论文标题
在高复合中的Brauer地图上的实验
Experiments on the Brauer map in High Codimension
论文作者
论文摘要
使用扭曲和正式的本地方法，我们证明了每个分离的代数空间，即仿射方案的（开/平坦）的推动力，具有足够的Azumaya代数。作为推论，我们表明，在温和的假设下，每个共同体brauer类都是由Azumaya代数从封闭的Codimension $ \ geq 3 $的封闭子集中所代表的，从而推广了Grothendieck的早期结果。接下来，我们表明$ \ mathrm {br}（x）= \ mathrm {br}'（x）$当$ x $是通过合同曲线从准标准方案获得的代数空间时。这是沿着各个维度起作用的Chow覆盖物沿Azumaya代数下降的第一种方法。作为推论，我们证明了共同体brauer类是任意分离的表面上的几何形状。在较高的维度中，我们获得了具有$ \ mathrm {br}（x）= \ mathrm {br}'（x）$的非Quasi-projective代数空间的第一个示例。
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 $\mathrm{Br}(X)=\mathrm{Br}'(X)$.
