Auslander-Reiten components with bounded short cycles

TL;DR

This paper investigates Auslander-Reiten components with bounded short cycles in artin algebras, providing combinatorial characterizations and linking them to tilted quotient algebras, ultimately characterizing representation-finite algebras.

Contribution

It introduces new combinatorial characterizations of almost acyclic Auslander-Reiten components and relates bounded short cycles to tilted quotient algebras, establishing finiteness results.

Findings

Finite number of such components in artin algebras

Each component is almost acyclic with finitely many DTr-orbits

Artin algebra is representation-finite iff its module category has bounded short cycles

Abstract

We 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 characteri\-zations of almost acyclic Auslander-Reiten components. Then, we show that an Auslander-Reiten component with bounded short cycles is closely related to the connec\-ting component of a tilted quotient algebra. 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.