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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 17th 2011

    I am about to create D-scheme, but currently the Lab is down and the server does not react to my login attempts…

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 17th 2011

    okay, I have created an entry D-scheme

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeJun 17th 2011
    • (edited Jun 17th 2011)

    There is such an entry already! In a variant, diffiety. See also D-geometry, and especially crystal.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 17th 2011

    Okay. You need to remember including the relevant redirects when you create an entry! I’ll inlcude them now.

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeJun 17th 2011
    • (edited Jun 17th 2011)

    Well, I am not sure. Maybe we indeed want to have D-scheme separate from diffiety. It is almost the same, but formally not entirely. I had that in mind then. But I stopped the activity on the D-geometry at the time, abruptly because there was no support/interest from the rest of the n-community at the time.

    Edit: why did you add toc for higher geometry – this is a notion of 1-categorical algebraic geometry. It can be viewed higher categorical, but it is optional. I would be much happier to consider it part of the cluster D-geometry as most of the practioners (cited references) do.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 17th 2011

    I have kept the entries separately, just cross-linked them.

    I had already renamed the toc to just “Geometry”. We just have a single such toc. I used to call it “Higher geoemtry”, but it should just be called “Geometry”.

    Around here everything is higher by default

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeJun 17th 2011

    Still, I will once replace this by lower circle D-geometry once. It is more closely related circle, but I have abosolutely no time to work tyhese days on it.

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeJun 17th 2011

    In any case, crystal of qcoh modules is equivalent to a D-module and crystal of schemes to a D-scheme. See the exposition by Lurie from Gaitsgory’s seminar linked at crystal.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 17th 2011

    Ah, Thanks. I know Lurie’s notes, but apparently I didn’t read them thoroughly.

    So I was proud to have figured out that the Jet scheme is just the change of base along the de Rham space projection. But sure enough that’s what Lurie says on p. 6 there. Thanks.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJun 17th 2011

    You know, now that it is officially not my original idea, I can tell you my opinion about it without being constrained by modesty: this is great! :-) This is way better than what BD write. This makes D-geometry entirely general abstract and work in vast generality away from the original examples.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 17th 2011

    I have added to D-scheme pointers to Lurie’s statement/proof of the assertion that 𝒟 X\mathcal{D}_X-schemes are just schemes over Π inf(X)\mathbf{\Pi}_{inf}(X).

    • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeJun 17th 2011
    • (edited Jun 17th 2011)

    This looks like somewhat related to the relative connections in the sense of p-connection. The idea of jet scheme as a crystal of schemes is I think from Grothendieck’s work on de Rham cohomology for alegbraic varieties. About de Rham space there is also fundamental idea in a paper of Kapranov in late 1980s where he had the idea for categories of complexes.

    • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeJun 17th 2011
    • (edited Jun 17th 2011)

    I am not user if I remember right but I think it is about this article

    • M.M.Kapranov, ON DG-MODULES OVER THE DE RHAM COMPLEX AND THE VANISHING CYCLES FUNCTOR

    I have the article, but I am not very familiar with it, I looked just at few related details at the time when I looked at that subject.

    Edit: after taking a look, maybe not quite that. Some other apsects related to the subject.

    This makes D-geometry entirely general abstract and work in vast generality away from the original examples.

    It makes sense whenever you have a resolution of diagonal.

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