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Sheaf Cohomology Sheaf Cohomology In this note we give the background needed to de…ne sheaf cohomology. In particular, we prove the following two facts. First, the category Ab(X) of sheaves on a topological space X has enough injectives. Second, if is the global function functor from Ab(X) to the category of Abelian groups, we show that is a left exact functor. Therefore, the right derived functors Rn() exist, and we can then de…ne the cohomology groups Hn(X; ) = Rn()( ) for any F F sheaf on X. Sheaf cohomology is one of the major tools of modern algebraic geometry, and F it is a good example to help justify learning homological algebra over an arbitrary Abelian category instead of restricting to categories of modules. As we will see, several facts come from producing an adjoint pair of functors, including the proof that is left exact. While one can prove that is left exact in a straightforward manner, we choose to prove it with the help of adjoint functors to illustrate the power of adjoint pairs. There are subtleties in working with sheaves and presheaves. Since every sheaf is a presheaf, we can view the category of sheaves as a subcategory of the category of presheaves. It is not hard to show that the global section functor is exact in the category of presheaves, but it is not exact in the category of sheaves. This is due to the fact that cokernels are di¤erent in the two categories even though morphisms are not. We will give an example to demonstrate this. In fact, sheaf cohomology is nontrivial exactly because the global section functor is not exact. Therefore, one can say that sheaf cohomology exists since cokernels are di¤erent in these two categories. We now recall the de…nitions of presheaf and sheaf. Let Top(X) be the category of open sets of a topological space X, where the morphisms are inclusions. That is, hom(U; V ) consists of the inclusion map U V if U V , and hom(U; V ) is empty otherwise. A presheaf ! (of Abelian groups) on X is a contravariant functor from Top(X) to the category of Abelian F groups with (?) = 0. Therefore, if is a presheaf on X, then for every open set U there F F is a group (U), and for every inclusion U V , there is a map resU;V : (V ) (U), such F F !F that for open sets U V W , the composition resU;V resV;W : (W ) (V ) (U) is F !F !F equal to resU;W . The maps resU;V is generally called restriction maps, and resU;V (f) is often written f V . This terminology is due to the following examples. j Example 1. Let X be a topological space. De…ne by (U) = C(U; C), the group of F F continuous functions from U to C, for any open set U of X. If U V , then restriction of functions gives a group homomorphism (V ) (U). Thus, is a presheaf. F !F F 1 Example 2. Let X = C. De…ne on X by (U) is the set of all analytic complex valued A A functions de…ned on U. Again, for V U, the map (U) (V ) is the restriction of A !A functions map. Then is a presheaf. A Example 3. Let X be an algebraic variety over an algebraically closed …eld k. De…ne (U) to be the ring of regular functions on U. That is, an element of (U) is a function O O f : U k such that for every x U there is an open neighborhood V of x such that f is ! 2 the quotient of polynomial functions on V . If U V , then restriction of functions gives the map (V ) (U). The presheaf is called the sheaf of regular functions on X. O !O O Example 4. Let X = x be a one point topological space. If is a presheaf on X, then f g F we have an Abelian group A = (X), and no other groups associated to since the only F F nonempty open subset of X is X itself. Furthermore, the only nonidentity restriction map is the trivial map A = (X) 0 = (?). Thus, a presheaf on is really nothing more F ! F F than an Abelian group. We now give the de…nition of a sheaf. De…nition 5. Let be a presheaf on X. Then is a sheaf provided that the following two F F conditions hold: (i) if U is an open set on X, if Ui is an open cover of U, and if f (U) f g 2 F such that f U = 0 for all i, then f = 0; and (ii) if Ui is an open cover of U and if there are j i f g fi (Ui) satisfying fi Ui Uj = fj Ui Uj for all i; j, then there is an f (U) with f Ui = fi 2 F j \ j \ 2 F j for all i. It is not hard to see that the presheaves of the examples above are in fact sheaves. In fact, the properties of the de…nition of a sheaf are motivated by examples involving functions, since functions clearly have the properties of the de…nition. Not all presheaves are sheaves, as we can see in the following example. Example 6. Let A be an Abelian group. If X is a topological space, then we may de…ne a presheaf A by A(U) = A for all U, and the restriction maps are all just the identity map on A. This presheaf is sometimes called the constant presheaf on X associated to A. It is clear that A ise a presheaf.e However, it need not be a sheaf. For example, suppose that X is the disjoint union of two open sets U1 and U2; that is, X is not connected. If a; b are distinct elementse of A, then a A(U1) and b A(U2) do not glue to give an element of A(X) even 2 2 though the compatibility axiom a U1 U2 = b U1 U2 is trivially true since U1 U2 = ? and j \ j \ \ A(?) = 0. It is true, however,e that if X eis connected, then A is a sheaf. e Example 7. Let A be an Abelian group, made into a topological space by giving it the e discrete topology. If X is a topological space, then we de…ne the constant sheaf associated to A by (U) = C(U; A), the group of continuous functions from U to A. The restriction A maps are given by ordinary function restriction. It is straightforward from the de…nition to see that is a sheaf. Note that if a A, then the constant function fa : X A given by A 2 ! fa(x) = a is an element of (X), and so fa U (U) for any U. In fact, if U is connected, A j 2 A then (U) = fa : a A = A. It is this fact that gives the name constant sheaf to . A f 2 g A 2 We will have categories of presheaves and of sheaves once we de…ne morphisms of presheaves. De…nition 8. A morphism ' : of presheaves is a natural transformation of functors. F!G That is, for every open set U, there is a group homomorphism ' : (U) (U), and for U F !G every inclusion V U, the following diagram commutes. ' (U) U - (U) F G res res ? ? (V ) - (V ) F 'V G Example 9. Let U be an open set of a topological space X. If is a sheaf on X, we get F another sheaf on X as follows. For V an open set in X, set (V ) = (U V ). If we G G F \ de…ne ' : by ' : (V ) (V ) = (U V ) is the restriction map, then the F!G V F !G F \ compatibility of the restriction maps shows that ' is a morphism of sheaves, as is indicated by the following commutative diagram. (V ) - (U V ) F F \ ? ? (W ) - (U W ) F F \ Example 10. Let A be an Abelian group and X a topological space. Looking at the constant sheaves and presheaves of Examples 6 and 7, we have a natural morphism of presheaves ' : A , given by ' (a) = fa U for any a A and any open U. !A U j 2 If and are sheaves, then a morphism ' : of sheaves is just a presheaf e F G F!G morphism. We will write homX ( ; ) for the group of all sheaf homomorphisms from F G F to . We thus have two categories associated to a topological space X, the category of G presheaves on X and that of sheaves on X. There is the obvious inclusion functor i from sheaves on X to presheaves on X. It is clear that i is an additive functor. Let ' : be a morphism of presheaves. Then we de…ne the following presheaves F!G ker('): U ker(' ); 7! U coker('): U coker(' ); 7! U im('): U im(' ): 7! U From these de…nitions it is easy to see that a sequence 0 0 of presheaves is !F!G!G! exact if and only if each sequence 0 (U) (U) (U) 0 is exact. In particular, if !F !G !H ! is the global section functor on presheaves; i.e., ( ) = (X), then is an exact functor. F F As we will see, the global section functor on sheaves is only left exact. The di¤erence between these categories with respect to comes from the following facts: if and are sheaves, F G 3 then ker(') is a sheaf; however, coker(') and im(') need not be sheaves.
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