On the top local cohomology modules
type: Item
instance of: scholarly article
title: On the top local cohomology modules
zbMATH Open document ID: 1244.13013
author: Le Thanh Nhan
author: Tran Do Minh Chau
published in: Journal of Algebra
publication date: 2012-05-23T00:00:00Z
reviewed by: Peter Schenzel
Mathematics Subject Classification ID: 13D45
Mathematics Subject Classification ID: 13E15
Mathematics Subject Classification ID: 13E10
zbMATH DE Number: 6037581
zbMATH Keywords: top local cohomology module
zbMATH Keywords: attached prime
zbMATH Keywords: co-support
zbMATH Keywords: multiplicity
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2011.08.027
OpenAlex ID: W2055661352
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DOI: 10.1016/J.JALGEBRA.2011.08.027
review text: Let \((R,\mathfrak{m})\) denote a local Noetherian ring. Let \(N\) be a finitely generated \(R\)-module. Then it is a well known fact that \(\text{Ann}_R (N/\mathfrak{p}N) = \mathfrak{p}\) for all \(\mathfrak{p} \in V(\text{Ann}_RN)\). Let \(A\) denote an Artinian \(R\)-module. Then the dual property (*) ``\(\text{Ann}_R (0:_A \mathfrak{p}) = \mathfrak{p}\) for all \(\mathfrak{p} \in V(\text{Ann}_RA)\)'' is true whenever \(R\) is complete but does not hold in general (see [\textit{Nguyen Tu Cung} and \textit{Le Thanh Nhan}, ``On the Noetherian dimension of Artinian modules'', Vietnam J. Math. 30, No. 2, 121--130 (2002; Zbl 1096.13523)]). See also \textit{H. Zöschinger} [Arch. Math. 95, No. 3, 225--231 (2010; Zbl 1206.13011)], where it is shown that (*) is satisfied for all Artinian \(R\)-moduls if and anly if \(R \to \hat{R}\) satisfies the Going Up Theorem.NEWLINENEWLINELet \(I \subset R\) denote and ideal and let \(M\) be a finitely generated \(R\)-module \(M\) and \(d = \dim M\). The authors investigate the property (*) for the case of the (Artinian) local cohomology module \(H^d_I(M)\). They use it in order to study the catenarity of the ring \(R/\text{Ann}_R H^d_I(M)\), the set of attached prime ideals, the co-support and the multiplicity of \(H^d_I(M)\).
description: scientific article; zbMATH DE number 6037581