Graded components of local cohomology modules over invariant rings-II

TL;DR

This paper investigates the structure of graded components of local cohomology modules over invariant rings formed by finite group actions on polynomial extensions of regular rings, providing new insights even over fields.

Contribution

It offers a comprehensive analysis of local cohomology modules over invariant rings, including new results for arbitrary homogeneous ideals and cases where the base ring is a field.

Findings

Stronger results for group actions in $GL_m(K)$

New insights into local cohomology over invariant rings

Analysis applicable to arbitrary homogeneous ideals

Abstract

Let $A$ be a regular ring containing a field $K$ 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 $i$. Let $G$ be a finite subgroup of $GL_m(A)$. Let $G$ act linearly on $R$ fixing $A$. Let $S = R^G$. In this paper we present a comprehensive study of graded components of local cohomology modules $H^i_I(S)$ where $I$ is an \emph{arbitrary} homogeneous ideal in $S$. We prove stronger results when $G \subseteq GL_m(K)$. Some of our results are new even in the case when $A$ is a field.