Cofinality of Galois Cohomology within Purely Quadratic Graded Algebras

TL;DR

This paper demonstrates that Galois cohomology algebras of fields are cofinal among purely quadratic graded-commutative algebras, with examples of bilinear maps not realizable by any field.

Contribution

It establishes the cofinality of Galois cohomology algebras within purely quadratic graded-commutative algebras and provides examples of non-realizable bilinear maps.

Findings

Galois cohomology algebras are cofinal in purely quadratic graded-commutative algebras.

Existence of bilinear maps not realizable by any field.

Connections to recent results on pro-p right-angled Artin groups and quadratic form theory.

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 $\cup\colon H^1(F)\times H^1(F)\to H^2(F)$. 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 $\cup\colon H^1(F)\times H^1(F)\to H^2(F)$. 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.