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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 $F : Scheme_{\mathcal{D}}(X) \to Scheme(X)$ 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?

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJun 17th 2011

Sorry, the math does not display.