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.Comment: v2:many minor revisions. Section 6 of v1 is removed. 34 page