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
Famous quotes containing the words connections with and/or connections:
“Growing up human is uniquely a matter of social relations rather than biology. What we learn from connections within the family takes the place of instincts that program the behavior of animals; which raises the question, how good are these connections?”
—Elizabeth Janeway (b. 1913)
“I have no connections here; only gusty collisions,
rootless seedlings forced into bloom, that collapse.
...
I am the Visiting Poet: a real unicorn,
a wind-up plush dodo, a wax museum of the Movement.
People want to push the buttons and see me glow.”
—Marge Piercy (b. 1936)