A local to global principle for higher zero-cycles

Abstract: We study a local to global principle for certain higher zero-cycles over global fields. We thereby verify a conjecture of Colliot-Th\'el`ene for these cycles. Our main tool are the Kato conjectures proved by Jannsen, Kerz and Saito. Our approach also allows to reprove the ramified global class field theory of Kato and Saito. Finally, we apply the Kato conjectures to study the $p$-adic cycle class map over henselian discrete valuation rings of mixed characteristic and to deduce finiteness theorems for arithmetic schemes in low degree.