  For schemes X over global or local fields, or over their rings of integers,
K. Kato stated several conjectures on certain complexes of Gersten-Bloch-Ogus
type, generalizing the fundamental exact sequence of Brauer groups for a global
field. He proved these conjectures for low dimensions. We prove Kato's
conjecture over number fields. In particular this gives a Hasse principle for
function fields F over a number field K, involving the corresponding function
fields over all completions of K. We get the same results over global fields K
of positive characteristic, for coefficients invertible in K. This was proved
earlier by M. Kerz and S. Saito, by another method. Finally we obtain a
conjecture of Kato over a finite field, and a generalization to finitely
generated fields K, assuming resolution of singularities or that the
coefficents are invertible in K. The latter case was again obtained earlier by
M. Kerz and S. Saito, by different methods.
