Auslander-Reiten theory in a Krull-Schmidt category

Abstract

We rst introduce the notion of an Auslander-Reiten sequence in a Krull-Schmidt category. This uni es the notion of an almost split sequence in an abelian category and that of an Auslander-Reiten triangle in a triangulated category. We then de ne the Auslander-Reiten quiver of a Krull-Schmidt category and describe the shapes of its semi-stable components. The main result generalizes those for an artin algebra and specializes to an arbitrary triangulated categories, in particular to the derived category of bounded complexes of nitely generated modules over an artin algebra of nite global dimension.