Module categories with infinite radical cube zero
type: Item
instance of: scholarly article
title: Module categories with infinite radical cube zero
zbMATH Open document ID: 0869.16006
author: Flávio U. Coelho
author: Héctor A. Merklen
author: Eduardo N. Marcos
author: Andrzej Skowroński
published in: Journal of Algebra
publication date: 1997-07-08T00:00:00Z
review text: Let \(A\) be an Artin algebra and let \(\text{mod }A\) be the category of finitely generated \(A\)-modules. Denote by \(\text{rad}(\text{mod }A)\) the radical of \(\text{mod }A\) and by \(\text{rad}^\infty(\text{mod }A)\) the intersection of all powers \(\text{rad}^i(\text{mod }A)\). It is well known that \(\text{rad}^\infty(\text{mod }A)=0\) if and only if \(A\) is representation finite, and it was shown by the authors [Commun. Algebra 22, No. 11, 4511-4517 (1994; Zbl 0812.16019)] that \((\text{rad}^\infty(\text{mod }A))^2=0\) characterizes finite type. In the paper under review, representation finite Artin algebras with \((\text{rad}^\infty(\text{mod }A))^3=0\) are studied and a rather complete description of their module categories is given. (1) If \(\Gamma(A)\) denotes the Auslander-Reiten quiver of a representation infinite Artin algebra \(A\) with \((\text{rad}^\infty(\text{mod }A))^3=0\), then all regular components of \(\Gamma(A)\) are pairwise orthogonal, they are generalized stable standard tubes and almost all of them have rank 1. The nonregular components may be quite complicated. (2) An Artin algebra \(A\) is tame concealed, if and only if \((\text{rad}^\infty(\text{mod }A))^3=0\) and there is a faithful regular component. Consequently an Artin algebra with \((\text{rad}^\infty(\text{mod }A))^3=0\) has tame concealed factor-algebras \(B=A/I\). The embedding \(\text{mod }B\to\text{mod }A\) is studied in section 4. It is shown that almost all tubes in \(\Gamma(B)\), the Auslander-Reiten quiver of \(B\), are full components in \(\Gamma(A)\). (3) In the final part it is shown that an Artin algebra \(A\) with \((\text{rad}^\infty(\text{mod }A))^3=0\) has only finitely many ideals \(I_1,\dots,I_m\) such that \(A/I_j\) is tame concealed. Consequently \(A\) is a domestic algebra.
reviewed by: Otto Kerner
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16P70
zbMATH DE Number: 923669
zbMATH Keywords: categories of finitely generated modules
zbMATH Keywords: radicals
zbMATH Keywords: representation finite Artin algebras
zbMATH Keywords: Auslander-Reiten quivers
zbMATH Keywords: regular components
zbMATH Keywords: generalized stable standard tubes
zbMATH Keywords: domestic algebras
MaRDI profile type: Publication
full work available at URL: jabr.1996.0204
OpenAlex ID: W2021337846
DOI: 10.1006/JABR.1996.0204
description: scientific article; zbMATH DE number 923669