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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJun 17th 2011
   • (edited Jun 17th 2011)
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   Author: Urs
   Format: MarkdownItexI am about to write something at [[jet bundle]] and elsewhere about the general abstract perspective. In chapter 2 of Beilinson-Drinfeld's [[Chiral Algebra]]s they have the nice characterization of the Jet bundle functor as the right adjoint to the forgetful functor $F : Scheme_{\mathcal{D}}(X) \to Scheme(X)$ from [[D-scheme]]s over $X$ to just schemes over $X$. Now, since D-modules on $X$ are quasicoherent modules on the [[de Rham space]] $\Pi_{inf}(X)$, I guess we can identify $$ Scheme_{\mathcal{D}}(X) $$ with $$ Schemes/\Pi_{inf}(X) $$ and hence the forgetful functor above is the pullback functor $$ \array{ F(E) &\to& E \\ \downarrow && \downarrow \\ X &\to& \Pi_{inf}(X) } $$ 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 $\mathbf{H}$ be a [[cohesive (infinity,1)-topos]] equipped with [[infinitesimal cohesion]] $\mathbf{H} \hookrightarrow \mathbf{H}_{th}$. For any $X \in \mathbf{X}$ we then have the canonical morphism $i : X \to \mathbf{\Pi}_{inf}(X)$. The **Jet bundle functor** is then simply the corresponding [[base change geometric morphism]] $$ Jet := (i^* \dashv i_*) : \mathbf{H}/X \to \mathbf{H}/\mathbf{\Pi}(X) $$ or rather, if we forget the $\mathcal{D}$-module structure on the coherent sheaves on the jet bundle, it is the comonad $i^* i_*$ induced by that. Does that way of saying it ring a bell with anyone?

   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 from D-schemes over to just schemes over .

   Now, since D-modules on are quasicoherent modules on the de Rham space , I guess we can identify

   with

   and hence the forgetful functor above is the pullback functor

   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 be a cohesive (infinity,1)-topos equipped with infinitesimal cohesion . For any we then have the canonical morphism .

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

   or rather, if we forget the -module structure on the coherent sheaves on the jet bundle, it is the comonad induced by that.

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

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 17th 2011
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   Author: Urs
   Format: MarkdownItexSorry, the math does not display.

   Sorry, the math does not display.