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

TL;DR

The paper addresses the relationship between Euler characteristics of de Rham and Koszul cohomology for graded local cohomology modules $M=H^i_I(R)$ over a standard graded polynomial ring $R$ in characteristic zero. It develops a framework for graded holonomic generalized Eulerian $A_{n+1}(K)$-modules and proves the key identity $χ^c(\mathbf{∂}, M) = (-1)^{n+1} χ^c(\mathbf{X}, M)$, reducing to the $n=0$ base case and then lifting via induction on the dimension of the support. The main contribution is a complete proof of this relation for the generalized Eulerian class, using localization, exact sequences, and the structure of holonomic $A_{n+1}(K)$-modules. This result links de Rham and Koszul invariants in the D-module framework and has potential implications for understanding Lyubeznik numbers and local cohomology in graded settings.

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 $χ^c(\mathbf{\partial}, M)$ ($χ^c(\mathbf{X}, M)$) be the Euler characteristics of de Rham cohomology (resp. Koszul cohomology) of $M$. We prove $χ^c(\mathbf{\partial}, M) = (-1)^{n+1}χ^c(\mathbf{X}, M)$.

Theorems & Definitions (20)

• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Definition 2.1
• Definition 2.2
• Proposition 2.4
• proof
• Theorem 3.2
• Lemma 3.3