arXiv:0910.2803 Hasse principles for higher-dimensional fields from arXiv Front: math.AG by Uwe Jannsen 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 (with infinite coefficients) 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 a conditional result over global fields K of positive characteristic, assuming resolution of singularities. This is unconditional for X of dimension at most 3, due to recent results on resolution. There are also applications to other cases considered by Kato.
nLab page on Hasse principle