When is Br(X) = H^2(X,G_m)? - MathOverflow

When is Br(X) = H^2(X,G_m)?

2010-03-01T08:30:38Z

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?

Answer by Qing Liu for When is Br(X) = H^2(X,G_m)?

2010-03-01T09:52:32Z

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 This is an unpublished result of Gabber, and J. de Jong wrote down a different proof.

Answer by Hailong Dao for When is Br(X) = H^2(X,G_m)?

2010-03-02T20:24:30Z

Dear norondion: It appears to me from your comments to Qing Liu's answer that you are interested in when this cohomological Brauer group is actually $0$. If that is true, then these two related MO questions (and Emerton's answer to one of them) may be of interest:

Two conjectures by Gabber.

Flat cohomology and Picard groups.

(Of course, the punctured spectrum of a regular local ring is a regular scheme. Also, you can probably get some statements for projective $X$ by looking at the local ring of the cone over $X$). My apology if this is not relevant.