On preprojective and preinjective partitions of a \(\Delta\)-good module category.
type: Item
instance of: scholarly article
title: On preprojective and preinjective partitions of a \(\Delta\)-good module category.
zbMATH Open document ID: 1084.16008
published in: Journal of Algebra
publication date: 2005-04-18T00:00:00Z
review text: Let \(\Lambda\) be a quasi-hereditary algebra. Then the category \({\mathcal F}(\Delta)\) of modules with a \(\Delta\)-filtration is functorially finite [see \textit{C. M. Ringel}, Math. Z. 208, No. 2, 209--224 (1991; Zbl 0725.16011)] and thus it has preprojective and preinjective partitions. For module categories, \textit{A. Skowroński} and \textit{S. O. Smalø} [see Arch. Math. 64, No. 1, 8--10 (1995; Zbl 0813.16007)] have shown that finite representation type is equivalent to every indecomposable module being preprojective or preinjective. An analogue of this result is shown in the present article for the category \({\mathcal F}(\Delta)\). On the way to the final result, the author studies components of the Auslander-Reiten quiver and degrees of irreducible maps. She transfers a variety of useful technology from module categories to the category \({\mathcal F}(\Delta)\). She also uses the validity of Brauer-Thrall II for certain bimodule problems.
reviewed by: Steffen Koenig
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16D10
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16G70
zbMATH DE Number: 2157750
zbMATH Keywords: quasi-hereditary algebras
zbMATH Keywords: preprojective partitions
zbMATH Keywords: finite representation type
zbMATH Keywords: almost split sequences
zbMATH Keywords: Artin algebras
zbMATH Keywords: categories of modules
zbMATH Keywords: good filtrations
zbMATH Keywords: cotilting modules
zbMATH Keywords: preinjective partitions
zbMATH Keywords: Auslander-Reiten quivers
zbMATH Keywords: irreducible maps
author: Zi Ting Zeng
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.09.023
OpenAlex ID: W2063309809
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cites work: Degrees of Irreducible Maps and the Shapes of Auslander-Reiten Quivers
cites work: Semi-Stable Components of an Auslander-Reiten Quiver
cites work: Q3887622
cites work: The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences
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cites work: Artin algebras with only preprojective or preinjective modules are of finite type
cites work: The Structure of Stable Components
DOI: 10.1016/J.JALGEBRA.2004.09.023
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description: scientific article; zbMATH DE number 2157750