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I have heard that Mumford has constructed an example of a normal complex algebraic surface $X$ such that $H^2_{et}(X,\mathbb{G}_m)$ contains a non-torsion element. But I cannot find the reference.

Then what is the surface $X$ and how to construct a non-torsion element in $H^2_{et}(X,\mathbb{G}_m)$?

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    $\begingroup$ I don't remember a reference for Mumford's example, but I do remember approximately how it works. If you have a cone over, say, a plane cubic curve, then the local Weil divisor class group at the vertex has plenty of non-torsion elements. Now find a surface $X$ with a singularity that is étale locally isomorphic to that, but not Zariski locally, where your non-torsion class doesn't extend to a Weil divisor on a Zariski neighbourhood. To see where your Brauer class comes from, let $Y \to X$ be a resolution and look at the Leray spectral sequence for cohomology with values in $\mathbf{G}_m$. $\endgroup$
    – Martin Bright
    Commented Feb 13, 2017 at 8:30
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    $\begingroup$ The reference in Dix Exposés is: D. Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity. Publications Math. 9 (1961), 5–22. $\endgroup$
    – Martin Bright
    Commented Feb 13, 2017 at 11:20

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In [Milne, Étale Cohomology], p. 146, Remark IV.2.8, there is a reference to [Grothendieck, Le groupe de Brauer. In: Dix Exposés sur la Cohomologie des Schémas], II.1.11b. (see https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/DixExp.pdf)

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    $\begingroup$ Giving an outline of the (or even the full) construction would be nice. That link you give has broken before, and may break again. $\endgroup$
    – David Roberts ♦
    Commented Feb 13, 2017 at 8:06

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