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Let $f: X \to S$ be a morphism of locally ringed spaces and $\triangle: X \to X \times_S X$ the corresponding diagonal morphism with kernel sheaf $\mathcal{I} = \ker \triangle^{\flat}$.

Definition: The $k$-th formal neighborhood of the diagonal of the morphism $X \to S$ is the sheaf over $X$ given by the pullback:

$$\triangle^{*}(\mathcal{O}_{X \times_S X} / \mathcal{I}^{k+1})$$

Question (vague): Is this the correct generalization of the jet bundle of a fibre bundle?

Special case (Less vague): Let $E \to X$ be a smooth vector bundle over a smooth manifold. Does the sheaf of sections of the jet bundle $\mathcal{J}^{k}(E)$ coincide with the sheaf given by the above procedure?

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    $\begingroup$ If $J^k = \Delta^{-1}(\mathcal{O}_{X\times X}/\mathcal{I}^{k+1})$, then $J^k$ is a sheaf of $\mathcal{O}_X$-bimodules. The jet bundle $\mathcal{J}^k(E)$ is the same as $J^k\otimes_{\mathcal{O}_X} E$ which is an $\mathcal{O}_X$-module using the left $\mathcal{O}_X$-action on $J^k$. Here it's important to take $\Delta^{-1}$ rather than $\Delta^*$ (inverse image of sheaves rather than pullback of $\mathcal{O}$-modules). $\endgroup$
    – Pavel Safronov
    Commented Feb 23, 2016 at 21:49
  • $\begingroup$ @PavelSafronov In the case where $E = M \times \mathbb{R}$ isn't this the same as the pullback of $\mathcal{O}$-modules? $\endgroup$
    – Cory
    Commented Feb 23, 2016 at 22:16
  • $\begingroup$ Concerning the vague question: from the background I come from the jet bundle of a fiber bundle talks about sections of the bundle, while the construction you gave, if applied to a fiber bundle $X\to S$, seems to be about vertical (along the fibers) jets of functions. These might be related, but they do not seem to be a priori the same. $\endgroup$
    – Michael Bächtold
    Commented Feb 24, 2016 at 8:52
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    $\begingroup$ @PavelSafronov I'm confused as well. I'm trying to connect the clearly related constructions of differential bi-modules (bi-modules supported on the diagonal with a full differential filtration) and jet bundles which are also related to this formal neighborhood of the diagonal stuff. In particular what is the sheaf charactrization of the jet bundle of a torsor. $\endgroup$
    – Cory
    Commented Feb 24, 2016 at 13:48
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    $\begingroup$ I've asked something related to this not long ago: mathoverflow.net/questions/230373/… $\endgroup$
    – Saal Hardali
    Commented Feb 26, 2016 at 11:38

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