Syzygy modules for Noetherian rings
type: Item
instance of: scholarly article
title: Syzygy modules for Noetherian rings
zbMATH Open document ID: 0857.16006
author: Maurice Auslander
author: Idun Reiten
published in: Journal of Algebra
publication date: 1997-01-14T00:00:00Z
review text: Let \(\Lambda\) be a two-sided Noetherian ring and \(\text{mod }\Lambda\) be the category of finitely generated \(\Lambda\)-modules. For each integer \(k>0\) we denote by \(\Omega^k(\text{mod }\Lambda)\) the full subcategory of \(\text{mod }\Lambda\) whose objects are the \(C\) in \(\text{mod }\Lambda\) which are the \(k\)th syzygies of modules in \(\text{mod }\Lambda\). While the projective resolutions of modules have been studied extensively, little is known about the subcategory \(\Omega^k(\text{mod }\Lambda)\) of \(\text{mod }\Lambda\). One of the few general properties known about these subcategories is that they are covariantly finite in \(\text{mod }\Lambda\). While in general covariantly finite subcategories of \(\text{mod }\Lambda\) have been proven to be of interest in various situations, they are particularly interesting when they are closed under module extension. The main theme of the paper under review is to describe when the subcategories \(\Omega^k(\text{mod }\Lambda)\) of \(\text{mod }\Lambda\) are extension closed. The main result is Theorem 0.1, which is too technical to be stated here. Interested readers are referred to the paper itself and the bibliography listed there, in particularly the following related papers by the same authors [Adv. Math. 86, No. 1, 111-152 (1991; Zbl 0774.16006); Proc. Tsukuba Workshop on Representations of Algebras, Lond. Math. Soc. Lect. Note Ser. 168, 1-42 (1992; Zbl 0774.16005); J. Pure Appl. Algebra 92, No. 1, 1-27 (1994; Zbl 0803.16016)].
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16G10
zbMATH DE Number: 923677
zbMATH Keywords: covariantly finite subcategories
zbMATH Keywords: extension closed subcategories
zbMATH Keywords: syzygies of modules
zbMATH Keywords: two-sided Noetherian rings
zbMATH Keywords: categories of finitely generated modules
zbMATH Keywords: projective resolutions
zbMATH Keywords: module extensions
reviewed by: Hua-Ping Yu
MaRDI profile type: Publication
full work available at URL: jabr.1996.0212
OpenAlex ID: W1970038355
DOI: 10.1006/JABR.1996.0212
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description: scientific article; zbMATH DE number 923677