Koszul algebras and quadratic duals in Galois cohomology

TL;DR

This paper demonstrates that Galois cohomology of certain pro-p groups is a Koszul algebra, confirming a conjecture and exploring its relation to quadratic duals and filtrations, with both conditional and unconditional results.

Contribution

It proves Galois cohomology has the PBW property and is Koszul under certain conjectural assumptions, and relates quadratic duals to group filtrations, advancing understanding of Galois cohomology structure.

Findings

Galois cohomology has the PBW property under conjectural assumptions.

Galois cohomology is a Koszul algebra in this context.

Quadratic duals relate to p-Zassenhaus filtrations.

Abstract

We investigate the Galois cohomology of finitely generated maximal pro-$p$ quotients of absolute Galois groups. Assuming the well-known conjectural description of these groups, we show that Galois cohomology has the PBW property. Hence in particular it is a Koszul algebra. This answers positively a conjecture by Positselski in this case. We also provide an analogous unconditional result about Pythagorean fields. Moreover, we establish some results that relate the quadratic dual of Galois cohomology with $p$-Zassenhaus filtration on the group. This paper also contains a survey of Koszul property in Galois cohomology and its relation with absolute Galois groups.