论文标题
在堆栈表面和非交通表面上
On stacky surfaces and noncommutative surfaces
论文作者
论文摘要
令$ \ mathbf {k} $为特征$ \ geq 7 $或零的代数封闭字段。令$ \ Mathcal {a} $为全局尺寸的驯服顺序$ 2 $上的普通表面$ x $上的$ \ mathbf {k} $，这样，$ \ peripatorName {z}（\ mathcal {a}）= \ natercal = \ nathcal = \ nathcal i {o} _ {o} _ {x} $ loctally a directal a $ a $ $ a \ n $ n $ {我们证明，在光滑的驯服代数堆栈上有一个$μ_n$ -gerbe $ \ Mathcal {x} $，其通用稳定剂很琐碎，带有粗大的空间$ x $，因此，在$ \ nathcal {x}上，一类1 twist的连贯的sheaves of coherent of coherent op coherent op coherent op coherent op coherent op coherent of coherent of coherents of coherent of coherent of SheAves SheAves SheAves SheAeves sheAves $ \ MATHCAL {A} $。此外，堆栈$ \ MATHCAL {X} $是通过一系列根堆栈，规范堆栈和Gerbes明确构建的。这将Reiten和van den Bergh的结果扩展到有限的特征和全球情况。作为应用程序，在特征$ 0 $中，我们证明了此类命令是Orlov的几何非共同方案，并且我们研究与Hochschild的同谋的关系和Connes的卷积代数。
Let $\mathbf{k}$ be an algebraically closed field of characteristic $\geq 7$ or zero. Let $\mathcal{A}$ be a tame order of global dimension $2$ over a normal surface $X$ over $\mathbf{k}$ such that $\operatorname{Z}(\mathcal{A})=\mathcal{O}_{X}$ is locally a direct summand of $\mathcal{A}$. We prove that there is a $μ_N$-gerbe $\mathcal{X}$ over a smooth tame algebraic stack whose generic stabilizer is trivial, with coarse space $X$ such that the category of 1-twisted coherent sheaves on $\mathcal{X}$ is equivalent to the category of coherent sheaves of modules on $\mathcal{A}$. Moreover, the stack $\mathcal{X}$ is constructed explicitly through a sequence of root stacks, canonical stacks, and gerbes. This extends results of Reiten and Van den Bergh to finite characteristic and the global situation. As applications, in characteristic $0$ we prove that such orders are geometric noncommutative schemes in the sense of Orlov, and we study relations with Hochschild cohomology and Connes' convolution algebra.
