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

Alexei N. Skorobogatov; Yuri G. Zahrin — 2014

Journal of the European Mathematical Society

Let $X$ and $Y$ be smooth and projective varieties over a field $k$ finitely generated over $Q$ , and let $\bar{X}$ and $\bar{Y}$ 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 $(\bar{X}) \oplus$ Br( $\bar{Y})$ has finite index in the Galois invariant subgroup of Br $(\bar{X} \times \bar{Y})$ . This implies that the cokernel of the natural map Br $(X) \oplus$ Br $(Y) \to$ Br $(X \times Y)$ is finite when $k$ is a number field. In this case we prove that the Brauer–Manin set of the product of...

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