Graded AR Sequences and the Huneke-Wiegand Conjecture

TL;DR

This paper investigates the Huneke-Wiegand Conjecture within the context of one-dimensional graded complete intersection rings, establishing conditions under which the conjecture holds for certain Cohen-Macaulay modules.

Contribution

It introduces a new criterion involving Auslander-Reiten sequences for verifying the Huneke-Wiegand Conjecture in specific graded complete intersection rings.

Findings

The conjecture holds for modules with an Auslander-Reiten sequence with multiple nonfree summands.

Applicable to numerical semigroup rings with embedding dimension at least three.

Provides a new approach to verify the conjecture in graded Cohen-Macaulay modules.

Abstract

We let R be a one-dimensional graded complete intersection, satisfying certain degree conditions which are satisfied whenever R is a numerical semigroup ring of embedding dimension at least three. We show that a graded maximal Cohen-Macaulay R-module M satisfies the Huneke-Wiegand Conjecture provided there exists an Auslander-Reiten sequence ending in M whose middle term has at least two nonfree direct summands.