The local structure of terminal Brauer classes on arithmetic surfaces was classified (2021), generalising the classification on geometric surfaces (2005). Part of the interest in these classifications is that it enables the minimal model program to be applied to the noncommutative setting of orders on surfaces. We give étale local structure theorems for terminal orders on arithmetic surfaces, at least when the degree is a prime $p>5$. This generalises the structure theorem given in the geometric case. They can all be explicitly constructed as algebras of matrices over symbols. From this description one sees that such terminal orders all have global dimension two, thus generalising the fact that terminal (commutative) surfaces are smooth and hence homologically regular.

对算术表面上终端 Brauer 类的局部结构进行了分类（2021 年），概括了几何表面的分类（2005 年）。这些分类的部分意义在于，它使最小模型程序能够应用于表面上阶次的非交换设置。我们给出了算术面上端阶的 étale 局部结构定理，至少当度数是素数 $p>5$ 时是这样。这概括了几何情况下给出的结构定理。它们都可以显式构造为符号上的矩阵代数。从这个描述中可以看出，这些终端阶都具有全局维度 2，从而概括了终端（交换）表面是光滑的，因此同理规则的事实。