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

TL;DR

This paper proves the Koszulity conjectures for certain maximal pro-p Galois groups under specific cohomological conditions, revealing their algebraic structure and proposing a refined conjecture.

Contribution

It establishes the validity of Koszulity conjectures for maximal pro-p Galois groups with finite H^1 and H^2 conditions, and describes their cohomology algebra structure.

Findings

Koszulity conjectures hold under given cohomological conditions

Cohomology algebra is quadratic dual to a graded algebra

Algebras decompose into products of elementary quadratic algebras

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.