Representation Theory – Indecomposable Representations for Group Ring $RG$ Over Commutative Ring $R$ with Characteristic $p$

Question

Given a field $k$ with characteristic $p$ and a finite cyclic $p$-group $G$ of order $p^a$, it is well-known that all the indecomposable representations of $kG$ are given by mapping a generator $x$ of $G$ to the Jordan matrix $J_s\in M_s(k)$ with all eigenvalues one for $1\leq s\leq p^a$. If we replace $k$ by a commutative ring $R$ with characteristic $p$, then what are the indecomposable representations of $RG$? Is it the same as in the situation of $kG$?

Answer 1

One might ask whether one can classify all indecomposable $RG$-modules when one knows all indecomposable $R$-modules but the example $R=K[x,y]/(x^2,y^2)$ shows that this is not possible.

The answer will in general be that one can not classify the indecomposable representations as those algebras are most often of "wild" representation type (see for example https://www.tandfonline.com/doi/abs/10.1080/00927879108824178 ). If $G$ is a non-trivial cyclic group and $R$ is a representation-infinite finite dimensional $K$-algebra (for example $R=K[x,y]/(x^2,y^2)$) then $RG \cong R \otimes_K K[x]/(x^n)$ for some $n$ and this will have wild representation type since the quiver of $RG$ will have at least three loops.

In the example of $R=K[x,y]/(x^2,y^2)$ one can classify all indecomposable $R$-modules but for $RG$ this is a wild problem already.