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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJun 17th 2011
   • (edited Jun 17th 2011)
   • PermaLink
   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 $F : Scheme_{\mathcal{D}}(X) \to Scheme(X)$ from D-schemes 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?

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

   Sorry, the math does not display.