Syzygies of Cohen-Macaulay modules and Grothendieck groups

TL;DR

This paper investigates the structure of Cohen-Macaulay modules over local rings with isolated singularities, linking syzygies, Grothendieck groups, and Auslander-Reiten sequences to establish finiteness results.

Contribution

It extends previous work by connecting Auslander-Reiten sequences to the finiteness of indecomposable syzygies in Cohen-Macaulay modules, generalizing Hiramatsu's Gorenstein case.

Findings

Finiteness of indecomposable syzygies under certain conditions

Relation of Auslander-Reiten sequences to Grothendieck groups

Extension of Hiramatsu's results to broader Cohen-Macaulay rings

Abstract

We study the converse of a theorem of Butler and Auslander-Reiten. We show that a Cohen-Macaulay local ring with an isolated singularity has only finitely many isomorphism classes of indecomposable summands of syzygies of Cohen-Macaulay modules if the Auslander-Reiten sequences generate the relation of the Grothendieck group of finitely generated modules. This extends a recent result of Hiramatsu, which gives an affirmative answer in the Gorenstein case to a conjecture of Auslander.