De Rahm cohomology of local cohomology modules-The graded case

TL;DR

This paper studies the graded De Rahm cohomology of local cohomology modules over polynomial rings with Weyl algebra actions, showing concentration in a specific degree and relating it to Koszul cohomology in singular cases.

Contribution

It proves the concentration of De Rahm cohomology in a specific degree for graded local cohomology modules and connects it to Koszul cohomology in isolated singularities.

Findings

De Rahm cohomology is concentrated in degree -ω.

Established a relation between De Rahm cohomology and Koszul cohomology in singular cases.

Extended understanding of local cohomology modules in graded Weyl algebra settings.

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 $\deg X_i = \omega_i $ and $\deg \partial_i = -\omega_i$ 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 $i \geq 0$. 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 $j \neq - \omega$. 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)$.