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nLab > Latest Changes: de Rham complex

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  1. Added an example (dR cohomology for spheres) with a very sketchy proof sketch. More examples, worked out in more detail, to follow. Might make a de Rham cohomology page and move the examples there.

    Mark Moon

    diff, v36, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 22nd 2020

    added statement of the relation to the PL de Rham complex (here)

    diff, v38, current

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 7th 2023

    Added:

    Explicitly, given a differential kk-form ω\omega, its de Rham differential dωd\omega can be computed as

    dω(v 0,…,v k)=∑ i(−1) iℒ v iω(v 0,…,v i−1,v i+1,…,v k)+∑ iUnknown characterj(−1) i+jω([v i,v j],v 0,…,v i−1,v i+1,…,v j−1,v j+1,…,v k),d\omega(v_0,\ldots,v_k)=\sum_i (-1)^i \mathcal{L}_{v_i} \omega(v_0,\ldots,v_{i-1},v_{i+1},\ldots,v_k)+\sum_{i<j}(-1)^{i+j}\omega([v_i,v_j],v_0,\ldots,v_{i-1},v_{i+1},\ldots,v_{j-1},v_{j+1},\ldots,v_k),

    where v iv_i are vector fields on XX, [−,−][-,-] is the Lie bracket of vector fields, and L −L_{-} is the Lie derivative of a smooth function with respect to a vector field.

    diff, v41, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 9th 2023
    • (edited Jun 9th 2023)

    added pointer to:

    diff, v42, current

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 9th 2024

    The last reference (Kapranov’s paper) links a local nLab file https://ncatlab.org/nlab/files/KapranovDGModuleVanishingCycle.pdf, which as far as I can see, is a work of literature and has no relationship to Kapranov’s paper.

    (Added on June 17, 2011 in Revision 12 by Urs.)

    The file itself is dated August 8, 2023, which means that it was somehow replaced recently?

    diff, v45, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 9th 2024
    • (edited Jan 9th 2024)

    Thanks for the alert.

    Apparently a passer-by used a pending link to upload some spam (Or is it a joke? Looks like it’s a private travel diary?!) I have removed the link.

    The document that I had meant to point to must have been:

    diff, v46, current

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 9th 2024

    Does the current software allow silent updates to files without any notifications?

    Perhaps it may be possible to modify it to post a notification to the nForum when a file is uploaded or reuploaded, just like for the pages themselves?

    Also, are the previous versions discarded when a file is uploaded? Given that some of the PDFs hosted on the nLab are no longer available anywhere else (and some never were), this seems like a major problem.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJan 9th 2024

    It’s impossible for ordinary users to delete a file on the server. I think in the present case there had been no file behind the link, hence the link had appeared in gray with a question mark, during all those years. I have a vague memory of trying to upload something and failing for some reason, but I forget what was going on

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