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

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).