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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 30th 2010
    • (edited Nov 30th 2010)

    added to tmf a section that gives an outline of the proof strategy for how to compute the homotopy groups of the tmftmf-spectrum from global sections of the E E_\infty-structure sheaf on the moduli stack of elliptic curves.

    A point which I wanted to emphasize is that

    1. The problem of constructing tmftmf as global sections of an \infty-structure sheaf has a tautological solution: take the underlying space to be SpectmfSpec tmf.

    2. From this tautological but useless solution one gets to the one that is used for actual computations by one single crucial fact:

      In the \infty-topos over the \infty-site of formal duals of E E_\infty-rings, the dual SpecMUSpec M U of the Thom spectrum, is a well-supported object. the terminal morphism

      SpecMU* Spec M U \to *

      in the \infty-topos is an effective epimorphism, hence a covering of the point.

    Using this we can pull back the tautological solution of the problem to the cover and then compute there. This is what actually happens in practice: the decategorification of the pullback of SpectmfSpec tmf to SpecMUSpec M U is the moduli stack of elliptic curves. And it is a happy coincidence that despite this drastic decategorification, there is still enough information left to compute 𝒪Spectmf\mathcal{O} Spec tmf on that.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 7th 2014

    I have considerably expanded the idea-section at tmf. Also I started some notes at Definition and construction – Decomposition via Arithmetic fracture squares, which is however very much stubby still.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 9th 2014

    Have added to tmf a section Maps to K-theory and to Tate K-theory.

    Also I have split the “Definition and Construction”-section into a Definition-section and a Construcion-section and added some actual (though basic) content to the Definition section (the Construction-section remains very piecemeal, naturally but nevertheless woefully).

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 14th 2014

    added a list of the low degree homotopy groups of tmf

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 20th 2015

    Corrected the indexing on the table in #4 (started at 1 instead of 0)

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 1st 2020

    Added the reference

    diff, v56, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2020
    • (edited Sep 7th 2020)

    added pointer to:

    diff, v57, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2020
    • (edited Sep 7th 2020)

    have given the statement about the Boardman homomorphism for tmftmf a little Properties-subsection (here) of its own (splitting it off from the subsection on stable homotopy groups).

    Will also give this a stand-alone entry: Boardman homomorphism in tmf, for ease of hyperlinking from elsewhere.

    diff, v58, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2020

    added DOI to

    diff, v61, current

    • CommentRowNumber10.
    • CommentAuthorGuest
    • CommentTimeJan 18th 2022

    Once nLab editing is open, someone should fix the mistake that (connective) tmf is defined as the global sections of a sheaf of E E_{\infty}-rings. That’s not true - it’s only known definition is as a connective cover of Tmf. For instance, see Behrens’ survey article in the Handbook. To quote the Hill-Lawson paper (p. 6): “Finally, the construction of the object tmf by connective cover remains wholly unsatisfactory, and this is even more true when considering level structure. In an ideal world, tmf should be a functor on a category of Weierstrass curves equipped with some form of extra structure. We await the enlightenment following discovery of what exact form this structure should take.”

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2022

    Thanks for the heads-up.

    It looks like in the “Definition”-Section 2 it’s stated correctly, at least after the words “more precisely”. Then section 3 is lacking the capitalization.