Using twisted sheaves, formal-local methods, and elementary transformations we show that separated algebraic spaces which are constructed as pushouts or contractions (of curves) have enough Azumaya algebras. This implies (1) Under mild hypothesis, every cohomological Brauer class is representable by an Azumaya algebra away from a closed subset of codimension $\ge 3$, generalizing an early result of Grothendieck and (2) $\mathrm{Br}<!--FUNCTION\; APPLICATION-->(X)={\mathrm{Br}<!--FUNCTION\; APPLICATION-->}^{\prime }(X)$ when $X$ is an algebraic space obtained from a quasiprojective scheme by contracting a curve. This result is valid for all dimensions but if we specialize to surfaces, it solves the question entirely: there are always enough Azumaya algebras on separated surfaces.

使用扭曲滑轮、形式局部方法和基本变换，我们证明了被构造为（曲线的）推出或收缩的分离代数空间具有足够的 Azumaya 代数。这意味着 (1) 在温和假设下，每个上同调 Brauer 类都可以用远离余维封闭子集的 Azumaya 代数表示$\ge 3$, 概括格洛腾迪克和 (2) 的早期结果$溴<!--FUNCTION\; APPLICATION-->(X)={溴<!--FUNCTION\; APPLICATION-->}^{\text{'}}(X)$什么时候$X$是通过收缩曲线从拟投影方案获得的代数空间。这个结果对所有维度都有效，但如果我们专门研究曲面，它完全解决了这个问题：在分离的曲面上总是有足够的 Azumaya 代数。