Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $\mathbb Q$, 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
$\Br(\ov X)\oplus \Br(\ov Y)$ has finite index in the Galois invariant subgroup
of $\Br(\ov X\times\ov Y)$. 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 varieties is the
product of their Brauer-Manin sets.

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The categories of this paper are: astro-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: q-fin
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The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
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The categories of this paper are: supr-con
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: eess
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The categories of this paper are: comp-gas
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The categories of this paper are: chao-dyn
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The categories of this paper are: mtrl-th
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The categories of this paper are: cs
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The categories of this paper are: nucl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: solv-int