Let and be smooth and projective varieties over a field finitely generated over , and let and 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 has finite index in the Galois invariant subgroup of . This implies that the cokernel of the natural map 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.
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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