For an affine variety $V$, the Brauer group $Br(V)$ and the cohomological Brauer group $Br'(V)$ (i.e. the torsion subgroup of $H^2_{et}(V,\mathbb{G}_m)$) coincide, by results of various authors including Hoobler, Gabber etc. In fact it is true for affine schemes, but I have a simpler setup, namely I have a smooth complex variety.
However, I have been told that in fact $Br'(V) = H^2_{et}(V,\mathbb{G}_m)$ for affine $V$, and I can't seem to find a reference for this fact. This implies that all $\mathbb{G}_m$-gerbes on affine varieties have torsion class in $H^2$, a fact which I find slightly surprising, but which I can't rule out. My dream is that there are non-torsion elements in $H^2$, but I can live with their absence.
What is a reference, if true, for the result that $H^2_{et}(V,\mathbb{G}_m)$ is a torsion group?
EDIT: not that it is a hugely reliable source, but wikipedia says (without citation):
If $F$ is a coherent sheaf (or $\mathbb{G}_m$) then the étale cohomology of $F$ is the same as Serre's coherent sheaf cohomology calculated with the Zariski topology (and if $X$ is a complex variety this is the same as the sheaf cohomology calculated with the usual complex topology).
which would indicate that $H^2_{et}(V,\mathbb{G}_m) \simeq H^2(V(\mathbb{C}),\mathcal{O}^\times)$ where the cohomology on the right is sheaf cohomology for the analytic topology on $V(\mathbb{C})$.
By the answer at https://mathoverflow.net/questions/171638, there is a theorem of Grothedieck [1] that implies that any non-singular variety $X$ has all cohomology groups $H^q_{et}(X,\mathbb{G}_m)$ torsion for $q\geq 2$. Why this is true is still a bit mysterious, even though I have hints to the proof.
[1] Le groupe de Brauer II, Proposition 1.4