An analog of Quillen-Venkov for eventually negligible classes and radicality for finite irreducible Coxeter groups

Let $G$ be a finite group, $M$ a $G$-module, and $F$ a field. A continuous group homomorphism $i:{\mathrm{\Gamma }}_L=\text{Gal}(L_{\text{sep}}\mathrm{/}L)\to G$ yields a homomorphism $i^{\ast }:H^d(G,M)\to H^d(L,M)$ for every $d\in \mathbb{N}$. An element $u\in H^d(G,M)$ is negligible over $F$ if $u\in \ker (i^{\ast })$ for every $i$ and every field extension $L\mathrm{/}F$. If $M={\mathbb{F}}_p$ for a prime $p$ with a trivial $G$-action, an element $u$ is eventually negligible over $F$ if $u^k$ is negligible over $F$ for some $k\ge 1$. The (eventually) negligible elements over $F$ form an ideal of the cohomology ring $H^{\ast }(G,{\mathbb{F}}_p)$, called the (eventually) negligible ideal. We prove an analog of Quillen-Venkov for eventually negligible elements. An element $u\in H^{\ast }(G,{\mathbb{F}}_p)$ is eventually negligible over $F$ if and only if the restriction of $u$ to the cohomology of each elementary abelian $p$-subgroup of $G$ is eventually negligible over $F$. Paired with applications of a Gysin-type long exact sequence, we identify generators of the negligible ideal over $\mathbb{Q}$ of the mod-2 cohomology of dihedral groups, cyclic $2$-groups, and generalized quaternion groups. We prove that the negligible ideal over $\mathbb{Q}$ of the mod-2 cohomology of every finite irreducible Coxeter group is radical. The result was previously known for Weyl groups; we prove it for the remaining non-Weyl types $I_2(m)$, $H_3$, and $H_4$.