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Given a smooth manifold $X$; by definition the topological Brauer group $B(X)$ is the group of classes of Azumaya algebras over the sheaf $\mathcal{O}_X$ of $\mathbb{C}$-valued functions on $X$.

If we replace the sheaf $\mathcal{O}_X$ by the sheaf of $C^{\infty}$-functions; will the resulting Brauer group be the same?

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    $\begingroup$ Yes. In both cases you get the torsion subgroup of $H^3(X,\mathbb{Z})$. This is discussed in this Bourbaki seminar by Grothendieck: he treats the topological case, but the argument works also for the sheaf of $\mathcal{C}^{\infty}$ functions. $\endgroup$
    – abx
    Oct 10, 2019 at 3:20
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    $\begingroup$ Yes, because the elements of the Brauer group can be described as equivalence classes of complex projective bundles, and the topological and smooth classification of bundles coincide for spaces as nice as a smooth manifold. $\endgroup$
    – David Roberts ♦
    Oct 10, 2019 at 5:20

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