## Nagoya Mathematical Journal

### de Rham cohomology of local cohomology modules: The graded case

Tony J. Puthenpurakal

#### Abstract

Let $K$ be a field of characteristic zero, and let $R=K[X_{1},\ldots,X_{n}]$. Let $A_{n}(K)=K\langle X_{1},\ldots,X_{n},\partial_{1},\ldots,\partial_{n}\rangle$ be the $n$th Weyl algebra over $K$. We consider the case when $R$ and $A_{n}(K)$ are graded by giving $\operatorname{deg}X_{i}=\omega_{i}$ and $\operatorname{deg}\partial_{i}=-\omega_{i}$ for $i=1,\ldots,n$ (here $\omega_{i}$ are positive integers). Set $\omega=\sum_{k=1}^{n}\omega_{k}$. Let $I$ be a graded ideal in $R$. By a result due to Lyubeznik the local cohomology modules $H^{i}_{I}(R)$ are holonomic $(A_{n}(K))$-modules for each $i\geq0$. In this article we prove that the de Rham cohomology modules $H^{*}(\partial ;H^{*}_{I}(R))$ are concentrated in degree $-\omega$; that is, $H^{*}(\partial ;H^{*}_{I}(R))_{j}=0$ for $j\neq-\omega$. As an application when $A=R/(f)$ is an isolated singularity, we relate $H^{n-1}(\partial ;H^{1}_{(f)}(R))$ to $H^{n-1}(\partial(f);A)$, the $(n-1)$th Koszul cohomology of $A$ with respect to $\partial_{1}(f),\ldots,\partial_{n}(f)$.

#### Article information

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

Dates
First available in Project Euclid: 26 January 2015

https://projecteuclid.org/euclid.nmj/1422282101

Digital Object Identifier
doi:10.1215/00277630-2857430

Mathematical Reviews number (MathSciNet)
MR3343837

Zentralblatt MATH identifier
1330.13028

#### 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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