  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.
