TITLE: For quasi-coherent D-Modules
QUESTION [7 upvotes]: It is well-know that the category of coherent D-modules over a smooth algebraic $k$-scheme is a Tannakian category. So it is equivalent to the category of finite representations of some affine group scheme G/k. My question is do we have the similar statement for quasi-coherent D-modules? I hope that the category of quasi-coherent D-modules is equivalent to the representation (not necessarily finite) category of some affine group schemes. Is that true, is there any reference for that?
I hope that any quasi-coherent D-module is the union of its coherent sub D-modules. If the answer to the above question is true then this is true. If the above is wrong. I still believe this is true. At least I think it is true for char$k$=0, where a quasi-coherent D-module is a quasi-coherent sheaf with a flat connection. If this is true, could you give me any reference?

REPLY [7 votes]: The proper Tannakian theory that has a chance to encompass categories of coherent or quasicoherent D-modules deals with tensor categories WITHOUT a faithful fiber functor - for example the categories of (quasi)coherent sheaves on quasicompact varieties, or more generally geoemtric stacks (see Lurie's article or this MO question). (Once you have D-modules that are not local systems, measuring them at a single point can't be faithful any more..)
Of course to talk about quasicoherent sheaves we need to drop rigidity since
they are not dualizable objects, but that's not a serious problem, since quasicoherent sheaves (eg D-modules) are unions of coherent ones (eg coherent D-modules, which does NOT mean coherent as O-modules). 
In any case in this context one can hope to recover not an affine group scheme but rather the underlying variety or stack again (the usual Tannakian story is the case of the classifying stack pt/G of an affine group scheme G). Here one builds a stack of all fiber functors (not necessarily faithful), and you can hope to reconstruct the original variety/stack by this procedure (the content of Lurie's theorem).
So how might this apply to D-modules? D-modules on X (coherent or quasicoherent) are the same as O-modules (coherent or quasicoherent) on the de Rham space (see e.g. this MO question). So it's reasonable to ask whether
Tannakian reconstruction rebuilds the de Rham space of X from the tensor category of D-modules on X. I believe the answer is yes - certainly there's a natural map
from the de Rham space of X to the Tannakian functor of D-modules (for every point
in X you get a fiber functor and infinitesimally nearby points give isomorphic fiber functors). I would imagine this is an equivalence but haven't thought it through.