Nagoya Mathematical Journal

de Rham cohomology of local cohomology modules: The graded case

Tony J. Puthenpurakal

Full-text: Open access


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

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

First available in Project Euclid: 26 January 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

local cohomology associated primes D-modules Koszul homology


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.

Export citation


  • [1] J.-E. Björk, Rings of Differential Operators, North-Holland Math. Library 21, North-Holland, Amsterdam, 1979.
  • [2] W. Bruns and J. Herzog, Cohen–Macaulay Rings, Cambridge Stud. Adv. Math. 39, Cambridge University Press, Cambridge, 1993.
  • [3] G. Lyubeznik, Finiteness properties of local cohomology modules (an application of D-modules to commutative algebra), Invent. Math. 113 (1993), 41–55.
  • [4] L. Ma and W. Zhang, Eulerian graded $\mathcal{D}$-modules, Math. Res. Lett. 21 (2014), 149–167.
  • [5] T. J. Puthenpurakal, De Rham cohomology of local cohomology modules, preprint, arXiv:1302.0116v2 [math.AC].
  • [6] T. J. Puthenpurakal and R. B. T. Reddy, de Rham cohomology of $H^{1}_{(f)}(R)$, where $V(f)$ is a smooth hypersurface in $\mathbb{P}^{n}$, preprint, arXiv:1310.4654v1 [math.AC].