Cofinality of Galois Cohomology within Purely Quadratic Graded Algebras

TL;DR

This work characterizes how Galois cohomology algebras $H^ullet(F)$ of fields $F$ containing a $p$-th root of unity embed into purely quadratic augmented $\\mathbb{F}_p$-bilinear structures. It introduces the κ-algebra framework and proves that augmented bilinear maps of field type are cofinal in all augmented bilinear maps, realized through intricate field-construction methods including Amitsur’s generic splitting fields and transfinite extensions. The authors relate these realizations to pro-$p$ right-angled Artin groups and show, in the $p=2$ case, that additional properties (like the common slot property) are necessary to characterize field realizability, yielding both positive realizations and explicit counterexamples. The results bridge Galois cohomology, quadratic form theory, and graph-based pro-$p$ groups, with implications for understanding which bilinear structures arise from fields and how these structures can be manipulated via generic splitting fields and power-series fields.

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.

Theorems & Definitions (37)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Remark 3.1
• Theorem 4.1
• proof
• Remark 4.2
• Proposition 5.1
• Proposition 5.2