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

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.