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

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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?

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## 2 Answers

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 This is an unpublished result of Gabber, and J. de Jong wrote down a different proof.

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Are there criteria when $H^2(X,\mathbf{G}_m)$ is torsion free? – norondion Mar 1 2010 at 14:37
In Lichtenbaum, Zeta-Functions of Varieties Over Finite Fields at s = 1, he proves that it is torsion for varieties over finite fields. – norondion Mar 1 2010 at 14:39
You mean X has an ample line bundle?  – Hailong Dao Mar 1 2010 at 14:49
Yes, thanks, I corrected it!  – Qing Liu Mar 2 2010 at 17:09
It is torsion for $X$ regular, Noetherian: Consider $0 \to \mathbf{G}_m \to j_*\mathbf{G}_m \to Div_X \to 0$ and the long exact sequence; since $X$ is regular, we have $Div_X = \oplus_{x \in X^{(1)})i_{x,*}\mathbf{Z}$. – norondion Mar 2 2010 at 17:35

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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.

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Thanks. Actually, I'm interested in cases where it is finite. – norondion Mar 6 2010 at 18:14
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