2-Cocycles and Azumaya algebras under birational transformations of algebraic schemes

F. A. Bogomolov and A. N. Landia
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F. A. Bogomolov: Steklov Mathematical Institute of the Academy of Sciences of USSR
A. N. Landia: Mathematical Institute of the Academy of Sciences of Georgian SSR

A chapter in Algebraic Geometry, 1990, pp 1-5 from Springer

Abstract: Abstract The basic question whether the injection Br $$\left( X \right) \to {H^2}{\left( {{X_,}\vartheta _x^*} \right)_{tors}}$$ is an isomorphism arose at the very definition of the Brauer group of an algebraic scheme X. Positive answers are known in the following cases: 1. the topological Brauer group Br $$\left( {{X_{top}}} \right) \cong {H^2}{\left( {X,\vartheta _{top}^*} \right)_{top}} \cong {H^3}{\left( {X,\mathbb{Z}} \right)_{top}} \cong {H^3}{\left( {X,\mathbb{Z}} \right)_{top}}$$ (J.-P. Serre); in the etale (algebraic) case the isomorphism is proved for 2. smooth projective surfaces (A. Grothendieck); 3. abelian varieties; 4. the union of two affine schemes (R. Hoobler, O. Gabber).

Date: 1990
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DOI: 10.1007/978-94-009-0685-3_1

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