Cofinality of Galois cohomology within purely quadratic graded algebras

Abstract

Let $p$ be a prime number. For a field $F$ containing a root of unity of order $p$, let $H^{\bullet }(F)=H^{\bullet }(F,{\mathbb{F}}_p)$ be the mod-$p$ Galois cohomology graded ${\mathbb{F}}_p$-algebra of $F$. By the Norm Residue Theorem, $H^{\bullet }(F)$ is a purely quadratic graded-commutative algebra, and is therefore determined by the cup product $\cup :H^1(F)\times H^1(F)\to H^2(F)$. We prove that the class of all Galois cohomology algebras $H^{\bullet }(F)$ is cofinal in the class of all purely quadratic graded-commutative ${\mathbb{F}}_p$-algebras $A_{\bullet }$, in the following sense: For every $A_{\bullet }$ there exists $F$ such that the bilinear map $A_1\times A_1\to A_2$, which determines $A_{\bullet }$, embeds in the cup product bilinear map $\cup :H^1(F)\times H^1(F)\to H^2(F)$.
We further provide examples of ${\mathbb{F}}_p$-bilinear maps which are not realizable by fields $F$ in this way. These are related to recent results by Snopce–Zalesskii and Blumer-Quadrelli–Weigel on the Galois theory of pro-$p$ right-angled Artin groups, as well as to a conjecture by Marshall on the possible axiomatization of quadratic form theory of fields.