Koszul algebras and quadratic duals in Galois cohomology

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.

39 pages

References in corpus (7)

• Koszul property and Bogomolov's conjecture
• Formality properties of finitely generated groups and Lie algebras
• Bloch-Kato pro-p groups and locally powerful groups
• Koszul duality and Galois cohomology
• Galois cohomology of a number field is Koszul
• Profinite groups with a cyclotomic $p$-orientation
• The Bloch-Kato Conjecture and Galois Theory

Cited by in corpus (11)

• On pro-$p$ groups with quadratic cohomology
• Enhanced Koszul properties in Galois cohomology
• Oriented pro-$\ell$ groups with the Bogomolov-Positselski property
• Pro-p groups with few relations and Universal Koszulity
• Mild pro-p groups and the Koszulity conjectures
• Massey products in Galois cohomology and the Elementary Type Conjecture
• Right-angled Artin pro-$p$ groups
• Oriented right-angled Artin pro-$\ell$ groups and maximal pro-$\ell$ Galois groups
• Chasing maximal pro-p Galois groups via 1-cyclotomicity
• One relator maximal pro-p Galois groups and the Koszulity conjectures
• Arithmetic properties encoded in the Galois module structure of $K^{\times }/K^{\times p^m}$