# Auslander-Reiten triangles and Grothendieck groups of triangulated categories

## TL;DR

This paper establishes a link between Auslander-Reiten triangles generating relations in Grothendieck groups and the finiteness of indecomposable objects in certain triangulated categories, providing a triangulated analogue to a classical theorem.

## Contribution

It proves a converse to Butler and Auslander-Reiten's theorem, showing that generation of relations by Auslander-Reiten triangles implies finiteness of indecomposables in Hom-finite Krull-Schmidt categories.

## Key findings

- Auslander-Reiten triangles generate relations in Grothendieck groups.
- Categories with such relations have finitely many indecomposables.
- Applications extend to Frobenius categories.

## Abstract

We prove that if the Auslander-Reiten triangles generate the relations for the Grothendieck group of a Hom-finite Krull-Schmidt triangulated category with a (co)generator, then the category has only finitely many isomorphism classes of indecomposable objects up to translation. This gives a triangulated converse to a theorem of Butler and Auslander-Reiten on the relations for Grothendieck groups. Our approach has applications in the context of Frobenius categories.