Galois cohomology of a number field is Koszul

TL;DR

This paper proves that the Milnor ring of any one-dimensional local or global field modulo a prime is a Koszul algebra, supporting conjectures about its algebraic structure using class field theory and Groebner basis computations.

Contribution

It establishes the Koszulity of Milnor rings for all one-dimensional local and global fields modulo a prime, extending previous conjectures.

Findings

Milnor rings are Koszul algebras over Z/l.

Module Koszulity properties hold under mild assumptions.

Supports existing Koszulity conjectures.

Abstract

We prove that the Milnor ring of any (one-dimensional) local or global field K modulo a prime number l is a Koszul algebra over Z/l. Under mild assumptions that are only needed in the case l=2, we also prove various module Koszulity properties of this algebra. This provides evidence in support of Koszulity conjectures that were proposed in our previous papers. The proofs are based on the Class Field Theory and computations with quadratic commutative Groebner bases (commutative PBW-bases).