When is Br(X) = H^2(X,G_m)? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T03:22:05Z http://mathoverflow.net/feeds/question/16745 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/16745/when-is-brx-h2x-g-m When is Br(X) = H^2(X,G_m)? norondion 2010-03-01T08:30:38Z 2013-04-25T10:41:12Z <p>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?</p> http://mathoverflow.net/questions/16745/when-is-brx-h2x-g-m/16749#16749 Answer by Qing Liu for When is Br(X) = H^2(X,G_m)? Qing Liu 2010-03-01T09:52:32Z 2013-04-25T10:41:12Z <p>When $X$ is quasi-projective over an affine scheme (or more generally if $X$ has an ample [<b>EDIT:</b> invertible] sheaf), then its Brauer group is isomorphic to the <b>torsion part</b> of $H^2(X, {\mathbb G}_m)$. This is an unpublished result of Gabber, and J. de Jong <a href="http://www.math.columbia.edu/~dejong/papers/2-gabber.pdf" rel="nofollow">wrote down a different proof</a>.</p> http://mathoverflow.net/questions/16745/when-is-brx-h2x-g-m/16912#16912 Answer by Hailong Dao for When is Br(X) = H^2(X,G_m)? Hailong Dao 2010-03-02T20:24:30Z 2010-03-02T20:24:30Z <p>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:</p> <p><a href="http://mathoverflow.net/questions/7537/two-conjectures-by-gabber-on-brauer-and-picard-groups" rel="nofollow">Two conjectures by Gabber</a>.</p> <p><a href="http://mathoverflow.net/questions/10501/flat-cohomology-and-picard-groups" rel="nofollow">Flat cohomology and Picard groups</a>.</p> <p>(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. </p>