Mild pro-p groups and the Koszulity conjectures

TL;DR

This paper proves that certain mild pro-p groups with quadratic cohomology have Koszul algebras, supporting conjectures relating to Galois groups and algebraic structures in field theory.

Contribution

It establishes the Koszulity of cohomology and associated graded algebras for mild pro-p groups with quadratic cohomology, confirming conjectures in Galois theory.

Findings

Koszulity of cohomology algebra for mild pro-p groups

Quadratic duality between cohomology and graded algebra

Validation of Koszulity conjectures for maximal pro-p Galois groups

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.