Reduction techniques for strongly graded rings and finite representation type. I: Bounded functors and graded equivalence
type: Item
instance of: scholarly article
title: Reduction techniques for strongly graded rings and finite representation type. I: Bounded functors and graded equivalence
zbMATH Open document ID: 0882.16009
author: Jeremy Haefner
published in: Journal of Algebra
publication date: 1998-03-02T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 1066138
zbMATH Keywords: strongly graded orders
zbMATH Keywords: finite groups
zbMATH Keywords: maximal orders
zbMATH Keywords: representation finiteness
zbMATH Keywords: representation finite local orders
zbMATH Keywords: crossed product orders
zbMATH Keywords: graded equivalences
reviewed by: Wolfgang Rump
MaRDI profile type: Publication
full work available at URL: jabr.1997.7031
OpenAlex ID: W2041510149
cites work: Q3922829
cites work: Group-Graded Rings, Smash Products, and Group Actions
cites work: PRIMARY ORDERS WITH A FINITE NUMBER OF INDECOMPOSABLE REPRESENTATIONS
cites work: Graded equivalence theory with applications
cites work: Graded Morita theory for infinite groups
cites work: Induced crossed products
cites work: The group of automorphisms of certain subalgebras of matrix algebras
cites work: Fixed rings of finite automorphism groups of associative rings
cites work: Crossed products over prime rings
cites work: Q3995022
cites work: Separable functors revisited
cites work: Q4061092
cites work: Categorical methods in graded ring theory
cites work: On Clifford systems and generalized crossed products
cites work: Homological dimension of skew group rings and crossed products
DOI: 10.1006/JABR.1997.7031
Recommended article: Q4005963
Recommended article: On local orders
Recommended article: Q3973369
Recommended article: On integral representations of crossed-product orders
Recommended article: Primary orders of finite representation type
Recommended article: Two-dimensional tame and maximal orders of finite representation type
Recommended article: On Gorenstein orders
Recommended article: Monoids of torsion-free modules over rings with finite representation type
Recommended article: Q4284173
Recommended article: Finite generalized crossed products over tame and maximal orders
description: scientific article; zbMATH DE number 1066138