Top dickson class and annihilators of cohomology over invariant rings

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 $R={\mathbb{F}}_q[V]=\text{Sym}(V^{\ast })$ and let $S=R^G$. Let ${\mathbf{d}}_{d,0}$ be the top Dickson class, i.e., ${\mathbf{d}}_{d,0}={\prod }_{0\mathrm{\ne }v\in V^{\ast }}v$. 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 $i\ge 1$. (b) We also show that ${\mathbf{d}}_{d,0}\in \sqrt{{\text{ann}}_S{H}_{S_{+}}^j(S)}$ for all $0\le j\le d-1$ (here $H_{S_{+}}^j(S)$ 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.