On stacky surfaces and noncommutative surfaces

Source abstract

Let k {\mathbf {k}} be an algebraically closed field of characteristic ≥ 7 \geq 7 or zero. Let A {\mathcal {A}} be a tame order of global dimension 2 2 over a normal surface X X over k {\mathbf {k}} such that Z ( A ) = O X Z({\mathcal {A}}) = {\mathcal {O}}_X is locally a direct summand of A {\mathcal {A}} . We prove that there is a μ N \mu _N -gerbe X {\mathcal {X}} over a smooth tame algebraic stack whose generic stabilizer is trivial, with coarse space X X such that the category of 1-twisted coherent sheaves on X {\mathcal {X}} is equivalent to the category of coherent sheaves of modules on A {\mathcal {A}} . Moreover, the stack X {\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 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.