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Auslander-Reiten Sequences for Gorenstein Rings of Dimension One

Auslander-Reiten Sequences for Gorenstein Rings of Dimension One Let R be a complete local Gorenstein ring of dimension one, with maximal ideal 𝔪 $\mathfrak {m}$ . 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 $\mathfrak {m}$ corresponding to a minimal prime ideal of R. We apply this result to determining the shape of some components of stable Auslander-Reiten quivers, which in the considered examples are shown to be tubes. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Algebras and Representation Theory Springer Journals

Auslander-Reiten Sequences for Gorenstein Rings of Dimension One

Roy, Robert
Algebras and Representation Theory , Volume 22 (4) – May 30, 2018
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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Science+Business Media B.V., part of Springer Nature
Subject
Mathematics; Commutative Rings and Algebras; Associative Rings and Algebras; Non-associative Rings and Algebras
ISSN
1386-923X
eISSN
1572-9079
DOI
10.1007/s10468-018-9805-5
Publisher site
See Article on Publisher Site

Abstract

Let R be a complete local Gorenstein ring of dimension one, with maximal ideal 𝔪 $\mathfrak {m}$ . 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 $\mathfrak {m}$ corresponding to a minimal prime ideal of R. We apply this result to determining the shape of some components of stable Auslander-Reiten quivers, which in the considered examples are shown to be tubes.

Journal

Algebras and Representation Theory – Springer Journals

Published: May 30, 2018

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