Minimal injective and flat resolutions of modules over Gorenstein rings
type: Item
instance of: scholarly article
title: Minimal injective and flat resolutions of modules over Gorenstein rings
zbMATH Open document ID: 0827.13005
published in: Journal of Algebra
publication date: 1995-12-11T00:00:00Z
review text: In the first part of this paper the author studies the minimal injective resolutions of modules of finite flat dimension over a Gorenstein ring \(R\). The main tools used are the Bass invariants \(\mu_i ({\mathfrak p}, M)\). For instance: Gorenstein rings are characterized in terms of vanishing of \(\mu_i ({\mathfrak p}, F)\), for any flat \(R\)-module \(F\) and any \(i \neq \text{ht} ({\mathfrak p})\); an \(R\)-module \(M\), where \(R\) is Gorenstein, is shown to be of finite flat dimension \(s\) iff \(\mu_i ({\mathfrak p}, M) \neq 0\) implies \(i \leq \text{ht} ({\mathfrak p}) \leq i + s\). In the second part of the paper, the author deals with minimal flat resolutions of modules. He defines new invariants \(\pi_i ({\mathfrak p}, M)\), which are analogous to Bass invariants, when ``injective resolution'' is replaced by ``flat resolution''. Some interesting characterization of the Gorenstein property is given, in terms of vanishing of \(\pi_i ({\mathfrak p}, X)\), when \(X\) is an injective \(R\)- module. Finally, strongly cotorsion modules \(G\) are defined by the condition \(\text{Ext}^1_R (X,C) = 0\), for all \(X\) of finite flat dimension. Strongly cotorsion modules over a Gorenstein ring are exactly those admitting a minimal flat resolution, while a strongly torsion free module \(M\) over a Gorenstein ring \(R\) (which means \(\text{Tor}^R_1 (X,M) = 0\) for all \(X\) of finite flat dimension) is characterized by: \(\mu_i ({\mathfrak p}, M) = 0\) if \(\text{ht} {\mathfrak p} > i\), \(i \geq 0\).
Mathematics Subject Classification ID: 13C11
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 783931
zbMATH Keywords: flat resolution
zbMATH Keywords: minimal injective resolution
zbMATH Keywords: modules of finite flat dimension
zbMATH Keywords: Gorenstein ring
zbMATH Keywords: Bass invariants
zbMATH Keywords: Gorenstein property
author: Jin Zhong Xu
reviewed by: Carla Massaza
MaRDI profile type: Publication
full work available at URL: jabr.1995.1196
OpenAlex ID: W2027220533
Wikidata QID: Q122930171
DOI: 10.1006/JABR.1995.1196
description: scientific article; zbMATH DE number 783931