Auslander-Reiten Sequences for Gorenstein Rings of Dimension One

Auslander-Reiten Sequences for Gorenstein Rings of Dimension One Algebr Represent Theor https://doi.org/10.1007/s10468-018-9805-5 Auslander-Reiten Sequences for Gorenstein Rings of Dimension One Robert Roy Received: 14 September 2017 / Accepted: 22 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 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 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. Keywords Commutative algebra · Auslander-reiten theory · Gorenstein Mathematics Subject Classification (2010) 13H10 · 16G70 1 Introduction The theory of Auslander-Reiten (AR) quivers is central in the study of artin algebras. Regarding AR theory for maximal Cohen-Macaulay modules over a complete Cohen- Macaulay local ring, the cases of finite AR quivers have been studied thoroughly (see [17]), but in the more common case of infinite type, shapes of AR quivers seem to be largely unknown. The paper [1] agrees with this assessment (cf. its introduction), and begins to bridge this gap. It establishes 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

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Publisher
Springer Netherlands
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
D.O.I.
10.1007/s10468-018-9805-5
Publisher site
See Article on Publisher Site

Abstract

Algebr Represent Theor https://doi.org/10.1007/s10468-018-9805-5 Auslander-Reiten Sequences for Gorenstein Rings of Dimension One Robert Roy Received: 14 September 2017 / Accepted: 22 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 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 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. Keywords Commutative algebra · Auslander-reiten theory · Gorenstein Mathematics Subject Classification (2010) 13H10 · 16G70 1 Introduction The theory of Auslander-Reiten (AR) quivers is central in the study of artin algebras. Regarding AR theory for maximal Cohen-Macaulay modules over a complete Cohen- Macaulay local ring, the cases of finite AR quivers have been studied thoroughly (see [17]), but in the more common case of infinite type, shapes of AR quivers seem to be largely unknown. The paper [1] agrees with this assessment (cf. its introduction), and begins to bridge this gap. It establishes

Journal

Algebras and Representation TheorySpringer Journals

Published: May 30, 2018

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