Noetherian dimension and co-localization of Artinian modules over local rings
type: Item
instance of: scholarly article
zbMATH Open document ID: 1307.13012
DOI: 10.1142/S1005386714000613
author: Le Thanh Nhan
author: Tran Do Minh Chau
published in: Algebra Colloquium
publication date: 2014-11-07T00:00:00Z
Mathematics Subject Classification ID: 13B21
Mathematics Subject Classification ID: 13E10
Mathematics Subject Classification ID: 13D45
Mathematics Subject Classification ID: 13J10
zbMATH DE Number: 6366788
zbMATH Keywords: local cohomology
zbMATH Keywords: secondary representations
zbMATH Keywords: attached primes
zbMATH Keywords: co-localization functor
MaRDI profile type: Publication
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cites work: The co-localization of an Artinian module
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cites work: Co-localization, co-support and local homology
cites work: KRULL DIMENSION FOR ARTINIAN MODULES OVER QUASI LOCAL COMMUTATIVE RINGS
cites work: Some Results on the Vanishing of Local Cohomology Modules
cites work: On the condition Going up for \({R \subset \hat{R}}\)
full work available at URL: s1005386714000613
OpenAlex ID: W2137969130
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title: Noetherian dimension and co-localization of Artinian modules over local rings
review text: The goal of this short paper is to establish conditions for the Noetherian local ring \(R\) when the Noetherian dimension for any local cohomology module \(H^i_{\mathfrak{m}}(M)\) supported at the maximal ideal is precisely equal to the Krull dimension of \(R\) modulo the annihilator of \(H^i_{\mathfrak{m}}(M)\). To do this, the authors use a co-localization functor \(F_{\mathfrak{p}}\) which is linear on Artinian \(R\)-modules, send Artinian \(R\)-modules to Artinian \(R_{\mathfrak{p}}\)-modules and \(F_{\mathfrak{p}}(A)\) is nonzero for all primes \(\mathfrak{p}\) containing the annihilator of \(A\). Their main result shows that \(N\)-\(\text{dim}_R(H^{i}_{\mathfrak{m}}(M))=\text{dim}_R(R/\text{Ann}_R(H^{i}_{\mathfrak{m}}(M))\) if and only if \(R\) is universally catenary and all its formal fibers are Cohen-Macaulay. The co-localization functor developed by \textit{A. S. Richardson} [Rocky Mt. J. Math. 36, No. 5, 1679--1703 (2006; Zbl 1134.13008)] satisfies the properties mentioned above for all local cohomology modules in the case that \(R\) is a complete ring but the authors are hoping that such a co-localization functor exists in the non complete case.
reviewed by: Janet Vassilev
description: scientific article; zbMATH DE number 6366788