De Rahm cohomology of local cohomology modules

TL;DR

This paper investigates the structure of local cohomology modules over polynomial rings in characteristic zero by computing their Koszul homology with respect to partial derivatives, revealing new insights into their algebraic properties.

Contribution

It computes the Koszul homology modules of local cohomology modules for specific classes of ideals, advancing understanding of their structure as holonomic modules.

Findings

Computed Koszul homology modules for certain ideals

Provided new structural insights into local cohomology modules

Extended previous results on holonomic modules

Abstract

Let $K$ be a field of characteristic zero, $R = K[X_1,...,X_n]$ and let $I$ be an ideal in $R$. Let $A_n(K) = K<X_1,...,X_n, \partial_1,..., \partial_n>$ be the $n^{th}$ Weyl algebra over $K$. 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 compute the Koszul homology modules $H_*(\partial_1,...,\partial_n ; H^*_I(R))$ for certain classes of ideals.