Experiments on the Brauer map in high codimension

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 show that, under mild hypothesis, cohomological Brauer class representable by an algebra away from closed subset codimension $\geq 3$, generalizing early result Grothendieck. Next, $\mathrm{Br}(X)=\mathrm{Br}'(X)$ when $X$ obtained quasi-projective scheme contracting curve. This first method descending along Chow cover works in all dimensions. corollary, classes are geometric on arbitrary surfaces. In higher dimensions, obtain examples non-quasi-projective spaces with $\mathrm{Br}(X)=\mathrm{Br}'(X)$.