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Abstract

Let $S$ be a commutative local ring of characteristic $p$, which is not a~field, $S^{*}$ the multiplicative group of $S$, $W$ a subgroup of $S^{*}$, $G$ a finite $p$-group, and $S^{\lambda}G$ a twisted group ring of the group $G$ and of the ring $S$ with a~$2$-cocycle $\lambda \in Z^{2}(G,S^{*})$. Denote by $\mathop{\rm Ind}\nolimits _{m}(S^{\lambda}G)$ the set of isomorphism classes of indecomposable $S^{\lambda}G$-modules of $S$-rank $m$. We exhibit rings $S^{\lambda}G$ for which there exists a function $f_{\lambda}: \mathbb{N} \rightarrow \mathbb{N}$ such that $f_{\lambda}(n)\geq n$ and $\mathop{\rm Ind}\nolimits_{f_{\lambda} (n)}(S^{\lambda}G)$ is an infinite set for every natural $n>1$. In special cases $f_{\lambda}(\mathbb{N})$ contains every natural number $m>1$ such that $\mathop{\rm Ind}\nolimits_{m}(S^{\lambda}G)$ is an infinite set. We also introduce the concept of projective $(S,W)$-representation type for the group $G$ and we single out finite groups of every type.