Kato conjecture and motivic cohomology over finite fields

Details Kato conjecture and motivic cohomology over finite fields Kato conjecture and motivic cohomology over finite fields

2009-10-15

arxiv.org/abs/0910.2815v1

Mathematics

Algebraic Geometry

38 pages

Scientific paper

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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