Journal of Algebraic Geometry

Author(s): Alexei N. Skorobogatov; Yuri G. Zarhin
Journal: J. Algebraic Geom. 17 (2008), 481-502.
Posted: December 10, 2007
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Abstract: Let

be a field finitely generated over the field of rational numbers, and $\operatorname{Br}(k)$ the Brauer group of

. For an algebraic variety

over

we consider the cohomological Brauer-Grothendieck group $\operatorname{Br}(X)$ . We prove that the quotient of $\operatorname{Br}(X)$ by the image of $\operatorname{Br}(k)$ is finite if

is a

surface. When

is an abelian variety over

, and $\overline{X}$ is the variety over an algebraic closure $\overline{k}$ of

obtained from

by the extension of the ground field, we prove that the image of $\operatorname{Br}(X)$ in $\operatorname{Br}(\overline{X})$ is finite.

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Alexei N. Skorobogatov
Affiliation: Department of Mathematics, South Kensington Campus, Imperial College, London, SW7 2BZ England, United Kingdom - Institute for the Information Transmission Problems, Russian Academy of Sciences, 19 Bolshoi Karetnyi, Moscow, 127994 Russia
Email: a.skorobogatov@imperial.ac.uk

Yuri G. Zarhin
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802 - Institute for Mathematical Problems in Biology, Russian Academy of Sciences, Pushchino, Moscow Region, Russia
Email: zarhin@math.psu.edu

PII: S 1056-3911(07)00471-7
Received by editor(s): May 13, 2006
Received by editor(s) in revised form: October 12, 2006
Posted: December 10, 2007

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