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 categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor galois-theory gauge-theory gebra geometric geometric-quantization geometry 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 homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics 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 string string-theory 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
    • CommentTimeJun 17th 2011
    • (edited Jun 17th 2011)

    I am about to write something at jet bundle and elsewhere about the general abstract perspective.

    In chapter 2 of Beilinson-Drinfeld’s Chiral Algebras they have the nice characterization of the Jet bundle functor as the right adjoint to the forgetful functor F:Scheme 𝒟(X)Scheme(X)F : Scheme_{\mathcal{D}}(X) \to Scheme(X) from D-schemes over XX to just schemes over XX.

    Now, since D-modules on XX are quasicoherent modules on the de Rham space Π inf(X)\Pi_{inf}(X), I guess we can identify

    Scheme 𝒟(X) Scheme_{\mathcal{D}}(X)


    Schemes/Π inf(X) Schemes/\Pi_{inf}(X)

    and hence the forgetful functor above is the pullback functor

    F(E) E X Π inf(X) \array{ F(E) &\to& E \\ \downarrow && \downarrow \\ X &\to& \Pi_{inf}(X) }

    aling the lower canonical morphism (“constant infinitesimal path inclusion”).

    This would mean that we have the following nice general abstract characterization of jet bundles:

    let H\mathbf{H} be a cohesive (infinity,1)-topos equipped with infinitesimal cohesion HH th\mathbf{H} \hookrightarrow \mathbf{H}_{th}. For any XXX \in \mathbf{X} we then have the canonical morphism i:XΠ inf(X)i : X \to \mathbf{\Pi}_{inf}(X).

    The Jet bundle functor is then simply the corresponding base change geometric morphism

    Jet:=(i *i *):H/XH/Π(X) Jet := (i^* \dashv i_*) : \mathbf{H}/X \to \mathbf{H}/\mathbf{\Pi}(X)

    or rather, if we forget the 𝒟\mathcal{D}-module structure on the coherent sheaves on the jet bundle, it is the comonad i *i *i^* i_* induced by that.

    Does that way of saying it ring a bell with anyone?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 17th 2011

    Sorry, the math does not display.

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)