# Group cohomology over ring

For any group $G$ and $G$-module ($\mathbb{Z}[G]$-module) $M$, we can define a group cohomology $H^{n}(G, M)$ as

$$H^{n}(G, M):=\mathrm{Ext}_{\mathbb{Z}[G]}^{n}(\mathbb{Z}, M).$$ However, I think one can replace $\mathbb{Z}$ with other rings $R$, if $M$ is $R$-module and $G$ acts on it (i.e. $M$ is $R[G]$-module). We can define $$H^{n}_{R}(G, M):=\mathrm{Ext}_{R[G]}^{n}(R, M).$$ Is there any reference about this cohomology group? Actually, there is an exercise about group cohomology of finite dimensional $\mathbb{F}_{p}$-vector space in Dummit-Foote Algebra (Exercise 20, 21 of chapter 17.2). In this case, it seems that we are computing cohomology group when $R=\mathbb{F}_{p}$, not $\mathbb{Z}$. Also, is this group is useful for number theory? Thanks in advance. ## 1 Answer

Actually, for every (commutative) ring $$R$$ we have an isomorphism of $$\mathbb Z$$-modules:

$$\DeclareMathOperator{\Ext}{Ext} \Ext^n_{R[G]}(R, M) \cong \Ext^n_{\mathbb Z[G]}(\mathbb Z, M),$$

see https://stacks.math.columbia.edu/tag/0DVD (they discuss there topological groups, but you can always take the discrete topology).