On stacky surfaces and noncommutative surfaces

Abstract:

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 $Z({\mathcal {A}}) = {\mathcal {O}}_X$ is locally a direct summand of ${\mathcal {A}}$. We prove that there is a $\mu _N$-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.