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Some Finiteness Results for Local Cohomology Modules with Respect to a Pair of Ideals

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Abstract

Suppose that \(R\) is a commutative Noetherian ring with identity, \(I\), \(J\) are ideals of \(R\), and let \(M\) be a finitely generated \(R\)-module. Let \(H^i_{I,J}(-)\) be the \(i\)th local cohomology functor with respect to \((I, J)\). In this paper, we show that the \(R\)-module

$$\mathrm{Hom}_R(R/I,H^1_{I,J}(M)/JH^1_{I,J}(M))$$

is always finitely generated. Moreover, we provide sufficient conditions such that the modules

$$\mathrm{Hom}_R(R/I,H^i_{I,J}(M)/JH^i_{I,J}(M)) \qquad \mathrm{or} \qquad \mathrm{Tor}^R_j(R/I,H^i_{I,J}(M)/JH^i_{I,J}(M))$$

are finitely generated.

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Acknowledgments

The authors are deeply grateful to the referee for careful reading of the manuscript and helpful suggestions.

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Authors and Affiliations

  1. Ferdowsi University of Mashhad, Mashhad, 1159-91775, Iran

    Batoul Naal & Kazem Khashyarmanesh

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  1. Batoul Naal
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  2. Kazem Khashyarmanesh
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Correspondence to Batoul Naal.

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Naal, B., Khashyarmanesh, K. Some Finiteness Results for Local Cohomology Modules with Respect to a Pair of Ideals. Math Notes 109, 335–346 (2021). https://doi.org/10.1134/S0001434621030020

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