On generalized standard Auslander-Reiten components having only finitely many non-directing modules.
type: Item
instance of: scholarly article
title: On generalized standard Auslander-Reiten components having only finitely many non-directing modules.
zbMATH Open document ID: 1078.16012
author: David Smith
published in: Journal of Algebra
publication date: 2004-10-01T00:00:00Z
review text: Let \(A\) be an Artin algebra, \(\text{mod\,}A\) denotes the category of all finitely generated right \(R\)-modules, and \(\Gamma(\text{mod\,}A)\) the Auslander-Reiten quiver of \(A\). A component \(\Gamma\) of \(\Gamma(\text{mod\,}A)\) is called semiregular if it does not contain both a projective module and an injective module, and is called generalized standard if \(\text{rad}^\infty(X,Y)=0\) for all modules \(X\) and \(Y\) in \(\Gamma\). \(\Gamma\) is convex if for every chain \(X_0\to X_1\to\cdots\to X_{n-1}\to X_n\) of non-zero non-isomorphisms between indecomposable modules in \(\text{mod\,}A\), with \(X_0,X_n\in\Gamma\), the \(X_i\) belong to \(\Gamma\) for \(i=1,2,\dots,n-1\). The author defines a component \(\Gamma\) of \(\Gamma(\text{mod\,}A)\) to be quasi-directed if it is generalized standard and contains only finitely many modules lying on cycles (of not necessarily irreducible morphisms). The author obtains complete characterizations of semiregular connected components of \(\Gamma(\text{mod\,}A)\) which are quasi-directed, and of non-semiregular connected components of \(\Gamma(\text{mod\,}A)\) which are quasi-directed and convex. Several applications and examples are given, related to classes of algebras studied extensively in the literature, including quasi-tilted algebras, laura algebras, or left supported algebras.
reviewed by: Nguyen Viet Dung
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 2104451
zbMATH Keywords: Artin algebras
zbMATH Keywords: Auslander-Reiten quivers
zbMATH Keywords: quasi-directed components
zbMATH Keywords: categories of finitely generated right modules
zbMATH Keywords: indecomposable modules
zbMATH Keywords: irreducible morphisms
zbMATH Keywords: semiregular connected components
zbMATH Keywords: quasi-tilted algebras
zbMATH Keywords: laura algebras
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.03.022
OpenAlex ID: W2067954169
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DOI: 10.1016/J.JALGEBRA.2004.03.022
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description: scientific article; zbMATH DE number 2104451