Auslander-Gorenstein resolution.
type: Item
instance of: scholarly article
title: Auslander-Gorenstein resolution.
zbMATH Open document ID: 1245.16008
author: Mitsuo Hoshino
author: Hirotaka Koga
published in: Journal of Pure and Applied Algebra
publication date: 2012-01-24T00:00:00Z
review text: Let \(\Lambda\) be a Noetherian ring with minimal injective resolution \(I^\bullet\). In the paper under review, \(\Lambda\) is said to satisfy the Auslander Condition if for every \(i\) one has \(\text{flat.dim}(I^i)\leq i\), and \(\Lambda\) is said to be an Auslander-Gorenstein ring if, in addition, \(\text{id}_\Lambda(\Lambda)=\text{id}_{\Lambda^{\text{op}}}(\Lambda)<\infty\). Auslander-Gorenstein rings, which have also been called \(m\)-Gorenstein rings in the literature, are a generalization of commutative Gorenstein rings. These arise naturally in various settings as described in the introduction of the paper under review. In this paper, the authors introduce the notion of Auslander-Gorenstein resolutions. The main theorem states that a ring is Auslander-Gorenstein if and only if it admits an Auslander-Gorenstein resolution over another Auslander-Gorenstein ring. Despite their prevalence, there are few known methods for constructing Auslander-Gorenstein rings. The main result of this paper may be viewed as providing one such construction. The utility of this method is illustrated by several examples in which Auslander-Gorenstein resolutions are explicitly computed.
reviewed by: Kosmas Diveris
Mathematics Subject Classification ID: 16E05
Mathematics Subject Classification ID: 16E65
Mathematics Subject Classification ID: 16E10
zbMATH DE Number: 5999994
zbMATH Keywords: Auslander-Gorenstein rings
zbMATH Keywords: injective resolutions
zbMATH Keywords: Auslander-Gorenstein resolutions
zbMATH Keywords: flat dimension
MaRDI profile type: Publication
full work available at URL: j.jpaa.2011.05.011
OpenAlex ID: W1977343088
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DOI: 10.1016/J.JPAA.2011.05.011
description: scientific article; zbMATH DE number 5999994