2011/12/14 by Skorobogatov, Alexei N., Zarhin, Yuri G.
#14F22 (Primary) 14G25 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1112.3089
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)⊕ \Br(\ov Y) has finite index in the Galois invariant subgroup of \Br(\ov X×\ov Y). 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.