Nagoya Mathematical Journal

de Rham cohomology of local cohomology modules: The graded case

Tony J. Puthenpurakal

Full-text: Open access

Abstract

Let K be a field of characteristic zero, and let R=K[X1,,Xn]. Let An(K)=KX1,,Xn,1,,n be the nth Weyl algebra over K. We consider the case when R and An(K) are graded by giving degXi=ωi and degi=ωi for i=1,,n (here ωi are positive integers). Set ω=k=1nωk. Let I be a graded ideal in R. By a result due to Lyubeznik the local cohomology modules HIi(R) are holonomic (An(K))-modules for each i0. In this article we prove that the de Rham cohomology modules H(;HI(R)) are concentrated in degree ω; that is, H(;HI(R))j=0 for jω. As an application when A=R/(f) is an isolated singularity, we relate Hn1(;H(f)1(R)) to Hn1((f);A), the (n1)th Koszul cohomology of A with respect to 1(f),,n(f).

Article information

Source
Nagoya Math. J., Volume 217 (2015), 1-21.

Dates
First available in Project Euclid: 26 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1422282101

Digital Object Identifier
doi:10.1215/00277630-2857430

Mathematical Reviews number (MathSciNet)
MR3343837

Zentralblatt MATH identifier
1330.13028

Subjects
Primary: 13D45: Local cohomology [See also 14B15]
Secondary: 13N10: Rings of differential operators and their modules [See also 16S32, 32C38]

Keywords
local cohomology associated primes D-modules Koszul homology

Citation

Puthenpurakal, Tony J. de Rham cohomology of local cohomology modules: The graded case. Nagoya Math. J. 217 (2015), 1--21. doi:10.1215/00277630-2857430. https://projecteuclid.org/euclid.nmj/1422282101


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References

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