The Brauer group and the Brauer–Manin set of products of varieties

Abstract

Let $X$ and $Y$ be smooth and projective varieties over a field $k$ finitely generated over $\mathbb{Q}$, and let $$ and $\bar{Y}$ be the varieties over an algebraic closure of $k$ obtained from $X$ and $Y$, respectively, by extension of the ground field. We show that the Galois invariant subgroup of $\mathrm{Br}(\bar{X})\oplus \mathrm{Br}(\bar{Y})$ has finite index in the Galois invariant subgroup of $\mathrm{Br}(\bar{X}\times \bar{Y})$. This implies that the cokernel of the natural map $\mathrm{Br}(X)\oplus \mathrm{Br}(Y)\to \mathrm{Br}(X\times Y)$ is finite when $k$ is a number field. In this case we prove that the Brauer–Manin set of the product of varieties is the product of their Brauer–Manin sets.

A correction to this paper is available.