On Artin algebras with almost all indecomposable modules of projective or injective dimension at most one.
type: Item
instance of: scholarly article
title: On Artin algebras with almost all indecomposable modules of projective or injective dimension at most one.
zbMATH Open document ID: 1035.16008
DOI: 10.2478/BF02475668
published in: Central European Journal of Mathematics
publication date: 2003-05-18T00:00:00Z
review text: For an Artin algebra \(\Lambda\), denote by \({\mathcal L}_\Lambda\) (and by \({\mathcal R}_\Lambda\)), the category of all indecomposable \(\Lambda\)-modules all whose predecessors (or successors, respectively) have projective (or injective, respectively) dimension at most one. Such categories were first studied by \textit{D. Happel}, \textit{I. Reiten} and \textit{S. O. Smalø} [Mem. Am. Math. Soc. 575 (1996; Zbl 0849.16011)], where the authors introduced the class of quasitilted algebras and, more recently, lead to the description of further classes of algebras, e.g. the shod algebras [\textit{F. U. Coelho} and \textit{M. A. Lanzilotta}, Manuscr. Math. 100, No. 1, 1-11 (1999; Zbl 0966.16001)], which are characterized by the property that the union \({\mathcal L}_\Lambda\cup{\mathcal R}_\Lambda\) contains all indecomposable \(\Lambda\)-modules. In the paper under review, the author gives a description of all algebras such that \({\mathcal L}_\Lambda\cup{\mathcal R}_\Lambda\) contains all but finitely many (up to isomorphism) indecomposable \(\Lambda\)-modules and derives some nice consequences. In a paralel and independent work, such class of algebras was also studied by \textit{I. Assem} and \textit{F. U. Coelho} [J. Algebra 269, No. 2, 456-479 (2003; Zbl 1045.16006)].
reviewed by: Flávio U. Coelho
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 1862504
zbMATH Keywords: Artin algebras
zbMATH Keywords: quasitilted algebras
zbMATH Keywords: generalized double tilted algebras
zbMATH Keywords: homological dimensions
zbMATH Keywords: Auslander-Reiten quivers
zbMATH Keywords: shod algebras
zbMATH Keywords: categories of modules
zbMATH Keywords: indecomposable modules
Wikidata QID: Q114041397
author: Andrzej Skowroński
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.2478/bf02475668
OpenAlex ID: W2067649346
description: scientific article; zbMATH DE number 1862504