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 cohomology groups and homology groups.

They are defined as follows:

Value of $n$	Definition of ${\hat {H}}^{n}(G,A)$
$n\geq 1$	Same as the cohomology group $H^{n}(G,A)$
$n\leq -2$	Same as the homology group $H_{m}(G,A)$ where $m+n=-1$.
$n=0$	Cokernel of the map $N:H_{0}(G,A)\to H^{0}(G,A)$ that is obtained by passing to homology/cohomology classes the following map at the level of cycles and cocycles: an element $a\in A$ (a 0-cycle) is sent to $\sum _{g\in G}\varphi (g)(a)$ (a 0-cocycle).
$n=-1$	Kernel of the map $N$ defined for the $n=0$ case.