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

Let X and Y be smooth and projective varieties over a field k finitely generated over Q, and let χ and γ 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(χ)Br(γ) has finite index in the Galois invariant subgroup of Br(χ γ ). This implies that the cokernel of the natural map Br(X) Br(Y) → Br(X × 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.