Annhilators of local cohomology modules over modular invariant rings and Dickson polynomials

TL;DR

This paper investigates the annihilators of local cohomology modules over invariant rings formed by Dickson polynomials, providing new insights and a simplified proof of the Landweber-Stong conjecture.

Contribution

The authors identify specific Dickson polynomials as annihilators of local cohomology modules over modular invariant rings, offering novel theoretical results and applications.

Findings

Dickson polynomials are contained in the annihilators of certain local cohomology modules.

The results lead to a simpler proof of the Landweber-Stong conjecture.

Applications demonstrate the relevance of these annihilators in algebraic invariant theory.

Abstract

Let $\mathbb{F}_q$ be a finite field with $q = p^s$ elements. Let $V$ be a $d$ dimensional vector space over $\mathbb{F}_q$ and let $G$ be a subgroup of $GL(V)$. Let $R = \mathbb{F}_q[V] = \text{Sym}_{\mathbb{F}_q}(V^*)$ and let $G$ act naturally on $R$. Set $S = R^G$. Let $\mathbf{d}_{d,0}, \mathbf{d}_{d, 1}, \ldots, \mathbf{d}_{d, d-1} \in S$ be the Dickson polynomials with $\deg \mathbf{d}_{d,i} = q^d - q^i$. Let $I$ be a homogeneous ideal of $S$ and let $H^i_I(S)$ be the $i^{th}$-local cohomology module of $S$ with respect to $I$. Let $J_i = \sqrt{\text{ann} H^i_I(S)}$. Assume $J_i \neq 0$ and $\dim S/J_i = d - g$. Then we show that $\mathbf{d}_{d,0}, \ldots, \mathbf{d}_{d, d - g + 1} \in J_i$. We give several applications of our results. An application is a considerably simpler proof of Landweber-Stong conjecture.