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Non-vanishing and cofiniteness of generalized local cohomology modules

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Abstract

In this paper, we show some results on the non-vanishing of the generalized local cohomology modules \(H^i_I(M,N)\). In a Cohen–Macaulay local ring \((R,\mathop {\mathfrak {m}})\), we prove, by using induction on \(\dim N\), that if M, N are two finitely generated R-modules with \({\text {id}}\,M<\infty \) and \({\text {Gid}}\,N<\infty \), then \(H^{\dim R-grade _R({\text {Ann}}_RN,M)}_{\mathop {\mathfrak {m}}}(M,N)\ne 0\). We also study the I-cofiniteness of the generalized local cohomology module \(H^i_{\mathop {\mathfrak {m}}}(M,N)\).

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Acknowledgements

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

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Authors and Affiliations

  1. Department of Mathematics-Informatics, Ho Chi Minh University of Education, Ho Chi Minh city, Vietnam

    Tran Tuan Nam

  2. Vietnam National University, Ho Chi Minh City, Vietnam

    Nguyen Minh Tri

  3. Department of Mathematics and Physics, University of Information Technology, Ho Chi Minh City, Vietnam

    Nguyen Minh Tri

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  1. Tran Tuan Nam
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  2. Nguyen Minh Tri
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Correspondence to Tran Tuan Nam.

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This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.04-2023.22.

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Nam, T.T., Tri, N.M. Non-vanishing and cofiniteness of generalized local cohomology modules. Period Math Hung 88, 461–474 (2024). https://doi.org/10.1007/s10998-023-00567-w

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