Let and be smooth and projective varieties over a field finitely generated over \( \Q \), and let \( \ov X \) and \( \ov Y \) be the varieties over an algebraic closure of obtained from and , 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 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. 749–769DOI 10.4171/JEMS/445