Graded components of local cohomology modules

Abstract

Let A be a regular ring containing a field of characteristic zero and let \(R = A[X_1,\ldots , X_m]\). Consider R as standard graded with \(\deg A = 0\) and \(\deg X_i = 1\) for all i. In this paper we present a comprehensive study of graded components of local cohomology s \(H^i_I(R)\) where I is an arbitrary homogeneous ideal in R. Our study seems to be the first in this regard.

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Acknowledgements

In 2008 I asked Prof. G. Lyubeznik whether de Rham cohomology of local cohomology modules will be interesting. He told me that it will be of interest. I (and co-authors) developed techniques to study de Rham cohomology and Koszul cohomology of local cohomology modules in a series of papers [12, 13, 15, 16, 19]. These techniques have proved to be fantastically useful in this paper. I thank Prof. G. Lyubeznik for his advice and to him this paper is dedicated.I thank the referee for many pertinent comments. I thank Prof. C. Huneke for giving me extra time to submit a revised version of this paper.

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Correspondence to Tony J. Puthenpurakal.

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Dedicated to Prof. Gennady Lyubeznik.

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Puthenpurakal, T.J. Graded components of local cohomology modules. Collect. Math. (2021). https://doi.org/10.1007/s13348-020-00311-4

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Keywords

  • Local cohomology
  • Graded local cohomology
  • Ring of differential operators
  • Weyl algebra
  • De Rham (and Koszul)

Mathematics Subject Classification

  • Primary 13 D45
  • 14 B15
  • Secondary 13 N10
  • 32C36