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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?

flag	asked Mar 1 2010 at 8:30 norondion 4,230●6●34

Qing Liu · Accepted Answer · 2013-04-25 10:41:12Z UTC

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 This is an unpublished result of Gabber, and J. de Jong wrote down a different proof.

edited Apr 25 at 10:41

answered Mar 1 2010 at 9:52

Qing Liu
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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

Hailong Dao · Answer 2 · 2010-03-02 20:24:30Z UTC

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.

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

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