Higher Chow groups with finite coefficients and refined unramified cohomology

TL;DR

This paper connects Bloch's higher cycle class map with refined unramified cohomology, establishing new exact sequences and localization properties, and conjectures its role as a motivic homology theory.

Contribution

It demonstrates the natural fit of Bloch's cycle class map into a long exact sequence involving refined unramified cohomology and proves localization sequence properties.

Findings

Refined unramified cohomology satisfies the localization sequence.

Cycle class map fits into a long exact sequence with refined unramified cohomology.

Conjecture that refined unramified cohomology is a motivic homology theory.

Abstract

In this paper we show that Bloch's higher cycle class map with finite coefficients for quasi-projective equi-dimensional schemes over a field fits naturally in a long exact sequence involving Schreieder's refined unramified cohomology. We also show that the refined unramified cohomology satisfies the localization sequence. Using this we conjecture in the end that refined unramified cohomology is a motivic homology theory and explain how this is related to the aforementioned results.