Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex-geometry computable-mathematics computer-science connection constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory history homological homological-algebra homology homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie lie-theory limit limits linear linear-algebra locale localization logic mathematics measure-theory modal-logic model model-category-theory monoidal monoidal-category-theory morphism motives motivic-cohomology nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 2nd 2010

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

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 15th 2010
    added References on the site of infinitesimally thickened Cartesian spaces.
    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeJun 15th 2010

    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 ?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 15th 2010

    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?

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeJun 15th 2010

    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 2A_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.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 15th 2010

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

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeJun 15th 2010
    • (edited Jun 15th 2010)

    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.

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeJun 15th 2010
    • (edited Jun 15th 2010)

    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.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 15th 2010

    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 nn with a multiplication operator is a differential operator of order (n1)(n-1).

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

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJun 15th 2010

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

    • CommentRowNumber11.
    • CommentAuthorzskoda
    • CommentTimeJun 15th 2010

    The condition says that the commutator of a differential operator of order nn with a multiplication operator is a differential operator of order (n1)(n-1).

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

    • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 16th 2010

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

    • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeJun 16th 2010

    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.

    • CommentRowNumber14.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 17th 2010

    (tongue removed from cheek)

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

    • CommentRowNumber15.
    • CommentAuthorzskoda
    • CommentTimeJun 17th 2010

    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.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)