Lower bounds for faithful, preinjective modules
type: Item
instance of: scholarly article
title: Lower bounds for faithful, preinjective modules
zbMATH Open document ID: 0608.16030
DOI: 10.1007/BF01172489
author: Luise Unger
published in: Manuscripta Mathematica
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155183
review text: Let A be a k-algebra and \(\tau\) its Auslander-Reiten translation. An interesting problem is to determine the faithful, indecomposable preinjective A-modules, i.e. modules of the form \(\tau^ nI\), with I injective. If A is the path algebra of a quiver \({\vec \Delta}\), there are only a finite number of indecomposable preinjectives which are not faithful. Hence, there is a minimum m(\({\vec \Delta})\) of the numbers n such that \(\tau^ nI\) is always faithful. The maximum of the m(\({\vec \Delta})'s\) for the different orientations of \(\Delta\) is denoted by m(\(\Delta)\). This paper solves the problem of determining m(\(\Delta)\) when \(\Delta\) is a tree of infinite representation type. If \(\Delta\) is a tame tree, it is shown that \(m(\Delta)=r(\Delta)+1\), where r(\(\Delta)\) is the replication number of \textit{R. Bautista} and \textit{S. Brenner} [see Proc. Lond. Math. Soc., III. Ser. 47, 429-462 (1983; Zbl 0505.16012)]. In the general cases m(\(\Delta)\) coincides with the diameter d(\(\Delta)\) of \(\Delta\), and it is always of the form \(m(\Delta)=d(\Delta)+c(\Delta)\) where c(\(\Delta)\) is a constant depending on certain structural properties of \(\Delta\). The paper is very clear and very well written.
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16P10
zbMATH DE Number: 3983425
zbMATH Keywords: Auslander-Reiten translation
zbMATH Keywords: indecomposable preinjective A-modules
zbMATH Keywords: path algebra
zbMATH Keywords: quiver
zbMATH Keywords: tree of infinite representation type
zbMATH Keywords: tame tree
zbMATH Keywords: replication number
reviewed by: Héctor A. Merklen
MaRDI profile type: Publication
cites work: Covering spaces in representation-theory
cites work: Tilted Algebras
cites work: Algebren, Darstellungsköcher, Überlagerungen und zurück
description: scientific article; zbMATH DE number 3983425