Local cohomology of modular invariant rings

TL;DR

This paper investigates the local cohomology of invariant rings under finite group actions, revealing how it relates to the cohomology of the polynomial ring and providing insights into invariants like the a-invariant and Hilbert series.

Contribution

It offers new results on the structure of local cohomology modules of invariant rings, including comparisons with the cohomology of the polynomial ring and analysis of invariants.

Findings

Relations between $H^n_{ } (R^G)$ and $H^n_{ } (R)^G$ established

Results on the $a$-invariant of invariant rings derived

Hilbert series of local cohomology modules characterized

Abstract

For $K$ a field, consider a finite subgroup $G$ of $\operatorname{GL}_n(K)$ with its natural action on the polynomial ring $R:=K[x_1,\dots,x_n]$. Let $\mathfrak{n}$ denote the homogeneous maximal ideal of the ring of invariants $R^G$. We study how the local cohomology module $H^n_{\mathfrak{n}}(R^G)$ compares with $H^n_{\mathfrak{n}}(R)^G$. Various results on the $a$-invariant and on the Hilbert series of $H^n_\mathfrak{n}(R^G)$ are obtained as a consequence.