Gorenstein injective, Gorenstein flat modules and the section functor
type: Item
instance of: scholarly article
title: Gorenstein injective, Gorenstein flat modules and the section functor
zbMATH Open document ID: 1120.13015
author: Reza Sazeedeh
published in: Journal of Pure and Applied Algebra
publication date: 2007-09-03T00:00:00Z
review text: Let \({\mathfrak{a}}\) be an ideal of a commutative noetherian ring \(R\) with a non-zero identity. For any two \(R\)-modules \(M\) and \(N\), the \(i\)-th generalized local cohomology module of \(M\) and \(N\) with respect to \({\mathfrak{a}}\) is defined by \(H^i_{{\mathfrak{a}}}(M,N):=\underset{n}{\varinjlim}\text{Ext}_R^i(M/{\mathfrak{a}}^n M,N)\). This notion was defined by J. Herzog in 1974. The aim of the paper under review is to investigate generalized local cohomology modules in conjunction with the theory of Gorenstein homological dimensions. The theory of Gorenstein homological dimensions was initiated by \textit{E. E. Enochs} and \textit{O. M. G. Jenda} [Math. Z. 220, No. 4, 611--633 (1995; Zbl 0845.16005)]. An \(R\)-module \(N\) is said to be Gorenstein injective if there exists an exact sequence \[ E_{\bullet}: \dots {\longrightarrow} E_1{\longrightarrow} E_0{\longrightarrow} E^0{\longrightarrow} E^1{\longrightarrow} \dots \] of injective \(R\)-modules such that \(N\cong \ker(E^0{\longrightarrow} E^1)\) and that \(\Hom_R(E,E_{\bullet})\) is exact for all injective \(R\)-modules \(E\). Also, an \(R\)-module \(N\) is said to be Gorenstein flat if there exists an exact sequence \[ F_{\bullet}: \dots {\longrightarrow} F_1{\longrightarrow} F_0{\longrightarrow} F^0{\longrightarrow} F^1{\longrightarrow} \dots \] of flat \(R\)-modules such that \(N\cong \ker(F^0{\longrightarrow} F^1)\) and that \(F_{\bullet}\otimes_RE\) is exact for all injective \(R\)-modules \(E\). Then, the notions Gorenstein injective (respectively, flat) dimension of an \(R\)-module \(N\) is defined as the infimum of the length of Gorenstein injective (respectively, flat) resolutions of \(N\). Assume that \(M\) is finitely generated and Krull dimension of \(R\) and projective dimension of \(M\) are finite. It is proved that any Gorenstein injective \(R\)-module is \(H^0_{{\mathfrak{a}}}(M,\cdot)\)-acyclic. Therefore for any \(R\)-module \(N\), the modules \(H^i_{{\mathfrak{a}}}(M,N)\)'s c…
reviewed by: Kamran Divaani-Aazar
Mathematics Subject Classification ID: 13D45
Mathematics Subject Classification ID: 13D07
zbMATH DE Number: 5186661
zbMATH Keywords: generalized local cohomology
zbMATH Keywords: homological dimensions
zbMATH Keywords: Auslander class
zbMATH Keywords: Bass class
MaRDI profile type: Publication
full work available at URL: j.jpaa.2007.04.013
OpenAlex ID: W2003435423
Wikidata QID: Q114155524
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DOI: 10.1016/J.JPAA.2007.04.013
description: scientific article; zbMATH DE number 5186661