Local cohomology associated to the radical of a group action on a noetherian algebra

TL;DR

This paper studies the radical ideal associated with a group action on a noetherian algebra, exploring its local cohomology, impact on singularities, and how it influences the Cohen-Macaulay property of invariant subalgebras.

Contribution

It introduces methods to identify elements of the radical, relates local cohomology to singularities, and establishes an equivalence between categories to analyze properties of invariant subalgebras.

Findings

Methods to compute elements of the radical

Relation between local cohomology and singularities

Inheritance of Cohen-Macaulay property by invariants

Abstract

An arbitrary group action on an algebra $R$ results in an ideal $\mathfrak{r}$ of $R$. This ideal $\mathfrak{r}$ fits into the classical radical theory, and will be called the radical of the group action. If $R$ is a noetherian algebra with finite GK-dimension and $G$ is a finite group, then the difference between the GK-dimensionsof $R$ and that of $R/\mathfrak{r}$ is called the pertinency of the group action. We provide some methods to find elements of the radical, which helps to calculate the pertinency of some special group actions. The $\mathfrak{r}$-adic local cohomology of $R$ is related to the singularities of the invariant subalgebra $R^G$. We establish an equivalence between the quotient category of the invariant $R^G$ and that of the skew group ring $R*G$ through the torsion theory associated to the radical $\mathfrak{r}$. With the help of the equivalence, we show that the invariant subalgebra $R^G$ will inherit certain Cohen-Macaulay property from $R$.