In Milne, Étale cohomology, it is proved that $\mathrm{Br}(X) = H^2(X,\mathbf{G}\_m)$ for $X$ regular of dimension $\leq 2$. Are there in the meantime further results for $X$ regular?



When is Br(X) = H^2(X,G_m)?
When $X$ is quasi-projective over an affine scheme (or more generally if $X$ has an ample [**EDIT:** invertible] sheaf), then its Brauer group is isomorphic to the **torsion part** of $H^2(X, {\mathbb G}\_m)$. This is an unpublished result of Gabber, and J. de Jong [wrote down a different proof](http://www.math.columbia.edu/~dejong/papers/2-gabber.pdf).


