Relations for Grothendieck groups of Gorenstein rings

TL;DR

This paper proves that a Gorenstein ring has finite representation type if its Auslander-Reiten sequences generate the relations in its Grothendieck group, confirming a conjecture by Auslander.

Contribution

It establishes a new criterion linking Auslander-Reiten sequences to the finite representation type of Gorenstein rings, confirming Auslander's conjecture.

Findings

Gorenstein ring is of finite representation type if Auslander-Reiten sequences generate Grothendieck group relations

Provides an affirmative answer to Auslander's conjecture

Connects Auslander-Reiten sequences with the structure of Grothendieck groups

Abstract

We consider the converse of the Butler, Auslander-Reiten's Theorem which is on the relations for Grothendieck groups. We show that a Gorenstein ring is of finite representation type if the Auslander-Reiten sequences generate the relations for Grothendieck groups. This gives an affirmative answer of the conjecture due to Auslander.