A remarkable property of the (co) syzygy modules of the residue field of a nonregular local ring
type: Item
instance of: scholarly article
title: A remarkable property of the (co) syzygy modules of the residue field of a nonregular local ring
zbMATH Open document ID: 0858.13010
DOI: 10.1016/0022-4049(95)00122-0
author: Alex Martsinkovsky
published in: Journal of Pure and Applied Algebra
publication date: 1997-03-25T00:00:00Z
review text: The author proves a characterization of the regular ones among the Noetherian local rings \(R\) by a surprisingly weak homological condition on the syzygy modules in a minimal free resolution of the residue class field \(k\) of \(R\): if there exists a surjective homomorphism from a direct sum of syzygy modules of \(k\) onto a module \(M\neq 0\) of finite projective dimension, then \(R\) is regular. There is also a dual theorem in which \(M\) is of finite injective dimension with non-zero socle, the surjection is replaced by an injection, and the syzygy modules are exchanged with their counterparts in a minimal injective resolution. The paper is dedicated to the memory of Maurice Auslander, and the author attributes a slightly more special case of the main result to him.
reviewed by: Winfried Bruns
Mathematics Subject Classification ID: 13H05
Mathematics Subject Classification ID: 13D02
zbMATH DE Number: 896562
zbMATH Keywords: regularity of local rings
zbMATH Keywords: syzygy modules
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00122-0
OpenAlex ID: W2025136540
cites work: Q3360288
cites work: The Radical of the Homotopy Lie Algebra
cites work: Homological dimensions and Macaulay rings
cites work: New homological invariants for modules over local rings. II
cites work: Q5534287
description: scientific article; zbMATH DE number 896562