Let $R$ be a commutative ring and let $G$ be a group (not necessarily finite). Let $\mathcal A$ be the category of (left) $R[G]$-modules that are finitely generated over $R$; one might call this the category of finite dimensional representations of $G$ over $R$. Consider the following statement $P(R,G)$:
- For all $M\in\mathcal A$, there is an epimorphism $N\to M$ in $\mathcal A$ where $N$ is free as an $R$-module.
For what conditions on $R$ and $G$ is $P(R,G)$ true?
For instance, if $R$ is a field, then $P(R,G)$ is true for all $G$, since any $M\in \mathcal A$ is already free as an $R$-module. Similarly, if $G$ is finite, then $P(R,G)$ is true for all $R$, since any $M\in\mathcal A$ will be a quotient of a finitely generated free $R[G]$-module, which is in turn finitely generated and free as an $R$-module (if I'm not messing anything up in my head...).
Thus, the question is only interesting in the hard case: representations of infinite groups over a ring that is not a field.
Partial progress on a question I asked recently indicates that $P(R,G)$ is true if $R$ is Noetherian and $G$ is free; at the very least, it seems to be true (I haven't checked the details) when $G=\mathbb Z$ and $R$ is Noetherian local, and when $G$ is free and $R$ is a PID.
