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    • 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)

    with

    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.

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