Let $G$ be a free group of finite rank. Consider any commutative ring $R$ containing $\mathbb{Z}$. Consider the group ring $RG$.


Q) What can we say about the Hochschild cohomology groups of $RG$ with coefficients in $R$?


I can find the $HH^1(RG)$ in terms of outer derivations but again I don't know the complete description of derivations on a free group. Also, I could not find a single element of $HH^n(RG)$ for $n\geq 2$.


Any comment, reference, or suggestion will be extremely helpful. Thanks in advance.



Hochschild cohomology of group ring of a free group
Hochschild cohomology of group rings is reducible to ordinary group cohomology. (Not really if you're interested in multiplicative structure, but that's another story). Everyting works over any commutative base ring, you can assume it's $\Bbb Z$ if you'd like.


Namely, there's an adjunction between $kG-bimod$ and $kG-mod$ (I mean left modules): forgetful right adjoint functor $\rho$ sends a bimodule $A$ to the module $\rho A$ where action is conjugation, and left adjoint $\lambda$ is $M \mapsto \lambda M := M \otimes\_k kG$. Right action is multiplication in $kG$, and left action is $h \cdot \sum\_G m \otimes g = \sum\_G hm \otimes hg$.


So, now we can get a string of identifications


$$HH^\*(kG, A) = Ext\_{bimod}(kG, A) = Ext\_{bimod}(\lambda k, A) = Ext\_{mod}(k, \rho A) = H^\*(G, \rho A)$$


In particular, Hochschild cohomological dimension of a group ring is equal to comohomological dimension of a group itself, and $HH^\*(kG, k) = H^\*(G, k)$.
(so if your group is free, there's no second cohomology at all)


Everything above is pretty much copied verbatim from chapter X of Homological algebra by Cartan and Eilenberg.


I'd suggest looking at Dan Burghelea and Sarah Witherspoons papers for further reading, but cannot provide explicit references for a while (because searching old papers from a phone is somewhat difficult).


