TITLE: Is the Brauer group functor a Zariski sheaf?
QUESTION [13 upvotes]: 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.

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 $\operatorname{Sch}$?

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.

REPLY [17 votes]: 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 and more generally this article.