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.
Categories: math.AC