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).
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