Quantization via Differential Operators on Stacks

1.1. We will define a D-module of differential operators on a smooth stack and construct a symbol map when the stack is good, but first we need to give definition of a D-module on a stack. D-modules are local for the smooth topology, so one’s naive guess for the definition of a D-module on a smooth Artin stack is correct. That is, a (left or right) D-module M on X is the assignment for each U πU −→X in Xsm of a (left or right) D-module MS on S and for each (f, α) a morphism in Xsm an isomorphism β : f MU ′ ' −→MU which satisfy the cocycle condition that whenever we have a composition of morphisms U f −→ U ′ f ′ −→ U ′′ that β ◦ f ∗(β′) = β′′. Let us denote by M (X ) the category of right D-modules on X and by M (X ) the category of left D-modules. For a D-module M on X , we denote by Γ(X ,M) the space of global sections of M considered as a quasi-coherent sheaf, where the functor Γ is defined for quasi-coherent sheaves by forming Hom(OX ,M), or equivalently, taking compatible families of sections. Now we will define the D-module of differential operators DX on X . Let f : U −→ X be a smooth map. Then let I be the left ideal of DU generated by the image of TU/X −→ U . Then define the pull-back (DX )U of DX to U to be DU/I. This is a D-module on U . One can immediately check that this defines a D-module on X for us. This is the D-module of differential operators and is denoted DX or D if there’s no confusion. It satisfies the property that HomM `(DX ,M) = Γ(X ,M) for any left D-module M . Note that DX is not a sheaf of rings (and there may be no sheaf of rings on Xsm such that D-modules are modules over it). However, the global sections of DX do form a ring because: