An analog of Quillen-Venkov for eventually negligible classes and radicality for finite irreducible Coxeter groups
Abstract
Let G be a finite group, M a G-module, and F a field. A continuous group homomorphism i : ΓL=Gal(Lsep/L) → G yields a homomorphism i* : Hd(G,M) → Hd(L,M) for every d∈ ℕ. An element u ∈ Hd(G,M) is negligible over F if u∈ker(i*) for every i and every field extension L/F. If M=𝔽p for a prime p with a trivial G-action, an element u is eventually negligible over F if uk is negligible over F for some k≥1. The (eventually) negligible elements over F form an ideal of the cohomology ring H*(G,𝔽p), called the (eventually) negligible ideal. We prove an analog of Quillen-Venkov for eventually negligible elements. An element u∈ H*(G,𝔽p) 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 ℚ of the mod-2 cohomology of dihedral groups, cyclic $2$-groups, and generalized quaternion groups. We prove that the negligible ideal over 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 I2(m), H3, and H4.
Related
- Quillen's finiteness conjecture for the cotangent complex in characteristic two
- Toward Quillen stratification for the odd primary Steenrod algebra
- The Quillen and sharbly Hopf algebra structures on Steinberg homology coincide
- Quillen's local parameter conjecture in dimension four
- Do I Even Need a Multilevel Model? Negligible Effect Significance Testing for Intraclass Correlation Coefficients
BibTeX
@misc{pradenne2026analog,
title = {{An analog of Quillen-Venkov for eventually negligible classes and radicality for finite irreducible Coxeter groups}},
author = {Constantin Cedillo Vayson de Pradenne and Matthew Gherman and Enoch Guo and Colin La},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19251},
eprint = {2610.19251},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19251},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An analog of Quillen-Venkov for eventually negligible classes and radicality for finite irreducible Coxeter groups
AU - Constantin Cedillo Vayson de Pradenne
AU - Matthew Gherman
AU - Enoch Guo
AU - Colin La
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.19251
UR - https://arxiv.org/abs/2610.19251
AB - Let G be a finite group, M a G-module, and F a field. A continuous group homomorphism i : Γ_L=Gal(L_sep/L) → G yields a homomorphism i^* : H^d(G,M) → H^d(L,M) for every d∈ ℕ. An element u ∈ H^d(G,M) is negligible over F if u∈ker(i^*) for every i and every field extension L/F. If M=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≥1. The (eventually) negligible elements over F form an ideal of the cohomology ring H^*(G,𝔽_p), called the (eventually) negligible ideal. We prove an analog of Quillen-Venkov for eventually negligible elements. An element u∈ H^*(G,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 ℚ of the mod-2 cohomology of dihedral groups, cyclic $2$-groups, and generalized quaternion groups. We prove that the negligible ideal over 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.
ER -