The Cohomology of Sheaves

A fast introduction to the the construction of the cohomology of sheaves pioneered by A. Grothendieck and J.L. Verdier. The approach is from the point of view of derived categories, though this concept is never mentioned. 1. Sheaves Let R commutative ring with 1. We denote by RMod the category of R-modules. For any topological space X we denote by OpenpXq the collection of open subsets. It can be organized as a category in which the morphisms are given by inclusions. A pre-sheaf of R-modules is a contravariant functor S : OpenpXq Ñ RMod, OpenpXq Q U ÞÑ ΓpU, Sq P RMod. For any inclusion U € V , we have a restriction map r SUV : ΓpV, Sq Ñ ΓpU, Sq. The module ΓpU, Sq is called the module of continuous sections of S over U . If s P ΓpV, Sq, V € U , we set s|U : rUV psq when there is no danger of confusion. A morphism of sheaves of R-modules S0 and S1 is a collection of morphisms of R-modules φU : ΓpU, S0q Ñ ΓpU, S1q compatible in the obvious way with the restriction maps. We obtain a category PshRpXq of presheaves of R-modules over X. For any morphism of presheaves φ : S0 Ñ S1 we can define a kernel presheaf kerφ and an image presheaf. For a sub presheaf S0 ãÑ S1 we can define the quotient presheaf, S1{S0. We can also define the direct sum of two presheves. Rigourously, PshRpXq is an Abelian category in an obvious fashion. A preseheaf S P PshRpXq is called a sheaf if for any open cover pUiqiPI of X, and for any collection of sections si P ΓpUi, Sq such that si|UiXUj si|UiXUj , @i, j there exists a unique section s P ΓpX, Sq such that s|Ui si, @i. To any presheaf S P PshRpXq we can associate a canonical sheaf rS P ShRpXq as follows. For x P X we define the stalk of S at x to be the inductive limit Sx : lim ÝÑUQx ΓpU, Sq. Note that we have natural morphisms γu : ΓpU, Sq Ñ Su, u P U. For s P ΓpU, Sq and u P U the element γupsq P Su is called the germ of s at u. We define ΓpU,rSq to be the submodule of ±uPU Su consisting of collections psuquPU satisfying the conditions su P Su, @u P U. @u P U, DV P OpenpUq, Ds P ΓpV, Sq such that u P V and γvpsq sv, @v P V . Date: Sometime during the spring of 2008. 1 2 LIVIU I. NICOLAESCU The correspondence U ÞÑ ΓpU,rSq defines a sheaf called the sheafification of S. If φ is a morphism of sheaves, its kernel is also a sheaf. However, its image is only a preshef, and we define the image sheaf to be the sheafification of the image presheaf. The quotient of two sheaves is a presheaf, and we define the quotient sheaf to be the sheafification of the quotient presheaf. We obtain in this fashion an (Abelian) category ShRpXq of sheaves of R-modules on X. 2. Complexes of sheaves We denote by C pXq the category of bonded from below complexes of sheaves, i.e., complexes of sheaves pS `nPZS, dq such that Sn 0 if n ! 0. Let K pXq be the category whose objects are bounded from below complex of sheaves on X, but whose morphisms are the homotopy classes of cochain maps. For a complex pA , dAq P C pXq and k P Z we denote by pA rks, dArksq the complex defined by Arks : A , dArks p 1q dA. The cohomology of a complex of sheaves pA , dAq is the direct sum of sheaves H ppAq à n HpAq, HpAq : kerpdA : A Ñ A q{ Impφ : A 1 Ñ Aq. A complex is called acyclic if its cohomology is trivial. A morphism of complexes is called a quasiisomorphism (qis for brevity) if it induces an isomorphism in cohomology. We will use the notation A φ ù B to denote a qis. Observe that any sheaf A P ShRpXq can be tautologically identified with a complex A where A0 A, An 0, @n 0. We will denote by rAs this complex. A resolution of A is then a qis rAs φ ù pS , dSq. A morphism of complexes of sheaves φ : pA , dAq Ñ pB , dBq determines a new complex Conepφq called the cone of φ defined by Conepφq : B `Ar1s and differential dφ : dB φ 0 dAr1s . The cone fits in the middle of a short exact sequence of complexes 0 Ñ B i Ñ Conepφq π Ñ Ar1s Ñ 0 where i and π denote respectively the canonical inclusion and projection. From the above short exact sequence we obtain the following result. Proposition 2.1 (The cone trick). Suppose φ : pA , dAq Ñ pB , dBq is a morphisms of complexes. We have a long exact sequence of sheaves φ ÝÑ HpBq i Ñ HpConepφqq π Ñ H pAq φ ÝÑ H pBq i Ñ . In particular, φ is a qis if and only if Conepφq is acyclic. [ \ Definition 2.2. Let A ,B P C pXq. (a) A left roof from A to B is a diagram of morphisms of complexes of sheaves of the form C s }} }= }= }= }= f !! B B B B B B B B