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nLab > Latest Changes: Introduction to Stable Homotopy Theory

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    • CommentRowNumber1.
    • CommentAuthorjesuslop
    • CommentTimeJul 25th 2023

    No changes. In “Adams spectral sequences”, It can be proper to justify “AbelianGroups simeq QCoh(Spec(Z))” by linking to affine Serre’s theorem.

    diff, v214, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJul 25th 2023
    • (edited Jul 25th 2023)

    I wasn’t aware (anymore) of the old entry affine Serre’s theorem. Somebody should merge whatever of value it contains into Serre-Swan theorem.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 25th 2023

    As a standalone statement, the affine Serre theorem certainly has value (“whatever of value it contains” suggests there’s something wrong with it; is there?).

    I think of the Serre-Swan theorem as more about certain projective modules, which typically lack certain limits and colimits, whereas coherent or quasicoherent modules extend them to repair that loss. So I wouldn’t think to put the latter inside the stomach of the former; instead I’d be more inclined to flesh out the affine Serre theorem article independently (although I’m hardly an expert).

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 25th 2023

    Actually, there may be nits to pick, particularly the title (does anyone call it the “affine Serre theorem”?). It’s not really given a name in Hartshorne, nor in the Stacks Project; there it’s just Lemma or Proposition 13.4 or whatever. Although I have seen the statement mentioned from time to time, as a basic ingredient in calculations, so maybe someone has given it a name after all.

    I may jigger around with it a little. It could use some references and maybe some nPOV-ifying.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 25th 2023

    I think the original reference is the same as Serre’s contribution to Serre-Swan:

1 to 5 of 5

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