Top dickson class and annihilators of cohomology over invariant rings

TL;DR

This paper demonstrates that a power of the top Dickson class in invariant rings annihilates various cohomology modules, revealing new algebraic properties of these invariants and their cohomological behavior.

Contribution

It establishes that the top Dickson class's power lies in the radical of annihilators of certain cohomology modules over invariant rings, a novel cohomological annihilation result.

Findings

The top Dickson class annihilates the cohomology modules H^i(G, R) for all i ≥ 1.

A fixed power of the top Dickson class annihilates local cohomology modules of S.

The results hold uniformly across various cohomological degrees.

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.