de Rham cohomology of local cohomology modules: The graded case

Abstract

Let $K$ be a field of characteristic zero, and let $R=K[X_1,\dots ,X_n]$. Let $A_n\left(K\right)=K\langle X_1,\dots ,X_n,{\partial }_1,\dots ,{\partial }_n\rangle$ be the $n$th Weyl algebra over $K$. We consider the case when $R$ and $A_n\left(K\right)$ are graded by giving $deg{X}_i={\omega }_i$ and $deg{\partial }_i=-{\omega }_i$ for $i=1,\dots ,n$ (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 $H_I^i\left(R\right)$ are holonomic $\left(A_n\right(K\left)\right)$-modules for each $i\ge 0$. In this article we prove that the de Rham cohomology modules $H^{\ast }(\partial ;H_I^{\ast }(R\left)\right)$ are concentrated in degree $-\omega$; that is, $H^{\ast }(\partial ;H_I^{\ast }(R){)}_j=0$ for $j\ne -\omega$. As an application when $A=R/\left(f\right)$ is an isolated singularity, we relate $H^{n-1}(\partial ;H_{\left(f\right)}^1(R\left)\right)$ to $H^{n-1}\left(\partial \right(f);A)$, the $(n-1)$th Koszul cohomology of $A$ with respect to ${\partial }_1\left(f\right),\dots ,{\partial }_n\left(f\right)$.