On stacky surfaces and noncommutative surfaces

Let k be an algebraically closed field of characteristic ≥ 7 or zero. Let A be a tame order of global dimension 2 over a normal surface X over k such that Z(A) = OX is locally a direct summand of A. We prove that there is a µN -gerbe 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 X is equivalent to the category of coherent sheaves of modules on A. Moreover, the stack 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.