Hasse principles for higher-dimensional fields

Uwe Jannsen

Published 2009-10-15, updated 2014-12-04Version 2

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.