Auslander-Reiten sequences and global dimensions

Abstract

1. IntroductionAlmost split sequences were introduced by Auslander and Reiten. Their significantimportance in the representation theory of Artin algebras and groups is well-knownin the literature. In [11] the relationship between projective dimensions of modulesappearing in an almost split sequence was studied. However, the relationship betweenthe global dimensions of endomorphism algebras of these modules is unknown. In thispaper we shall establish an inequality for global dimensions of these endomorphismalgebras. This reveals also that almost split sequences may play a role in the studyof representation dimensions of algebras, or the global dimensions of endomorphismalgebras of certain modules. Our main result isTheorem 1.1. Let Abe an Artin algebra, let M be a generator-cogenerator for A-mod, and let sbe a positive integer. Then:(1) If α