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Graded Components of Local Cohomology Modules II

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Abstract

Let A be a commutative Noetherian ring containing a field of characteristic zero. Let R = A[X1,…,Xm] be a polynomial ring and Am(A) = A〈X1,…,Xm, ∂1,…,∂m〉 be the m th Weyl algebra over A, where ∂i = ∂/∂Xi. Consider standard gradings on R and Am(A) by setting \(\deg z=0\) for all z ∈ A, \(\deg X_{i}=1\), and \(\deg \partial _{i} =-1\) for i = 1,…,m. We present a few results about the behavior of the graded components of local cohomology modules \({H_{I}^{i}}(R)\), where I is an arbitrary homogeneous ideal in R. We mostly restrict our attention to the vanishing, tameness, and rigidity properties. To obtain this, we use the theory of D-modules and show that generalized Eulerian Am(A)-modules exhibit these properties. As a corollary, we further get that components of graded local cohomology modules with respect to a pair of ideals display similar behavior.

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Acknowledgements

The first author thanks SERB, Department of Science and Technology, Government of India, for the project grant MATRICS (Project No. MTR/2017/000585). We thank the referee for a careful reading and many pertinent remarks.

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  1. Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400 076, India

    Tony J. Puthenpurakal & Sudeshna Roy

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  1. Tony J. Puthenpurakal
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  2. Sudeshna Roy
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Correspondence to Tony J. Puthenpurakal.

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Puthenpurakal, T.J., Roy, S. Graded Components of Local Cohomology Modules II. Vietnam J. Math. 52, 1–24 (2024). https://doi.org/10.1007/s10013-022-00555-6

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