# Kato conjecture and motivic cohomology over finite fields

## TL;DR

This paper proves Kato's conjecture on the homology of a complex related to Galois cohomology over finite fields, assuming resolution of singularities, and applies it to motivic cohomology finiteness questions.

## Contribution

It establishes the conjecture over finite fields assuming resolution of singularities and derives unconditional results for smooth projective varieties.

## Key findings

- Proves Kato's conjecture over finite fields with resolution of singularities.
- Unconditional vanishing of homology up to degree 4 for smooth projective varieties.
- Applications to finiteness of motivic cohomology groups.

## Abstract

For an arithmetical scheme X, K. Kato introduced a certain complex of Gersten-Bloch-Ogus type whose component in degree a involves Galois cohomology groups of the residue fields of all the points of X of dimension a. He stated a conjecture on its homology generalizing the fundamental exact sequences for Brauer groups of global fields. We prove the conjecture over a finite field assuming resolution of singularities. Thanks to a recently established result on resolution of singularities for embedded surfaces, it implies the unconditional vanishing of the homology up to degree 4 for X projective smooth over a finite field. We give an application to finiteness questions for some motivic cohomology groups over finite fields.