On the Brauer group of an arithmetic scheme. II

摘要

Let π: X → Spec A be an arithmetic model of a regular smooth projective variety V over a number field k. We prove the finiteness of H~1 (Spec A, R~1π_* G_m) under the assumption that π_*G_m = G_m for the etale toopology. (This assumption holds automatically if all geometric fibres of π are reduced and connected.) If a prime l does not divide Card ([NS(V k-bar)]_(tors)), V(k) ≠ O, and the Tate conjecture holds for divisors on V, then the l-primary component Br'(X)(l) is finite. We also study finiteness properties of the Brauer group of a Calabi-Yau variety V of dimension ≥ 2 over a number field.