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Let $X$ be a smooth projective algebraic variety over $\mathbb{C}$. Then I think that someone (Serre?) showed that the Cohomological Etale Brauer Group agrees with the torsion part of the Analytic Brauer Group $H^{2}(X,\mathcal{O}^{\times})$. This latter group is calculated in the classical (metric) topology on the associated complex manifold with the sheaf of nowhere vanishing holomorphic functions.

However there can easily be non-torsion elements in $H^{2}(X,\mathcal{O}^{\times})$: for instance consider the image in $H^{3}(X,\mathbb{Z}) \cap (H^{(2,1)}(X) \oplus H^{(1,2)}(X))$.

Could there be a topology more refined than etale but defined algebraically which can see these non-torsion classes? Notice that one can also ask the question for any $H^{i}(X,\mathcal{O}^{\times})$. For $i=0,1$ the Zariski and etale work fine.

Why do things break down for $i>1$?

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I'd be surprised if such a topology were in the literature. (I'm no expert on the Brauer group, but once I thought a little about it.) So, it's unlikely you'll get a yes answer to your question. To give a no answer you'd of course have to turn it into a precise, mathematical yes/no question. It would probably be interesting if you could.

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I think the article by B. Toen "Derived Azumaya algebras and generators for twisted derived categories", arXiv:1002.2599, gives a pointer to a possible answer to your question.

EDIT I have weakened the assertion...

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    $\begingroup$ I don't think so. Toën shows how to realise non-torsion classes in the étale cohomological Brauer group in terms of generalised Azumaya algebras. This, I believe, has very little to do with the analytic non-torsion classes which generally exist even when the étale group is torsion. $\endgroup$
    – Torsten Ekedahl
    Commented Sep 19, 2011 at 9:46
  • $\begingroup$ @Torsten Ekedahl. You are right: it doesn't answer the question literally. My idea is that B. Toën provides an interpretation for the non-torsion elements in the étale case and that his method (which is very categorical) might be adaptable to the analytic case. Of course, I might have overlooked a fundamental obstruction here. $\endgroup$
    – Damian Rössler
    Commented Sep 19, 2011 at 10:56
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    $\begingroup$ You may of course be right but I think the point is that the reason for non-torsion is completely different. Non-torsion in etale cohomology come from singularities so if the variety is non-singular all classes are torsion. The analytic non-torsion classes appear for completely different reason even in the non-singular case. A worth-while analogy is perhaps logarithmic transformations. Algebraically, they are all done by using torsion classes, analytically you can use an class. $\endgroup$
    – Torsten Ekedahl
    Commented Sep 19, 2011 at 15:19
  • $\begingroup$ @Torsten Ekedahl. Right. It had slipped my mind that the cohomological Brauer group is torsion for any regular scheme (I knew this for the "Azumaya" one). I wonder how this difference comes into play in Toën's work. $\endgroup$
    – Damian Rössler
    Commented Sep 19, 2011 at 16:37
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    $\begingroup$ Yes. I have introduced a $\Delta$-flat topology on $X$ which includes derivations. Then the spectral sequence for the change of sites map $X_{\Delta}\to X_{et}$ captures part of how the additional Hodge structure appears in $H^2(X_{\Delta},\cal{O^*})$. See arXiv:1003.1421 for the beginnings of this theory. $\endgroup$
    – Ray Hoobler
    Commented Dec 28, 2014 at 17:45

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