# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1701.02794