Top local cohomology modules and Gorenstein injectivity with respect to a semidualizing module
type: Item
instance of: scholarly article
title: Top local cohomology modules and Gorenstein injectivity with respect to a semidualizing module
zbMATH Open document ID: 1246.13021
author: Maryam Salimi
author: Elham Tavasoli
author: Siamak Yassemi
published in: Archiv der Mathematik
publication date: 2012-05-23T00:00:00Z
review text: Let \((R,\mathfrak m)\) be a Cohen-Macaulay complete local ring. It is known that \(R\) is Gorenstein if and only if the top local cohomology module \(H_{\mathfrak m}^{\dim R}(R)\) is isomorphic to \(E(R/\mathfrak{m})\). Recently, Yoshizawa proved that the following are equivalent: i) \(R\) is a Gorenstein; ii) \(H_{\mathfrak m}^{\dim R}(R)\) is an injective \(R\)-module; iii) \(H_{\mathfrak m}^{\dim R}(R)\) is a Gorenstein injective \(R\)-module. (See Theorem 2.6 in the paper: [\textit{T. Yoshizawa}, ``On Gorenstein injectivity of top local cohomology modules'', Proc. Am. Math. Soc. 140, No. 6, 1897--1907 (2012; Zbl 1251.13017)]). Let \(C\) be a semidualizing \(R\)-module. In the paper under review, it is shown that the following statements are equivalent: i) \(C\) is a dualizing \(R\)-module; ii) \(H_{\mathfrak m}^{\dim R}(R)\) is a \(C\)-injective \(R\)-module; iii) \(H_{\mathfrak m}^{\dim R}(R)\) is a \(G_C\)-injective \(R\)-module. iv) \(H_{\mathfrak m}^{\dim R}(R)\) has finite \(G_C\)-injective dimension. The authors did not assume that \(R\) is Cohen-Macaulay, but one should be aware that their given argument for the implication \(iv)\Longrightarrow i)\) is correct only when \(R\) is Cohen-Macaulay.
reviewed by: Kamran Divaani-Aazar
Mathematics Subject Classification ID: 13D05
Mathematics Subject Classification ID: 13D45
zbMATH DE Number: 6037954
zbMATH Keywords: local cohomology
zbMATH Keywords: semidualizing modules
MaRDI profile type: Publication
full work available at URL: s00013-012-0371-5
OpenAlex ID: W1998266089
Wikidata QID: Q122941626
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DOI: 10.1007/S00013-012-0371-5
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description: scientific article; zbMATH DE number 6037954