Graded components of local cohomology modules

TL;DR

This paper investigates the structure of graded components of local cohomology modules over polynomial rings with a focus on arbitrary homogeneous ideals, providing new insights into their behavior in characteristic zero.

Contribution

It offers the first comprehensive analysis of graded components of local cohomology modules for arbitrary homogeneous ideals in polynomial rings over regular rings containing a field of characteristic zero.

Findings

Detailed description of graded components of local cohomology modules

New results on the structure and properties of these modules

Foundational work for further research in local cohomology

Abstract

Let $A$ be a regular ring containing a field of characteristic zero and let $R = A[X_1,\ldots, X_m]$. Consider $R$ as standard graded with $deg \ A = 0$ and $deg \ X_i = 1$ for all $i$. In this paper we present a comprehensive study of graded components of local cohomology modules $H^i_I(R)$ where $I$ is an \emph{arbitrary} homogeneous ideal in $R$. Our study seems to be the first in this regard.