Motivic Cohomology and K-groups of varieties over higher local fields

TL;DR

This paper investigates the structure of K-groups and higher Chow groups of varieties over higher local fields, establishing divisibility, finiteness, and divisibility properties, with implications for reciprocity maps and étale cohomology.

Contribution

It proves divisibility and finiteness results for K-groups and higher Chow groups over higher local fields, advancing understanding of their arithmetic and cohomological properties.

Findings

K-groups above a certain degree are divisible-by-finite.

Prime-to-p torsion in higher Chow groups is finite.

Kernel of tame reciprocity map is uniquely p'-divisible.

Abstract

For quasi-projective varieties over a higher local field $k_N$, we prove that its $K$-groups, above a suitable degree, are divisible-by-finite. We also prove the finiteness of the prime-to-$p$ torsion subgroup of certain higher Chow groups for smooth projective varieties over such fields, where $p$ denotes the final residue characteristic of $k_N$. As an application, we show that the kernel of the tame reciprocity map is uniquely $p'$-divisible. A key ingredient in achieving these results is the finiteness of \'etale cohomology groups over such fields.