On a Relation between Euler characteristics of \MakeLowercase{de} Rham cohomology and Koszul cohomology of graded local cohomology modules

Abstract

Let $K$ be a field of characteristic zero. Let $R = K[X_0, X_1,\ldots,X_n]$ be standard graded. Let $A_{n+1}(K)$ be the $(n + 1)^{th}$ Weyl algebra over $K$. Let $I$ be a homogeneous ideal of $R$ and let $M = H^i_I(R)$ for some $i \geq 0$. By a result of Lyubeznik, $M$ is a graded holonomic $A_{n +1}(K)$-module for each $i \geq 0$. Let $\chi^c(\mathbf{\partial}, M)$ ($\chi^c(\mathbf{X}, M)$) be the Euler characteristics of de Rham cohomology (resp. Koszul cohomology) of $M$. We prove $\chi^c(\mathbf{\partial}, M) = (-1)^{n+1}\chi^c(\mathbf{X}, M)$.