Let $G$ be a group and $R$ a commutative unital ring. We define the $R$-cohomological dimension of $G$ to be $$cd_R(G) := \sup \{ n : H^n(G, M) \neq 0 \text{ for some } R[G]\text{-module } M \}.$$ I have read somewhere that this may be equivalently defined as $$cd_R'(G) := \sup \{ n : H^n(G, R[G]) \neq 0 \},$$ but in a recent discussion on a previous post it was pointed out to me that this is only true if $cd_R(G) < \infty$. Is this the case? What are some counterexamples? Could you point me to a reference?
I would like to understand this in general but I am especially interested in a specific case: $G$ is finitely presented, $R$ is finite, and $cd_R'(G) \leq 1$. Can I deduce in this case that $cd_R(G) = cd_R'(G)$, or even just that $cd_R(G) \leq 1$ as well?
One class of groups that have $\operatorname{cd}_R(G)=\infty$ but $\operatorname{cd'}_R(G)<\infty$ are groups with a finite index subgroup $H$ such that $\operatorname{cd}_R(G)<\infty$, and with an element $x$ of finite order whose order is not invertible in $R$. Such groups necessarily have torsion. I don't know whether there are other examples that are torsion free, but there may be examples that are well known to experts.
For example, this includes (taking $H$ to be the trivial subgroup) all finite groups $G$ with $|G|$ not invertible in $R$. But also many infinite groups.
The references in the following proof are to Brown's Cohomology of Groups. Actually I state results for general $R$ that are only stated for the case $R=\mathbb{Z}$ in Brown, but the generalizations are straightforward.
First, a group with $\operatorname{cd}_R(G)<\infty$ has no elements $x$ with finite order not invertible in $R$ [Corollary (2.5). In the case $R=\mathbb{Z}$ this just says it is torsion free, and relies on the fact that $H^n(C,\mathbb{Z})$ is nonzero for infinitely many $n$, where $C=\langle x\rangle$ is the cyclic subgroup generated by $x$. But if $|C|$ is not invertible in $R$ then $H^n(C,R)$ is nonzero for infinitely many $n$, and the proof goes through.]
Next, by Shapiro's Lemma [Proposition (6.2)], $$H^n(G,\operatorname{coind}^G_H R[H])\cong H^n(H, R[H]),$$ which is zero for all but finitely many $n$, and since $|G:H|<\infty$, $\operatorname{coind}^G_H R[H]\cong R[G]$, so $\operatorname{cd'}_R(G)<\infty$.
One positive result is [Proposition (2.3)]: $$\operatorname{cd}_R(G)=\sup\left\{n\mid H^n(G,F)\neq0\text{ for some free $R[G]$ module $F$}\right\}$$ if $\operatorname{cd}_R(G)<\infty$, but I don't see why this free module could necessarily be taken to be $R[G]$. Maybe there are examples where $F$ necessarily has infinite rank that are well known to experts.