# Twisted group rings and cohomology

Let $R$ be a commutative ring with unity and let $G$ be a finite group. Let $\gamma \in Z^{2}(G,R^{\times})$ be a $2$-cocycle. The twisted group ring (of $G$ over $R$ with respect to $\gamma$) $R *_{\gamma} G$ is an $R$-algebra defined as follows: it is the free $R$-module with basis $\{ u_g \}_{g \in G}$ where the multiplication on the basis is defined via $u_{g}u_{h} = \gamma(g,h)u_{gh}$ for all $g,h \in G$ and $ru_g =u_gr$ for all $g \in G$ and $r \in R$. (Note that this also goes under different names such as crossed product order. Moreover, the term twisted group ring sometimes has a different but related meaning.)

The structure of $R *_{\gamma} G$ only depends on the class of $\gamma$ in $H^2(G,R^{\times})$. It would appear that this result is well-known to experts (see page 2 of this preprint, for example https://arxiv.org/abs/1611.00499). However, I can't seem to find a proof is this in the literature.

Is there a nice a reference (textbook or article) that gives an introduction to such twisted group rings and in particular proves the above fact on the relation with $H^2(G,R^{\times})$? Ideally, it should be for general $R$ rather than the special case that $R$ is a field.

Note that I don't need a proof (I know how to do this), but rather a reference to a proof, etc.

• If $\gamma$ and $\gamma'$ differ by $\partial c$, then the map $u_g \mapsto c_g u_g$ is an isomorphism between the two group rings. – LSpice Jul 10 '18 at 11:29
• (Meaning, it's probably too short to expect to find a reference.) – LSpice Jul 10 '18 at 11:36
• @LSpice I agree that the proof is very short. But it would still be nice to have a reference that gives an introduction to twisted group rings and their basic properties. – Henri Johnston Jul 10 '18 at 12:06
• Here is a more general construction I don't know where to find in the literature: $\gamma$ can actually be generalized to an action of $G$ on $R$ regarded as an $\text{Ab}$-enriched category with one object. Equivalently, to a homomorphism from $G$ to the automorphism $2$-group of $R$, whose $\pi_0$ is $\text{Out}(R)$ and whose $\pi_1$ is $Z(R)^{\times}$ ($R$ need not be commutative and $G$ need not be finite). The $2$-cocycle case is the special case where the induced map on $\pi_0$ is trivial. The twisted group ring is given by taking the homotopy quotient by this action. – Qiaochu Yuan Jul 10 '18 at 17:04

Concretely, Pierce shows that if $E/F$ is a Galois extension with Galois group $G$, then there is a bijection between elements of the Brauer group $B(E/F)$, which you can think of as isomorphism classes of simple central $F$-algebras with $E$ as maximal subfield, and $H^2(G,E^\times)$, among other things.
The construction goes as you suggest: to every $2$-cocycle one can associate such an $F$-algebra by taking the group algebra $FG$ with a crossed multiplication to obtain a map $Z^2(G,E^\times)\to B(E/F)$, which one then checks is trivial on boundaries and descends to the desired isomorphism.