# 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

## Access Full Article

top## Abstract

top## How to cite

topSkorobogatov, 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 -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.