Auslander-Reiten sequences for Gorenstein rings of dimension one

TL;DR

This paper investigates Auslander-Reiten sequences in the context of one-dimensional Gorenstein rings, revealing how specific endomorphisms generate these sequences and analyzing their impact on the structure of AR quivers.

Contribution

It establishes a link between AR-sequences and endomorphisms of the maximal ideal in Gorenstein rings of dimension one, providing new insights into their structure.

Findings

AR-sequences are generated by particular endomorphisms of the maximal ideal.

The shape of some components of Auslander-Reiten quivers is characterized.

The results connect module theory with ring endomorphisms in Gorenstein rings.

Abstract

Let $R$ be a complete local Gorenstein ring of dimension one, with maximal ideal m. We show that if $M$ is a Cohen-Macaulay $R$-module which begins an AR-sequence, then this sequence is produced by a particular endomorphism of m corresponding to a minimal prime ideal. We apply this result to determining the shape of some components of Auslander-Reiten quivers.