De Rahm cohomology of local cohomology modules-The graded case

Abstract

Let $K$ be a field of characteristic zero, $R=K[X_1,...,X_n]$. Let $A_n(K)=K<X_1,...,X_n,{\partial }_1,...,{\partial }_n>$ be the $n^{th}$ Weyl algebra over $K$. We consider the case when $R$ and $A_n(K)$ is graded by giving $°X_i={\omega }_i$ and $°{\partial }_i=-{\omega }_i$ for $i=1,...,n$ (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 $i\ge 0$. In this article we prove that the De Rahm cohomology modules $H^*(\bP ; H^*_I(R))$ is concentrated in degree $-\omega$, i.e., $H^*(\bP ; H^*_I(R))_j = 0$ for $j\ne -\omega$. As an application when $A=R/(f)$ is an isolated singularity we relate $H^{n-1}(\bP ; H^1_{(f)}(R)$ to $H^{n-1}(\partial (f);A)$, the $(n-1{)}^{th}$ Koszul cohomology of $A$ \wrt \ ${\partial }_1(f),...,{\partial }_n(f)$.

References in corpus (2)

• Eulerian graded $D$-modules
• De Rahm Cohomology of Local Cohomology modules II