Stable Module Category - Connections With Cohomology

Connections With Cohomology

Now we suppose that R = kG is a group algebra for some field k and some group. G. One can show that there exist isomorphisms

for every positive integer n. The group cohomology of a representation M is given by where k has a trivial G-action, so in this way the stable module category gives a natural setting in which group cohomology lives.

Furthermore, the above isomorphism suggests defining cohomology groups for negative values of n, and in this way, one recovers Tate cohomology.