One relator maximal pro-p Galois groups and the Koszulity conjectures

Abstract

Let $p$ be a prime number and let $K$ be a field containing a root of 1 of order $p$. If the absolute Galois group $G_K$ satisfies $\dim H^1(G_K,{\mathbb{F}}_p)<\infty$ and $\dim H^2(G_K,{\mathbb{F}}_p)=1$, we show that L.~Positselski's and T.~Weigel's Koszulity conjectures are true for $K$. Also, under the above hypothesis we show that the ${\mathbb{F}}_p$-cohomology algebra of $G_K$ is the quadratic dual of the graded algebra ${\mathrm{gr}}_{\bullet }{\mathbb{F}}_p[G_K]$, induced by the powers of the augmentation ideal of the group algebra ${\mathbb{F}}_p[G_K]$, and these two algebras decompose as products of elementary quadratic algebras. Finally, we propose a refinement of the Koszulity conjectures, analogous to I. Efrat's Elementary Type Conjecture.

Updated version

References in corpus (8)

• Norm varieties and algebraic cobordism
• Koszul property and Bogomolov's conjecture
• The Kummerian Property and Maximal Pro-$p$ Galois Groups
• Bloch-Kato pro-p groups and locally powerful groups
• Koszul algebras and quadratic duals in Galois cohomology
• On pro-$p$ groups with quadratic cohomology
• Profinite groups with a cyclotomic $p$-orientation
• The Bloch-Kato Conjecture and Galois Theory

Cited by in corpus (3)

• On pro-$p$ groups with quadratic cohomology
• Enhanced Koszul properties in Galois cohomology
• Two families of pro-p groups that are not absolute Galois groups