De Rahm Cohomology of Local Cohomology modules II

TL;DR

This paper computes the Euler characteristic of De-Rahm cohomology for local cohomology modules over polynomial rings, extending understanding of their algebraic and topological properties in characteristic zero.

Contribution

It provides explicit calculations of De-Rahm cohomology Euler characteristics for local cohomology modules associated with certain prime ideals in polynomial rings.

Findings

Euler characteristic formulas for specific prime ideals

Extension of De-Rahm cohomology understanding for local cohomology modules

Connections between local cohomology and De-Rahm cohomology in characteristic zero

Abstract

Let $K$ be an algebraically closed field of characteristic zero and let $R = K[X_1,\ldots,X_n]$. Let $I$ be an ideal in $R$. Let $A_n(K)$ be the $n^{th}$ Weyl algebra over $K$. By a result of Lyubeznik, the local cohomology modules $H^i_I(R)$ are holonomic $A_n(K)$-modules for each $i \geq 0$. In this paper we compute the Euler characteristic of De-Rahm cohomology of $H^{\htt P}_P(R)$ for certain classes of prime ideals $P$ in $R$.