Kato conjecture and motivic cohomology over finite fields

2009/10/15 by Uwe Jannsen, Jannsen, Uwe, Shuji Saito +1
Mathematics · #14F20 #14F42 #14G18 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.0910.2815

openalex publication_date 2009/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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