Derived Functor - Applications

Applications

Sheaf cohomology. If X is a topological space, then the category of all sheaves of abelian groups on X is an abelian category with enough injectives. The functor which assigns to each such sheaf L the group L(X) of global sections is left exact, and the right derived functors are the sheaf cohomology functors, usually written as H i(X,L). Slightly more generally: if (X, O_X) is a ringed space, then the category of all sheaves of O_X-modules is an abelian category with enough injectives, and we can again construct sheaf cohomology as the right derived functors of the global section functor.

Étale cohomology is another cohomology theory for sheaves over a scheme.

Ext functors. If R is a ring, then the category of all left R-modules is an abelian category with enough injectives. If A is a fixed left R-module, then the functor Hom(A,-) is left exact, and its right derived functors are the Ext functors Ext_Ri(A,B).

Tor functors. The category of left R-modules also has enough projectives. If A is a fixed right R-module, then the tensor product with A gives a right exact covariant functor on the category of left R-modules; its left derivatives are the Tor functors TorR_i(A,B).

Group cohomology. Let G be a group. A G-module M is an abelian group M together with a group action of G on M as a group of automorphisms. This is the same as a module over the group ring ZG. The G-modules form an abelian category with enough injectives. We write MG for the subgroup of M consisting of all elements of M that are held fixed by G. This is a left-exact functor, and its right derived functors are the group cohomology functors, typically written as H i(G,M).