On the Brauer group of an arithmetic scheme.~II

S. G. Tankeev

Izvestiya. Mathematics , Tome 67 (2003) no. 5, pp. 1007-1029

Résumé

Let $\pi\colon X\to\operatorname{Spec}A$ be an arithmetic model of a regular smooth projective variety $V$ over a number field $k$. We prove the finiteness of $H^1(\operatorname{Spec} A,R^1\pi_\ast\operatorname{G}_m)$ under the assumption that $\pi_\ast\operatorname{G}_m=\operatorname{G}_m$ for the étale topology. (This assumption holds automatically if all geometric fibres of $\pi$ are reduced and connected.) If a prime $l$ does not divide $\operatorname{Card}([\operatorname{NS}(V\otimes \bar k)]_{\mathrm{tors}})$, $V(k)\ne\varnothing$, and the Tate conjecture holds for divisors on $V$, then the $l$-primary component $\operatorname{Br}'(X)(l)$ is finite. We also study finiteness properties of the Brauer group of a Calabi–Yau variety $V$ of dimension $\geqslant 2$ over a number field.