Reduction techniques for strongly graded rings and finite representation type. I: Bounded functors and graded equivalence
Let \(G\) be a finite group and \(R\) a Dedekind domain with a global field of fractions \(K\). For a strongly \(G\)-graded \(R\)-order \(\Lambda\) with 1-component \(\Lambda_1\) a maximal order, a criterion for representation-finiteness is given. Here, strongly graded means that \(\Lambda_\sigma\Lambda_\tau=\Lambda_{\sigma\tau}\) for \(\sigma,\tau\in G\). By a number of reductions, the criterion is obtained from \textit{Yu. A. Drozd} and \textit{V. V. Kirichenko}'s characterization of representation-finite local orders [Izv. Akad. Nauk SSSR, Ser. Mat. 37, 715-736 (1973; Zbl 0291.16005)]. Firstly, Jones' theorem allows to assume \(R\) to be a complete discrete valuation domain, and similarly to the group ring case, \(G\) can be replaced by a \(p\)-group. Next it is shown that \(\Lambda\) is Morita equivalent to a ring direct sum of crossed product orders which are again strongly graded, and without loss of generality, the maximal order \(\Lambda_1\) may be assumed to be prime. In this case, the author shows that \(\Lambda\) itself is primary, i.e. \(\Lambda/\text{Rad }\Lambda\) is simple, and thus Drozd and Kirichenko's result applies. Moreover, for such orders \(\Lambda\), the Morita equivalence between \(\Lambda_1\) and some maximal order \(\Delta\) in a skewfield gives rise to a graded equivalence between \(\Lambda\) and a similar order with 1-component \(\Delta\). In a (forthcoming) second part of the paper, the author will further reduce \(\Lambda\) to an order with \(\Lambda_1=\Delta\) commutative.
scientific article; zbMATH DE number 1066138
scientific article