Cohomological Hasse principle and motivic cohomology for arithmetic schemes

TL;DR

This paper proves a significant part of Kato's conjecture on the cohomological Hasse principle for arithmetic schemes, impacting the understanding of motivic cohomology and zeta functions.

Contribution

It establishes the prime-to-characteristic case of the cohomological Hasse principle, advancing the theory of arithmetic schemes and motivic cohomology.

Findings

Proved the prime-to-characteristic part of the cohomological Hasse principle.

Implications for finiteness of motivic cohomology.

Connections to special values of zeta functions.

Abstract

In 1985 Kazuya Kato formulated a fascinating framework of conjectures which generalizes the Hasse principle for the Brauer group of a global field to the so-called cohomological Hasse principle for an arithmetic scheme. In this paper we prove the prime-to-characteristic part of the cohomological Hasse principle. We also explain its implications on finiteness of motivic cohomology and special values of zeta functions.