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de Rham cohomology of local cohomology modules: The graded case

Part of: Commutative algebra: Homological methods Differential algebra

Published online by Cambridge University Press:  11 January 2016

Tony J. Puthenpurakal
Tony J. Puthenpurakal*
Affiliation:
Department of Mathematics, Indian Institute of Technology, Bombay Powai, Mumbai 400 076, India, [email protected]
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Abstract

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Let K be a field of characteristic zero, and let R = K[X1,… ,Xn]. Let An(K) = K⟨X1,… ,Xn,∂1,… ,∂n⟩ be the nth Weyl algebra over K. We consider the case when R and An(K) are graded by giving deg Xi = ωi and deg ∂i = –ωi for i = 1,…,n (here ωi are positive integers). Set . Let I be a graded ideal in R. By a result due to Lyubeznik the local cohomology modules are holonomic (An(K))-modules for each i≥0. In this article we prove that the de Rham cohomology modules are concentrated in degree —ω; that is, for j ≠ –ω. As an application when A = R/(f) is an isolated singularity, we relate to Hn-1(∂(f);A), the (n – 1)th Koszul cohomology of A with respect to ∂1(f),…, ∂n(f).

MSC classification

Secondary: 13D45: Local cohomology 13N10: Rings of differential operators and their modules
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 217 , March 2015 , pp. 1 - 21
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

References

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