Timeline for Is the formal neighborhood of the diagonal a generalization of the Jet bundle?
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10 events
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| Jun 9, 2017 at 11:32 | comment | added | Malkoun | I may be wrong but, thinking in terms of the tangent bundles only, if the OP wants a natural/canonical isomorphism, one would in particular like to have a natural isomorphism between the tangent bundle of the diagonal, and its normal bundle in the product. However, without additional structure, there is no canonical such isomorphism. | |
| Nov 20, 2016 at 1:21 | comment | added | Fang Hung-chien | I do not know too much about jet bundles, but I think mathoverflow.net/questions/20940/sheaves-of-principal-parts may be helpful. | |
| Feb 26, 2016 at 11:38 | comment | added | Saal Hardali | I've asked something related to this not long ago: mathoverflow.net/questions/230373/… | |
| Feb 24, 2016 at 13:48 | comment | added | Cory | @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. | |
| Feb 24, 2016 at 10:17 | comment | added | Pavel Safronov | Perhaps I misunderstood the question. If a vector bundle $E\rightarrow X$ is supposed to be a particular example of $X\rightarrow S$ (the appearance of $X$ threw me off), then clearly the two constructions give sheaves on different spaces (the first one on the target of the map and the second one on the source). What is the proposed relation between these? | |
| Feb 24, 2016 at 8:52 | comment | added | Michael Bächtold | 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. | |
| Feb 23, 2016 at 22:16 | comment | added | Cory | @PavelSafronov In the case where $E = M \times \mathbb{R}$ isn't this the same as the pullback of $\mathcal{O}$-modules? | |
| Feb 23, 2016 at 21:49 | comment | added | Pavel Safronov | 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). | |
| Feb 23, 2016 at 18:58 | review | First posts | |||
| Feb 23, 2016 at 18:59 | |||||
| Feb 23, 2016 at 18:54 | history | asked | Cory | CC BY-SA 3.0 |