Geodesic
The Brauer group and the Brauer–Manin set of products of varieties
Journal of the European Mathematical Society, Tome 16 (2014) no. 4, pp. 749-769

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Let X and Y be smooth and projective varieties over a field k finitely generated over Q, and let $$ and Yˉ 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(Xˉ)⊕Br(Yˉ) has finite index in the Galois invariant subgroup of Br(Xˉ×Yˉ). 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.
DOI : 10.4171/jems/445
Classification : 14-XX, 00-XX
Keywords: Brauer group, Brauer–Manin obstruction 
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     title = {The {Brauer} group and the {Brauer{\textendash}Manin} set of products of varieties},
     journal = {Journal of the European Mathematical Society},
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Alexei N. Skorobogatov; Yuri G. Zarhin. The Brauer group and the Brauer–Manin set of products of varieties. Journal of the European Mathematical Society, Tome 16 (2014) no. 4, pp. 749-769. doi: 10.4171/jems/445

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