Auslander-Reiten sequences and global dimensions.
type: Item
instance of: scholarly article
title: Auslander-Reiten sequences and global dimensions.
zbMATH Open document ID: 1114.16009
author: Wei Hu
author: Changchang Xi
published in: Mathematical Research Letters
publication date: 2007-02-05T00:00:00Z
review text: The authors prove the following result: Theorem. Let \(A\) be an Artin algebra, \(M\) a generator-cogenerator for \(A\)-mod. Then: (1) If \(\alpha_i\colon V_i\to M'_i\), \(i=1,\dots,s\), are left almost split morphisms in \(A\)-mod with all \(M'_i\) being in the additive closure of \(M\), then the global dimension of the endomorphism algebra of \(M\oplus\bigoplus_{i=1}^s V_i\) does not exceed the global dimension of the endomorphism algebra of \(M\). (2) If \(\beta_i\colon M'_i\to V_i\), \(i=1,\dots,s\), are right almost split morphisms in \(A\)-mod with all \(M'_i\) being in the additive closure of \(M\), then the global dimension of the endomorphism algebra of \(M\oplus\bigoplus_{i=1}^s V_i\) does not exceed the global dimension of the endomorphism algebra of \(M\).
reviewed by: Volodymyr Mazorchuk
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 18G20
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 5121804
zbMATH Keywords: Auslander-Reiten sequences
zbMATH Keywords: global dimension
zbMATH Keywords: coherent functors
zbMATH Keywords: generators
zbMATH Keywords: cogenerators
zbMATH Keywords: Artin algebras
zbMATH Keywords: endomorphism algebras
zbMATH Keywords: representation dimension
zbMATH Keywords: finitistic dimension
MaRDI profile type: Publication
DOI: 10.4310/MRL.2006.V13.N6.A4
description: scientific article; zbMATH DE number 5121804