Some finiteness results for local cohomology modules with respect to a pair of ideals
type: Item
instance of: scholarly article
title: Some finiteness results for local cohomology modules with respect to a pair of ideals
zbMATH Open document ID: 1469.13039
DOI: 10.1134/S0001434621030020
author: Batoul Naal
author: K. Khashyarmanesh
published in: Mathematical Notes
publication date: 2021-07-08T00:00:00Z
review text: Let \(M\) denote a finitely generated module over a Noetherian ring \(R\). For two ideals \(I, J\) \textit{R. Takahashi} et al. [J. Pure Appl. Algebra 213, No. 4, 582--600 (2009; Zbl 1160.13013)]) introduced the local cohomology \(H^i_{I, J}(M)\) for the pair of ideals. Note that for \(J = R\) it coincides with the ordinary local cohomology \(H^i_I(M)\). As a main result the authors prove: Suppose that \(H^i_{I, J}(M)/JH^i_{I, J}(M)\) is finitely generated for all \(i < n\) and that \(\operatorname{Tor}_1^R(R/J,L) = 0\) for any \((I,J)\)-torsion module \(L\). Then \(\operatorname{Hom}_R(R/I,H^n_{I, J}(M)/JH^n_{I, J}(M))\) is finitely generated, in particular \(\operatorname{Ass}_R H^n_{I, J}(M)/JH^n_{I, J}(M)\) is a finite set. This generalizes the classical situation for ordinary local cohomology, i.e. \(J = R\), discussed in several articles. Moreover, the authors study sufficient conditions for the finiteness of \(\operatorname{Hom}_R(R/I,H^i_{I, J}(M)/JH^i_{I, J}(M) )\).
reviewed by: Peter Schenzel
Mathematics Subject Classification ID: 13D45
zbMATH DE Number: 7369709
zbMATH Keywords: local cohomology with respect to pair of ideals
zbMATH Keywords: associated prime ideals
zbMATH Keywords: filter regular element
MaRDI profile type: Publication
cites work: Affine duality and cofiniteness
cites work: A finiteness result for associated primes of local cohomology modules
cites work: A FINITENESS RESULT FOR LOCAL COHOMOLOGY MODULES
cites work: On annihilators and associated primes of local cohomology modules
cites work: Bass Numbers of Local Cohomology Modules
cites work: On the finiteness properties of extension and torsion functors of local cohomology modules
cites work: A generalization of the cofiniteness problem in local cohomology modules
cites work: Local cohomology based on a nonclosed support defined by a pair of ideals
cites work: Some results on local cohomology modules defined by a pair of ideals
cites work: Some splitting theorems for local cohomology modules with respect to a pair of ideals
cites work: Q4819620
cites work: Filter regular sequences and the finiteness of local cohomology modules
cites work: Local cohomology. An algebraic introduction with geometric applications
full work available at URL: s0001434621030020
OpenAlex ID: W3182990082
description: scientific article; zbMATH DE number 7369709