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 Fp the finite field on p elements, we also have that H*(G,M) = Ext*
Fp(Fp, 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.
Read more about this topic: Ext Functor
Famous quotes containing the words interesting and/or examples:
“What is most interesting and valuable in it, however, is not the materials for the history of Pontiac, or Braddock, or the Northwest, which it furnishes; not the annals of the country, but the natural facts, or perennials, which are ever without date. When out of history the truth shall be extracted, it will have shed its dates like withered leaves.”
—Henry David Thoreau (1817–1862)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)