly, we have σ 2 F +(U) and so we set θU (s) := σ: It is an exercise to verify that this defines θ. Finally, we also leave as an exercise the proof of the mapping property. 3 2.3 Kernels and Images Definition 2.11. If F is a presheaf, then a subpresheaf G of F is a presheaf for which G (U) is a subgroup of F (U) for all open U ⊆ X and for which the restriction morphisms are induced from those of F .A subsheaf of a sheaf is simply a subpresheaf which is also a sheaf.