Auslander–Reiten components with bounded short cycles

Abstract

ABSTRACTWe study Auslander–Reiten components of an artin algebra with bounded short cycles, namely, there exists a bound for the depths of maps appearing on short cycles of non-zero non-invertible maps between modules in the given component. First, we give a number of combinatorial characterizations of almost acyclic Auslander–Reiten components. Then, we shall show that an Auslander–Reiten component with bounded short cycles is obtained, roughly speaking, by gluing the connecting components of finitely many tilted quotient algebras. In particular, the number of such components is finite and each of them is almost acyclic with only finitely many DTr-orbits. As an application, we show that an artin algebra is representation-finite if and only if its module category has bounded short cycles. This includes a well known result of Ringel’s, saying that a representation-directed algebra is representation-finite.