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

TL;DR

This paper investigates the Brauer group and Brauer-Manin set of product varieties over fields finitely generated over Q, establishing finiteness results and the product structure of the Brauer-Manin set for such varieties over number fields.

Contribution

It proves the finiteness of the cokernel of the Brauer group map for products over number fields and shows the Brauer-Manin set of a product equals the product of the sets, extending understanding of these invariants.

Findings

The Galois invariant subgroup of the Brauer group of the product has finite index in the sum of the factors.

The cokernel of the natural Brauer group map for products over number fields is finite.

The Brauer-Manin set of a product equals the product of the individual sets.

Abstract

Let $X$ and $Y$ be smooth and projective varieties over a field $k$ finitely generated over $\mathbb Q$, 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.