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

Alexei N. Skorobogatov; Yuri G. Zahrin

Journal of the European Mathematical Society (2014)

  • Volume: 016, Issue: 4, page 749-768
  • ISSN: 1435-9855

Abstract

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Let X and Y be smooth and projective varieties over a field k finitely generated over Q , and let X ¯ and 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 ( X ¯ ) Br( Y ¯ ) has finite index in the Galois invariant subgroup of Br ( X ¯ × 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.

How to cite

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Skorobogatov, Alexei N., and Zahrin, Yuri G.. "The Brauer group and the Brauer–Manin set of products of varieties." Journal of the European Mathematical Society 016.4 (2014): 749-768. <http://eudml.org/doc/277447>.

@article{Skorobogatov2014,
abstract = {Let $X$ and $Y$ be smooth and projective varieties over a field $k$ finitely generated over $Q$, and let $\overline\{X\}$ and $\overline\{Y\}$ 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 $(\overline\{X\}) \oplus $ Br($\overline\{Y\})$ has finite index in the Galois invariant subgroup of Br$(\overline\{X\}\times \overline\{Y\})$. This implies that the cokernel of the natural map Br$(X)\oplus $ Br$(Y)\rightarrow $ 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.},
author = {Skorobogatov, Alexei N., Zahrin, Yuri G.},
journal = {Journal of the European Mathematical Society},
keywords = {Brauer group; Brauer–Manin obstruction; universal torsors; Brauer-Manin obstruction; universal torsors},
language = {eng},
number = {4},
pages = {749-768},
publisher = {European Mathematical Society Publishing House},
title = {The Brauer group and the Brauer–Manin set of products of varieties},
url = {http://eudml.org/doc/277447},
volume = {016},
year = {2014},
}

TY - JOUR
AU - Skorobogatov, Alexei N.
AU - Zahrin, Yuri G.
TI - The Brauer group and the Brauer–Manin set of products of varieties
JO - Journal of the European Mathematical Society
PY - 2014
PB - European Mathematical Society Publishing House
VL - 016
IS - 4
SP - 749
EP - 768
AB - Let $X$ and $Y$ be smooth and projective varieties over a field $k$ finitely generated over $Q$, and let $\overline{X}$ and $\overline{Y}$ 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 $(\overline{X}) \oplus $ Br($\overline{Y})$ has finite index in the Galois invariant subgroup of Br$(\overline{X}\times \overline{Y})$. This implies that the cokernel of the natural map Br$(X)\oplus $ Br$(Y)\rightarrow $ 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.
LA - eng
KW - Brauer group; Brauer–Manin obstruction; universal torsors; Brauer-Manin obstruction; universal torsors
UR - http://eudml.org/doc/277447
ER -

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