On the Brauer group of an arithmetic scheme

For an Enriques surface $V$ over a number field $k$ with a $k$-rational point we prove that the $l$-component of $\operatorname{Br}(V)/{\operatorname{Br}(k)}$ is finite if and only if $l\ne 2$. For a regular projective smooth variety satisfying the Tate conjecture for divisors over a number field, we find a simple criterion for the finiteness of the $l$-component of $\operatorname{Br}'(V)/{\operatorname{Br}(k)}$. Moreover, for an arithmetic model $X$ of $V$ we prove a variant of Artin's conjecture on the finiteness of the Brauer group of $X$. Applications to the finiteness of the $l$-components of Shafarevich–Tate groups are given.