Constructing the preprojective components of an algebra
type: Item
instance of: scholarly article
title: Constructing the preprojective components of an algebra
zbMATH Open document ID: 0848.16010
author: Stanisław Kasjan
author: José A. de la Peña
published in: Journal of Algebra
publication date: 1996-10-24T00:00:00Z
review text: Let \(k\) be an algebraically closed field, \(Q\) be a finite connected quiver without oriented cycles and consider the finite dimensional algebra \(A=kQ/I\), where \(I\) is an admissible ideal of \(kQ\). The Auslander-Reiten quiver \(\Gamma_A\) of \(A\) has as vertices representatives of isoclasses of finite dimensional indecomposable \(A\)-modules, and there are as many arrows from \(X\) to \(Y\) in \(\Gamma_A\) as \(\dim_k(\text{rad}(X,Y)/\text{rad}^2(X,Y))\). A component \(\mathcal P\) of \(\Gamma_A\) is said to be preprojective if \(\mathcal P\) has no oriented cycles and each module \(X\in{\mathcal P}\) has only finitely many predecessors in the path order of \(\mathcal P\). The importance of having such components is that for each \(X\) on them, it holds that \(\text{Ext}^i_A(X,X)=0\), for \(i>0\), \(\text{End }X\) is local and the isomorphism class of \(X\) is uniquely determined by the composition factors. Several classes of algebras have preprojective components, such as algebras with separation condition and quasitilted algebras. In this paper, the authors give an algorithmic procedure to construct all preprojective components in \(\Gamma_A\).
reviewed by: Flávio U. Coelho
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16G70
zbMATH DE Number: 868021
zbMATH Keywords: finite connected quivers
zbMATH Keywords: finite dimensional algebras
zbMATH Keywords: Auslander-Reiten quivers
zbMATH Keywords: finite dimensional indecomposable modules
zbMATH Keywords: preprojective components
zbMATH Keywords: quasitilted algebras
MaRDI profile type: Publication
full work available at URL: jabr.1996.0037
OpenAlex ID: W2042728426
Wikidata QID: Q131317018
DOI: 10.1006/JABR.1996.0037
description: scientific article; zbMATH DE number 868021