  Using twisted and formal-local methods, we prove that every separated
algebraic space which is the (open/flat) pushout of affine schemes has enough
Azumaya algebras. As a corollary we show that, under mild hypothesis, every
cohomological Brauer class is representable by an Azumaya algebra away from a
closed subset of codimension $\geq 3$, generalizing an early result of
Grothendieck. Next, we show that $\mathrm{Br}(X)=\mathrm{Br}'(X)$ when $X$ is
an algebraic space obtained from a quasi-projective scheme by contracting a
curve. This is the first method of descending an Azumaya algebra along a Chow
cover which works in all dimensions. As a corollary, we prove that
cohomological Brauer classes are geometric on arbitrary separated surfaces. In
higher dimensions, we obtain the first examples of non-quasi-projective
algebraic spaces with $\mathrm{Br}(X)=\mathrm{Br}'(X)$.
