For any scheme $X$, let $\operatorname{Br}X$ denote the (Azumaya) Brauer group of $X$, namely the Morita equivalence classes of Azumaya $\mathcal{O}\_{X}$-algebras.



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> Is the functor $$\operatorname{Br} : \operatorname{Sch}^{\operatorname{op}} \to \operatorname{Ab}$$ sending $X \mapsto \operatorname{Br}X$ a sheaf for the Zariski topology on $\operatorname{Sch}$?
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In other words, if $X = \bigcup\_{i \in I} U\_{i}$ is a Zariski open covering of a scheme $X$, is the sequence $$0 \to \operatorname{Br}X \to \prod\_{i \in I} \operatorname{Br}U\_{i} \to \prod\_{i\_{1},i\_{2} \in I} \operatorname{Br}(U\_{i\_{1}} \cap U\_{i\_{2}}) $$ exact?


Thoughts: My naive guess is "no" since $\operatorname{Br}$ is not an etale sheaf and $\operatorname{Pic}$ is not a Zariski sheaf. Exercise 8(f) of <http://www-personal.umich.edu/~bhattb/teaching/mat731fall2011/ex6.pdf> shows that the functor $U \mapsto \mathrm{H}\_{et}^{2}(U,\mathbb{G}\_{m})$ is a Zariski sheaf if $X$ is regular Noetherian. See also the discussion on Milne's "Etale Cohomology", IV, Remark 2.10.



Is the Brauer group functor a Zariski sheaf?
No, $\mathrm{Br}$ is not a Zariski sheaf: it is possible for a non-trivial Azumaya algebra on a variety to become trivial when restricted to a Zariski cover. This can happen even for a normal surface with rational singularities. Some references:


* M. Ojanguren, "A non-trivial locally trivial algebra", *J. Algebra* 29, 1974
* F. DeMeyer and T. Ford, "Nontrivial, locally trivial Azumaya algebras", *Contemporary Mathematics* 124, 1992
* My answer to [this question](https://mathoverflow.net/q/201341/3753) and more generally [this article](http://arxiv.org/abs/1201.4299).


