Page "Derived functor" Paragraph 34

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 M < sup > G </ sup > 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 < sup > i </ sup >( G, M ).