Top dickson class and annihilators of cohomology over invariant rings

Abstract

Let $\mathbb{F}_q$ denote the finite field with $q = p^r$ elements. Let $V$ be a finite dimensional vector space of dimension $d$ over $\mathbb{F}_q$ and let $G \subseteq GL(V)$ be a group. Let $R = \mathbb{F}_q[V] = \text{Sym}(V^*)$ and let $S = R^G$. Let $\mathbf{d}_{d,0} $ be the top Dickson class, i.e., $\mathbf{d}_{d,0} = \prod_{0\neq v \in V^*}v$. Surprisingly (a power of) $\mathbf{d}_{d,0}$ annihilates many cohomological modules. (a) Let $H^i(G, R)$ be the $i^{th}$-group cohomology of $R$ considered as a $S$-module. Set $J_i = \text{ann}_S \ H^i(G, R)$. We show that $\mathbf{d}_{d,0} \in \sqrt{J_i}$ for all $i \geq 1$. (b) We also show that $\mathbf{d}_{d,0} \in \sqrt{ \text{ann}_S \ H^j_{S_+}(S)}$ for all $ 0 \leq j \leq d - 1$ (here $H^j_{S_+}(S)$ is the $j^{th}$ local cohomology of $S$ with respect to $S_+$). As an application we get that there exists a fixed power of $\mathbf{d}_{d,0} $ which works as a cohomological annihilator.