Group Cohomology

Let G be a group. We can form the group ring Z[G] over G; by definition it is the set of formal finite sums ∑ aigi, where ai ∈ Z, gi ∈ G, and multiplication is defined in the obvious manner. We shall call an abelian group A a G-module if it is a left Z[G]-module. This means, of course, that there exists a homomorphismG→ AutZ(A). We can also makeA into a right Z[G]-module simply by writing ag := g−1a for a ∈ A, g ∈ G. This is important for tensor products. An example of a G-module is any abelian group with trivial action by G. For instance, we shall in the future denote by Z the integers with trivial G-action. Finally, if A and B are G-modules, then a G-homomorphism between them is a map φ : A→ B which is a Z[G] homomorphism. The set of G-homomorphisms between A and B is denoted by HomG(A,B). It is a left exact functor of A and B, covariant in B and contravariant in A. As usual its derived functors are denoted by Ext. Let A be a G-module. Then we define the cohomology groups as