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This question was motivated by the answers in D-module as quasi coherent sheaves on deRham stack. What I am interested in is the case of D-module on flag variety of Lie algebra. So,in this case, if we realize the D-module as quasi coherent sheaves on De Rham stack X^DR. Then the stack X^DR is a scheme or not?

Why do I ask this question? Because from the point of view of noncommutative algebraic geometry, if we realize D-module on flag variety of Lie algebra as a quasi coherent sheaves on "space", then this space is actually a noncommutative separated scheme.

So I guess, this DeRham stack will be a scheme in this special case

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up vote 10 down vote accepted

The de Rham space of a scheme is essentially never a scheme or algebraic space (unless I guess you're Spec of an Artin ring, in which case you'll get a discrete set of points). In particular this applies to the flag variety. I'm not sure which perspective of NC AG you're taking, but certainly if you define the field as the study of Grothendieck categories, or pretriangulated dg categories, etc then D-modules are a very nice (nonproper) noncommutative space. (An interesting comment on this is found at the end of Kontsevich's letter here. Also from the point of view of "function theory" D-modules on a scheme are great, i.e. all functors are given by integral transforms, sheaves on a (fiber) product are tensor product of categories of sheaves, etc, see here.) But I don't think this says anything about the de Rham stack in a classical commutative sense..

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The category of D-modules on commutative scheme as a noncommutative space is a noncommutative scheme. – Shizhuo Zhang Jan 16 '10 at 0:23
But this is a trivial example, the first non-trivial example is D-modules on quantized flag variety is a noncommutative scheme – Shizhuo Zhang Jan 16 '10 at 0:24

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