Tate cohomology group

Definition
Suppose $$G$$ is a finite group, and $$A$$ is an abelian group, and $$\varphi:G \to \operatorname{Aut}(A)$$ is a homomorphism of groups, making $$A$$ into a $$G$$-module (i.e., $$G$$ acts on $$A$$).

The Tate cohomology groups $$\hat{H}^n(G,A)$$ are a collection of groups for $$n$$ varying over all integers (both positive and negative) that combine information about the defining ingredient::cohomology groups and defining ingredient::homology groups.

They are defined as follows: