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

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 -