On the theory and applications of \(G\)-dimension with respect to a semidualizing module
type: Item
instance of: scholarly article
title: On the theory and applications of \(G\)-dimension with respect to a semidualizing module
zbMATH Open document ID: 1548.1304
author: Cleto B. Miranda-Neto
author: Thyago S. Souza
published in: Journal of Algebra and its Applications
publication date: 2024-09-11T00:00:00Z
review text: Let \(R\) denote a commutative Noetherian ring. A finitely generated \(R\)-module \(C\) is called semi-dualizing whenever the homothety map \(R \to \operatorname{Hom}_R(C,C)\) is an isomorphism and \(\operatorname{Ext}^i_R(C,C) = 0\) for all \(i> 0\). Let \(C\) denote a semi-dualizing module. Then a finitely generated \(R\)-module \(M\) is called totally \(C\)-reflexive if the following holds: (i) \(M \to \operatorname{Hom}_R(\operatorname{Hom}_R(M,C),C)\) is an isomorphism. (ii) \(\operatorname{Ext}^i_R(M, C) = 0\) for all \(i > 0\). (iii) \(\operatorname{Ext}^i_R( \operatorname{Hom}(M.C),C) = 0\) for all \(i > 0\). - An \(R\)-module \(M\) has finite \(G_C\)-dimension if there is a finite resolution by totally \(C\)-reflexive modules. The well-defined number of members in a shortest such resolution is called \(G_C\)-dimension. In their paper the authors focus on the investigation of the corresponding relative properties of a module being totally \(C\)-reflexive, \(C_k\)-torsionless, and \(C_k\)-syzygy (especially when connected to the finiteness of the \(G_C\)-dimension). Moreover, they introduce the relative concept of \(C_q\)-Gorensteiness, extending the property of \(q\)-Gorensteiness investigated by \textit{V. Maşek} [Commun. Algebra 28, No. 12, 5783--5811 (2000; Zbl 1002.13005)]. They also consider \(C\)-duals and their Cohen-Macaulayness and initiate the study of \(C\)-valued derivation modules over a local ring of low depth.
reviewed by: Peter Schenzel
Mathematics Subject Classification ID: 13D05
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 13D07
Mathematics Subject Classification ID: 13E15
Mathematics Subject Classification ID: 13N15
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13D45
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 13C14
zbMATH DE Number: 7910528
zbMATH Keywords: Gorenstein dimension
zbMATH Keywords: semidualizing module
zbMATH Keywords: torsionless module
zbMATH Keywords: syzygy module
zbMATH Keywords: derivation module
MaRDI profile type: Publication
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DOI: 10.1142/S0219498824501342
description: scientific article; zbMATH DE number 7910528