Graded components of local cohomology modules of invariant rings

TL;DR

This paper investigates the structure of graded components of local cohomology modules of invariant rings under finite group actions, extending previous results to a new setting involving invariant rings of regular domains.

Contribution

It provides a comparative analysis of local cohomology modules of invariant rings, generalizing earlier work on polynomial rings to rings of invariants under finite automorphism groups.

Findings

Establishes properties of graded components of local cohomology modules of invariant rings.

Provides analogs of previous results for rings of invariants.

Enhances understanding of local cohomology in invariant theory.

Abstract

Let $A$ be a regular domain containing a field $K$ of characteristic zero, $G$ be a finite subgroup of the group of automorphisms of $A$ and $B=A^G$ be the ring of invariants of $G$. Let $S= A[X_1,\ldots, X_m]$ and $R= B[X_1, \ldots, X_m]$ be standard graded with $\ deg \ A=0$, $\ deg \ B=0$ and $\ deg \ X_i=1$ for all $i$. Extend the action of $G$ on $A$ to $S$ by fixing $X_i$. Note $S^G=R$. Let $I$ be an arbitrary homogeneous ideal in $R$. The main goal of this paper is to establish a comparative study of graded components of local cohomology modules $H_I^i(R)$ that would be analogs to those proven in a previous paper of the first author for $H_J^i(S)$ where $J$ is an arbitrary homogeneous ideal in $S$.