Infinite radicals in standard Auslander-Reiten components
type: Item
instance of: scholarly article
title: Infinite radicals in standard Auslander-Reiten components
zbMATH Open document ID: 0809.16016
author: Shiping Liu
published in: Journal of Algebra
publication date: 1994-08-17T00:00:00Z
review text: Let \(A\) be a finite dimensional algebra over an algebraically closed field \(k\) and \(\text{mod }A\) the category of finitely generated left \(A\)- modules. Denote by \(\text{rad}^ \infty(\text{mod }A)\) the infinite power of the Jacobson radical \(\text{rad}(\text{mod }A)\) of \(\text{mod }A\) and by \(\Gamma_ A\) the Auslander-Reiten quiver of \(A\). Recall that a component \(\mathcal C\) of \(\Gamma_ A\) is called standard if the full subcategory of \(\text{mod }A\) formed by modules from \(\mathcal C\) is equivalent to the mesh-category \(k({\mathcal C})\) of \(\mathcal C\) [see \textit{C. M. Ringel}, Tame algebras and integral quadratic forms (Lect. Notes Math. 1099, 1984; Zbl 0546.16013)], and \(\mathcal C\) is called generalized standard if \(\text{rad}^ \infty (X,Y) = 0\) for all modules \(X\) and \(Y\) in \(\mathcal C\) [see \textit{A. Skowroński}, J. Math. Soc. Japan 46, 517-543 (1994)]. In the paper, the author proves that every standard component \(\mathcal C\) of \(\Gamma_ A\) is generalized standard. In particular, one obtains that if \(\Gamma_ A\) is standard then \(A\) is of finite representation type. We note also that it has been proved in [\textit{A. Skowroński}, Bull. Pol. Acad. Sci. 42, 157-163 (1994)] that every generalized standard component \(\mathcal C\) of \(\Gamma_ A\) containing not both a projective and an injective module is standard.
reviewed by: Andrzej Skowroński
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16G30
zbMATH DE Number: 614226
zbMATH Keywords: finite dimensional algebra
zbMATH Keywords: category of finitely generated left \(A\)- modules
zbMATH Keywords: Jacobson radical
zbMATH Keywords: Auslander-Reiten quiver
zbMATH Keywords: mesh-category
zbMATH Keywords: standard component
zbMATH Keywords: finite representation type
MaRDI profile type: Publication
full work available at URL: jabr.1994.1150
OpenAlex ID: W2064327160
DOI: 10.1006/JABR.1994.1150
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description: scientific article; zbMATH DE number 614226