Ext Functor - Interesting Examples

Interesting Examples

If Z is the integral group ring for a group G, then Ext*
Z(Z, M) is the group cohomology H*(G,M) with coefficients in M.

For F_p the finite field on p elements, we also have that H*(G,M) = Ext*
F_p(F_p, M), and it turns out that the group cohomology doesn't depend on the base ring chosen.

If A is a k-algebra, then Ext*
A ⊗_k Aop(A, M) is the Hochschild cohomology HH*(A,M) with coefficients in the A-bimodule M.

If R is chosen to be the universal enveloping algebra for a Lie algebra, then Ext*
R(R, M) is the Lie algebra cohomology with coefficients in the module M.