Mild pro-p groups and the Koszulity conjectures

Abstract

Let $p$ be a prime, and ${\mathbb{F}}_p$ the field with $p$ elements. We prove that if $G$ is a mild pro-$p$ group with quadratic ${\mathbb{F}}_p$-cohomology algebra $H^{\bullet }(G,{\mathbb{F}}_p)$, then the algebras $H^{\bullet }(G,{\mathbb{F}}_p)$ and $\mathrm{gr}{\mathbb{F}}_p[[G]]$ - the latter being induced by the quotients of consecutive terms of the $p$-Zassenhaus filtration of $G$ - are both Koszul, and they are quadratically dual to each other. Consequently, if the maximal pro-$p$ Galois group of a field is mild, then Positselski's and Weigel's Koszulity conjectures hold true for such a field.

Final revised version, to be published on «Expositiones Mathematicæ»

References in corpus (4)

• Algebraic invariants for right-angled Artin groups
• Norm varieties and algebraic cobordism
• Koszul property and Bogomolov's conjecture
• Bloch-Kato pro-p groups and locally powerful groups