Group cohomology valued in a bimodule

Question

The usual setup for group cohomology of a group $G$ is as follows. One takes a $G$-module $M$, and considers the space of all maps $$\ell : G \times \cdots \times G \longrightarrow M $$ together with the coboundary operator $$(\delta\ell)(g_1, \dots, g_{n+1}) = g_1 \cdot \ell(g_2, \dots, g_{n+1}) +\sum_{i=1}^{n}(-1)^i\ell(g_1, \dots, g_ig_{i+1}, \dots, g_{n+1}) + (-1)^n \ell(g_1, \dots, g_n)$$ where $\ell$ is a $n$-cochain.

It seems that this has a completely generalization to the case that $M$ is a $G$-$G$-bimodule, so an abelian group $M$ with commuting left and right actions of $G$. In this case, one just has to replace the last term in the formula for the boundary operator with $$(-1)^n \ell(g_1, \dots, g_n) \cdot g_{n+1}.$$ The case discussed above is then the the special case where the right action on $M$ is trivial.

Is there a particular reason I cannot find this general description anwhere in the literature? It may be that this just never comes up in practice (although I know examples where it does), or maybe it is somewhat useless because it is impossible to compute?

Or is there some general reason that I am unaware of, like, does this general version somehow reduce to the standard version?

Answer 1

It looks to me like your cochain complex agrees with the Hochschild cochain complex of the group ring $Z[G]$ with coefficients in the bimodule $M$. I think Hochschild cohomology is the generalization you're looking for.

One has to be a bit careful if $M$ isn't flat over $Z$—there is a variant of Hochschild cohomology called "Shukla cohomology" which is intended to deal with these flatness issues. Let me avoid those subtleties and simply assume that $M$ is a $k$-linear representation of $G$, where $k$ is a field. Then the cohomology of your cochain complex is the Hochschild cohomology $HH^*(k[G]; M)$ of the group algebra $k[G]$.

You can ask what the this Hochschild cohomology turns out to be, when the right action of $G$ on $M$ is nontrivial, so you aren't just getting ordinary group cohomology. At least for finite $G$, with coefficients in $k[G]$ itself, the Hochschild cohomology $HH^*(k[G]; k[G])$ splits as a direct sum of the group cohomology of centralizer subgroups of $G$: $$HH^*(k[G]; k[G]) \cong \bigoplus_{\langle g\rangle} H^*(C(g); k), $$ where the direct sum is taken over conjugacy classes in $G$, and $C(g)$ is the centralizer subgroup of $g$ in $G$. I think I remember correctly that this was originally proven by Burghelea in characteristic zero, and Loday in arbitrary characteristic, but both wrote about homology rather than cohomology, since they were motivated by cyclic homology. For a clear statement of this splitting for Hochschild cohomology (rather than homology), see Witherspoon's book "Hochschild cohomology for algebras," section 9.5.