Bounds for indecomposable torsion-free modules
type: Item
instance of: scholarly article
title: Bounds for indecomposable torsion-free modules
zbMATH Open document ID: 1168.13006
author: Nicholas R. Baeth
author: Melissa R. Luckas
published in: Journal of Pure and Applied Algebra
publication date: 2009-04-16T00:00:00Z
review text: Let \(M\) be a finitely generated torsion-free module over a one-dimensional reduced noetherian ring \(R\) with finitely generated normalization \(\widetilde{R}\) (i.e., \(R\) is a ring-order). The rank of \(M\) is the tuple of vector-space dimensions \(\dim_{R_P}M_P\), where \(P\) ranges over the minimal primes of \(R\). A ring-order \(R\) is said to have bounded representation type if there exists a bound \(N_R\) on the ranks of all indecomposable finitely generated torsion-free \(R\)-modules. \(M\) is said to have constant rank if all the dimensions \(\dim_{R_P}M_P\) are the same. Following a long list of results (see loc. cit.) this article studies what bounds and ranks occur for indecomposable torsion-free \(R\)-modules, when \(R\) is of bounded representation type. The main result (Main Theorem 1.2) of this paper states the following: If \(R\) is a local ring-order of bounded representation type, then \(R\) has at most three minimal primes. Moreover, if \(M\) is an indecomposable torsion-free finitely generated \(R\)-module then: (1) if \(R\) is a domain, then \(\text{rank}(M)\in\{1,2,3\}\); (2) if \(R\) has exactly two minimal primes then \(\text{rank}(M)\in\{(0,1),(1,0),(1,1),(1,2),(2,1), (2,2)\}\); (3) if \(R\) has exactly three minimal primes then, with a suitable ordering of the minimal primes, \[ \text{rank}(M)\in\{(0,0,1),(0,1,0),(1,0,0),(0,1,1),(1,0,1),(1,1,0),(1,1,1),(2,1,1)\}. \] Note that \(R\) is not necessarily equicharacteristic with perfect residue field and characteristic not 2,3 or 5 as previous results required [see \textit{N. R. Baeth}, J. Pure Appl. Algebra 208, No. 3, 923--940 (2007; Zbl 1106.13010); \textit{M. Arnavut, M. Luckas} and \textit{S. Wiegand} [J. Pure Appl. Algebra 208, No. 2, 739--760 (2007; Zbl 1111.13007)]. Essential to the argument is to observe (Theorem 2.2 and [\textit{N. Cimen, R. Wiegand} and \textit{S. Wiegand}, One-dimensional rings of finite Cohen-Macaulay type. in: Abelian groups and modules, Kluwer Academic, Doldre…
reviewed by: Stefan O. Tohǎneanu
Mathematics Subject Classification ID: 13C14
Mathematics Subject Classification ID: 13H99
zbMATH DE Number: 5544379
zbMATH Keywords: ring-order
zbMATH Keywords: Cohen-Macaulay type
zbMATH Keywords: Artinian pair
zbMATH Keywords: indecomposable torsion-free module
MaRDI profile type: Publication
full work available at URL: j.jpaa.2008.11.032
OpenAlex ID: W2004560462
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DOI: 10.1016/J.JPAA.2008.11.032
description: scientific article; zbMATH DE number 5544379