Divisibility phenomena in motivic Bloch--Ogus theory

TL;DR

This paper investigates divisibility properties of unramified motivic cohomology and Milnor K-groups for smooth projective varieties over various fields, extending known results and exploring filtrations.

Contribution

It generalizes divisibility results in motivic cohomology and Milnor K-theory to broader classes of fields and schemes, including finite and separably closed fields.

Findings

Unramified classes in Milnor K-groups are n-divisible over separably closed fields.

Most steps in the Bloch--Ogus filtration are l-divisible up to torsion for finite or separably closed fields.

Generalizations to quasi-projective schemes are established.

Abstract

Let X be a smooth projective variety over a field k. For k separably closed, we prove that the subgroup of unramified classes in the Milnor K-group $K^M_i(k(X))$ of the function field of X is contained in the subgroup of n-divisible elements of $K^M_i(k(X))$ for any integer n invertible in k. This generalizes to a statement for unramified motivic cohomology of arbitrary bidegree. We further show that whenever k is finite or separably closed and l is a prime invertible in k, then all but the last step in the Bloch--Ogus filtration of the motivic cohomology of X are l-divisible up to torsion. Generalizations of this last result to arbitrary quasi-projective k-schemes are also proven.