An Auslander-Reiten principle in derived categories
type: Item
instance of: scholarly article
title: An Auslander-Reiten principle in derived categories
zbMATH Open document ID: 1365.13026
author: Maiko Ono
author: Yuji Yoshino
published in: Journal of Pure and Applied Algebra
publication date: 2017-01-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1805.05676
review text: The Auslander-Reiten principle or (the AR duality) is a theorem which states that if \(R\) is a Cohen-Macaulay local ring of Krull dimension \(d\) such that it has only an isolated singularity, and \(M,\;N\) are maximal Cohen-Macaulay modules over \(R\) then there exists a natural isomorphism \(\underline{\mathrm{Hom}}_R(M,N)^\vee\cong \mathrm{Ext}^1_R(N,\tau M)\). In the main result of the present paper (Theorem 2.3) the author proves a general principle. This is a theorem for chain complexes of modules. This theorem is used to obtain: the standard version for the AR-duality (Section 3) or a generalized form proved (see Section 4) by \textit{O. Iyama} and \textit{M. Wemyss} [Invent. Math. 197, No. 3, 521--586 (2014; Zbl 1308.14007)]. The paper is closed with a discussion about the Auslander-Reiten conjecture for modules over Gorenstein rings.
reviewed by: Simion Sorin Breaz
Mathematics Subject Classification ID: 13D09
Mathematics Subject Classification ID: 13C14
Mathematics Subject Classification ID: 16G50
zbMATH DE Number: 6676875
zbMATH Keywords: Auslander-Reiten duality
zbMATH Keywords: Cohen-Macaulay local ring
zbMATH Keywords: complex of modules
MaRDI profile type: Publication
OpenAlex ID: W2963673399
arXiv ID: 1805.05676
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DOI: 10.1016/J.JPAA.2016.09.009
description: scientific article