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In a note by deJong showing the cohomological and ordinary Brauer groups coincide for separated quasicompact schemes with ample line bundle, it is mentioned that Gabber had an unpublished proof of the main result therein, but by a different method. From elsewhere I read that this proof dates back to the 90s. Is this proof available anywhere? If not, could someone give an outline?

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I presume that it is a combination of the affine case (proved by R. Hoobler, 1980, see projecteuclid.org/euclid.pjm/1102780616) and Gabber's extension of the Bloch-Ogus theorem. –  ACL Feb 25 at 18:30
    
Why not ask Gabber (or perhaps de Jong) about a manuscript? –  Jason Starr Feb 25 at 18:55
    
@JasonStarr - ok, I'll ask de Jong. –  David Roberts Feb 25 at 21:17
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@ACL - o.O Bloch-Ogus sounds a bit scary to me, but I'll have a look. –  David Roberts Feb 25 at 21:18

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$\newcommand{\Z}{\mathbb{Z}}\newcommand{\G}{\mathbb{G}}$

I emailed Johan de Jong, and he sent me the following, which I reproduce with his permission (lightly edited to only keep mathematical content). Gabber's proof is not written up anywhere, from what de Jong told me.

The argument is different because I lectured about the proof from my write-up in front of Gabber and he then subsequently told me it is different. After the lecture he also explain his proof. Roughly what he does is two things:

(1) Prove it for regular quasi-projective schemes. This involves carefully choosing a maximal order over the scheme which is Azumaya by modifying along an ample divisor (on some affine open you can already do the thing).

(2) Prove the following amazing theorem: Suppose that $X$ is a scheme which has an ample invertible sheaf. Suppose that $\alpha$ is a cohomology class in $H^i(X, \Z/n\Z)$ for some $n > 0$ and since $i \ge 0$. Then one can find a quasi-projective scheme $Y$ smooth over $\Z$ and a morphism $X \to Y$ and a $\beta \in H^i(Y, \Z/n\Z)$ which pulls back to $\alpha$. Moreover, the same thing can be done with torsion classes in $H^2(X, \G_m)$. To do this you may immediately assume that $X$ is of finite type over $\Z$ by a limit argument. Then you can embed $X$ into some smooth $P$, e.g., projective space. Now on an open you can sort of extend the class (after replacing $P$ by a blow up in closed subscheme and maybe an etale neighbourhood). Then, and this was all explained to me at a party later in the day [...] you kind of keep blowing up until $\alpha$ extends.

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Sounds like a great party! –  Jason Starr Feb 26 at 3:43

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