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.
Read more about this topic: Stable Module Category
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