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

### Yuri G. Zarhin

Pennsylvania State University, University Park, USA### Alexei N. Skorobogatov

Imperial College London, UK

## Abstract

Let $X$ and $Y$ be smooth and projective varieties over a field $k$ finitely generated over $\Q$, and let $\ov X$ and $\ov Y$ 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 $\Br(\ov X)\oplus \Br(\ov Y)$ has finite index in the Galois invariant subgroup of $\Br(\ov X\times\ov Y)$. This implies that the cokernel of the natural map $\Br (X)\oplus\Br (Y)\to\Br(X\times Y)$ 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.

## Cite this article

Yuri G. Zarhin, Alexei N. Skorobogatov, The Brauer group and the Brauer–Manin set of products of varieties. J. Eur. Math. Soc. 16 (2014), no. 4, pp. 749–769

DOI 10.4171/JEMS/445