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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 2nd 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded to [[CartSp]] a section that lists lots of notions of (generalized) geometry modeled on this category.

   added to CartSp a section that lists lots of notions of (generalized) geometry modeled on this category.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 15th 2010
   • PermaLink
   Author: Urs
   Format: Htmladded <a href="http://ncatlab.org/nlab/show/CartSp#References">References</a> on the site of infinitesimally thickened Cartesian spaces.

   added References on the site of infinitesimally thickened Cartesian spaces.
3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeJun 15th 2010
   • PermaLink
   Author: zskoda
   Format: MarkdownYou were asking [here](http://www.math.ntnu.no/~stacey/Vanilla/nForum/comments.php?DiscussionID=1402&Focus=11921) about some of your sharp observations about **infinitesimal thickenings**. I have tried to help with references to something what sounds very similar in noncommutative geometry. Did you make some comparison ?

   You were asking here about some of your sharp observations about infinitesimal thickenings. I have tried to help with references to something what sounds very similar in noncommutative geometry. Did you make some comparison ?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 15th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI looked at it, but I don't yet have a good understanding of the standard theory of differential bimodules that is being generalized there. What's the motivating example here?

   I looked at it, but I don’t yet have a good understanding of the standard theory of differential bimodules that is being generalized there.

   What’s the motivating example here?

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeJun 15th 2010
   • PermaLink
   Author: zskoda
   Format: MarkdownItexThe motivating example is the standard theory of [[regular differential operators]] (on complex analytic manifols, on rigid analytic varieties or for algebraic schemes). for Gorthendieck the motivation was the crystalline cohomology. The reg. dif. op. is the case of the endomorphism object of a ring object (say of the structure sheaf). There is a duality between the infinitesimals and regular differential operators. [[D-modules]] are the modules over the sheaf of regular differential operators. It is amazing that the local role which is in the commutative case give to the [[Weyl algebra]] in some interesting case of modern geometry (related to [[integrable systems]] of Calogero-Moser type) is played by the [[preprojective algebra]]s, which have similar generators and relations. But the modules for the preprojective algebras correspond to A-infinity modules (actually $A_2$-modules) for the Weyl algebra; both approaches are studied by Yuri Berest. So somehow the higher categorical interpretation and noncommutativization of the algebra of differential operators meet. But somehow the preprojective algebras are a different noncommutative extension than those in work of Lunts and Rosenberg. But both have the property of localization: the rings of differential operators are compatible with a good class of localizations. Flat localizations having right adjoint in the case of Lunts and Rosenberg and stably flat maps of algebras in the case of preprojective algebras. Godo comparison is not known.

   The motivating example is the standard theory of regular differential operators (on complex analytic manifols, on rigid analytic varieties or for algebraic schemes). for Gorthendieck the motivation was the crystalline cohomology. The reg. dif. op. is the case of the endomorphism object of a ring object (say of the structure sheaf). There is a duality between the infinitesimals and regular differential operators. D-modules are the modules over the sheaf of regular differential operators. It is amazing that the local role which is in the commutative case give to the Weyl algebra in some interesting case of modern geometry (related to integrable systems of Calogero-Moser type) is played by the preprojective algebras, which have similar generators and relations. But the modules for the preprojective algebras correspond to A-infinity modules (actually $A_2$-modules) for the Weyl algebra; both approaches are studied by Yuri Berest. So somehow the higher categorical interpretation and noncommutativization of the algebra of differential operators meet. But somehow the preprojective algebras are a different noncommutative extension than those in work of Lunts and Rosenberg. But both have the property of localization: the rings of differential operators are compatible with a good class of localizations. Flat localizations having right adjoint in the case of Lunts and Rosenberg and stably flat maps of algebras in the case of preprojective algebras. Godo comparison is not known.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJun 15th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexSo what's the basic example of differential bimodule that i should be thinking of? If I am thinking of a space $Spec A$, is it the $A$-bimodule $A \otimes A$?

   So what’s the basic example of differential bimodule that i should be thinking of? If I am thinking of a space $Spec A$, is it the $A$-bimodule $A \otimes A$?

7. • CommentRowNumber7.
   • CommentAuthorzskoda
   • CommentTimeJun 15th 2010
   • (edited Jun 15th 2010)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexNo, the differential part of End A, what is the ring Diff A of regular differential operators on A, which is just a small part of all the Z-linear endomorphisms of A. For Spec of affine n-space it is the Weyl algebra.

   No, the differential part of End A, what is the ring Diff A of regular differential operators on A, which is just a small part of all the Z-linear endomorphisms of A. For Spec of affine n-space it is the Weyl algebra.

8. • CommentRowNumber8.
   • CommentAuthorzskoda
   • CommentTimeJun 15th 2010
   • (edited Jun 15th 2010)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexin particular the A itself sits as 0-th order differential operators on A. You have filtration by the degree, a differential bimodule is the one which is equal to the union of its own differential filtration, called the differential part.

   in particular the A itself sits as 0-th order differential operators on A. You have filtration by the degree, a differential bimodule is the one which is equal to the union of its own differential filtration, called the differential part.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJun 15th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexAh, that helps! Thanks. Ah, now it's obvious. The filtration is by "order of the differential opetor". The condition says that the commutator of a differential operator of order $n$ with a multiplication operator is a differential operator of order $(n-1)$. I'll create [[differential bimodule]] in a moment, so that I won't forget...

   Ah, that helps! Thanks. Ah, now it’s obvious. The filtration is by “order of the differential opetor”. The condition says that the commutator of a differential operator of order $n$ with a multiplication operator is a differential operator of order $(n-1)$.

   I’ll create differential bimodule in a moment, so that I won’t forget…

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJun 15th 2010
    • PermaLink
    Author: Urs
    Format: MarkdownItexOh, silly me, it already exists, of course. :-) But I add the standard example now.

    Oh, silly me, it already exists, of course. :-) But I add the standard example now.

11. • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeJun 15th 2010
    • PermaLink
    Author: zskoda
    Format: MarkdownItex> The condition says that the commutator of a differential operator of order $n$ with a multiplication operator is a differential operator of order $(n-1)$. Hard to believe that nobody noticed this as a characterising property to define the differential operators until Grothendieck entered the area.

    The condition says that the commutator of a differential operator of order $n$ with a multiplication operator is a differential operator of order $(n-1)$.

    Hard to believe that nobody noticed this as a characterising property to define the differential operators until Grothendieck entered the area.

12. • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 16th 2010
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexSurely it was noticed (in a special case) in quantum mechanics? $[x,\frac{d}{dx}] = i\hbar$ :P

    Surely it was noticed (in a special case) in quantum mechanics? $[x,\frac{d}{dx}] = i\hbar$ :P

13. • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeJun 16th 2010
    • PermaLink
    Author: zskoda
    Format: MarkdownItexNo, it was not. It is not only that the commutator is lower order differential operators but more importantly that no other operator except for differential operator of order n has that property. Moreover, the fact is true for differential operators on any variety/manifold.

    No, it was not. It is not only that the commutator is lower order differential operators but more importantly that no other operator except for differential operator of order n has that property. Moreover, the fact is true for differential operators on any variety/manifold.

14. • CommentRowNumber14.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 17th 2010
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItex(tongue removed from cheek) What other sort of operators are considered in that setup then? What's an example?

    (tongue removed from cheek)

    What other sort of operators are considered in that setup then? What’s an example?

15. • CommentRowNumber15.
    • CommentAuthorzskoda
    • CommentTimeJun 17th 2010
    • PermaLink
    Author: zskoda
    Format: MarkdownItexYou look at Hom(E,F), of k'linear map of sheaves, where E and F are two O-modules. For this bimodule you can look its differential part. If E and F is simply O we have End(O) -- just all linear operators. Its differential part are regular differential operators, but you can also look at differential operators in similar way from one vector bundle to another vector bundle as usually PDE people look, and do the same kind of Gorthendieck styleO-linearization at sheaf theoretic level. This is explained at the beginning of the book of Ogus and Berthelot on crystalline cohomology.

    You look at Hom(E,F), of k’linear map of sheaves, where E and F are two O-modules. For this bimodule you can look its differential part. If E and F is simply O we have End(O) – just all linear operators. Its differential part are regular differential operators, but you can also look at differential operators in similar way from one vector bundle to another vector bundle as usually PDE people look, and do the same kind of Gorthendieck styleO-linearization at sheaf theoretic level. This is explained at the beginning of the book of Ogus and Berthelot on crystalline cohomology.