What is the motivation to introduce Tate-cohomology groups ?
Let $G$ be a Galois group and $M$ be a $G-$module. Let $H^n(G,M)$ be usual Galois cohomology.
In group cohomology theory, we often introduce Tate-cohomology $\hat{H^n}(G,M)$in the area of number theory.
For $n\ge 1$, notion of usual Galois cohomology and Tate-cohomology coincides, but when $n=0$, it does not.
What is the motivation to introduce the notion, Tate-cohomology ?