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In commutative algebraic geometry, differential operator commutes with localization. I wonder whether there is anybody who consider the differential operator in noncommutative geometry. Say, differential operator on noncommutative ring.

I wonder whether one can still get the proposition that differential operator commutes with localizations.

I think there might be correct notions for noncommutative D-module?

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See papers of Kapranov. – Oren Ben-Bassat Jan 25 '10 at 15:21
Please read the "How to ask a good question" page. I'll only point out that your 'question', "I think there might be correct notions for noncommutative D-module?", isn't actually a question. – Scott Morrison Jan 26 '10 at 1:02
"See papers of Kapranov" sounds amazingly close to "It's in Euler already" :P – Mariano Suárez-Alvarez Feb 18 '10 at 19:50

The Ginzburg's paper quoted above has a specific setup which is related to the examples like the "preprojective algebras" related to quivers. There is a recipe attaching an algebra to a noncommutative algebra by generators and relations reminiscent of the Weyl algebra. This recipe is local not only under flat localization, but even under stably flat maps. Yuri Berest has studied these aspects with hopes to globalize that definition to the nonaffine situations.

The usual Grothendieck definition may work in some noncommutative cases, e.g. possibly when the only noncommutative variables are nilpotent and alike. I do not know which definition in Kapranov, Oren means in his comment, but I think he points to similar cases related to nilpotent thickenings. Already for quantum groups this does not suffice.

Lunts and Rosenberg have tried to find a definition which would go along the Grothendieck's geometric picture: dealing with resolutions of diagonal. This has been studied in two of their Max Planck Bonn preprints, in very abstract categorical framework, and the results are global in the language of categories of quasicoherent sheaves on noncommutative schemes so to speak. Then they wrote two other papers on the same topic with down-to-earth recipes in the case of rings, modules and graded case. These differential operators correspond to filtrations reminding the Grothendieck's case of filtration by order, but being corrected in an improtant subtle detail. Their basic property is standard behaviour under flat localization functors as you suggest. There is also an arxiv paper by Tomasz Maszczyk who uses a variant of nc algebraic geometry based on bimodules and monoidal categories, and he rederives the same definition of the ring of regular differential operators as Lunts and Rosenberg do, with different geometric insight based on the duality between the infinitesimals and differential operators.

For the references look at nlab page on differential operators in nc geometry which I just started writing

nlab:regular differential operator in noncommutative geometry

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nice answers, and welcome to mathoverflow – Shizhuo Zhang Feb 18 '10 at 20:34
By the way, I am now reading the paper of Lunts-Rosenberg's paper written in categorical language and will give a talk in the seminar here. I just took a look at Maszczyk's paper but never read carefully. Could you explain more about his work? Maybe I should ask a question about his work in this site – Shizhuo Zhang Feb 18 '10 at 20:39
I admire Tomasz Maszczyk's work a lot and understood some parts of it, when I discussed with him. Currently he is working on vast generalization and update of his work. I think his paper could speak more itself than my explanations at the present moment. – Zoran Skoda Feb 19 '10 at 19:03

Check out the paper arXiv:0710.3392 - it talks about such generalization.

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