Twisted group rings of strongly unbounded representation type

Leonid F. Barannyk¹ ; Dariusz Klein²

¹ Institute of Mathematics Pedagogical University of S/lupsk Arciszewskiego 22b 76-200 S/lupsk, Poland
² Institute of Mathematics Pedagogical University of S/lupsk Arciszewskiego 22b, 76-200 S/lupsk, Poland

Colloquium Mathematicum, Tome 100 (2004) no. 2, pp. 265-287

Résumé

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.

Détail

DOI : 10.4064/cm100-2-8

Keywords: commutative local ring characteristic which field * multiplicative group subgroup * finite p group lambda twisted group ring group ring cocycle lambda * denote mathop ind nolimits lambda set isomorphism classes indecomposable lambda g modules s rank exhibit rings lambda which there exists function lambda mathbb rightarrow mathbb lambda geq mathop ind nolimits lambda lambda infinite set every natural special cases lambda mathbb contains every natural number mathop ind nolimits lambda infinite set introduce concept projective representation type group single out finite groups every type