Hasse principles for higher-dimensional fields

Abstract

For rather general excellent schemes $X$, K. Kato defined complexes of Gersten-Bloch-Ogus type involving the Galois cohomology groups of all residue fields of $X$. For arithmetically interesting schemes, he developed a fascinating web of conjectures on some of these complexes, which generalize the classical Hasse principle for Brauer groups over global fields, and proved these conjectures for low dimensions. We prove Kato’s conjecture over number fields in any dimension. This gives a cohomological Hasse principle for function fields $F$ over a number field $K$, involving the corresponding function fields $F_v$ over the completions $K_v$ of $K$. For global function fields $K$ we prove the part on injectivity for coefficients invertible in $K$. Assuming resolution of singularities, we prove a similar conjecture of Kato over finite fields, and a generalization to arbitrary finitely generated fields.