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Most surely I will tag this by reference request: I am sure very much is known about this question, I am just too ignorant to even guess where to look. What makes me feel especially foolish is the suspicion that I've actually seen the answers and just cannot remember where.

I am only aware of two instances of what I want to know, but only aware: these represent complicated cases, and I would like to see into simpler ones.

First, I heard that tmf (or TMF?) is the homotopy inverse limit of all elliptic ring spectra. (As an aside: does one really need all of them? Is there some small diagram of elliptic spectra that suffices to obtain it?)

Second, I heard that the orthogonal K-theory is the homotopy fixed point spectrum for an involution on the complex K-theory. And that (maybe) what is called Galois theory of ring spectra represents many of them as homotopy fixed points under actions of finite groups on better understood ones. And that actually algebraic K-theory of any ring spectrum is itself a homotopy inverse limit of some kind.

Hopefully simpler instances that I would like to read about somewhere:

Are there interesting explicit diagrams of Eilenberg-MacLane ring spectra whose homotopy inverse limits produce something interesting, like the same complex K-theory? Here I am aware of the construction of Snaith producing BU from a chain of K(Z,2)-s but this seems to be direct rather than inverse limit, and I think it does not say anything about the ring structure. Or does it?

In general, can one move up the chromatic levels by forming homotopy limits? Can what is called Lubin-Tate theory in this context be formulated in these terms? Can complex cobordism be obtained as homotopy limit of some "smaller" ring spectra? And what about the sphere spectrum?

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    $\begingroup$ First: the higher real K-theories $EO_n$ are homotopy fixed points of the action of a maximal finite subgroup of the Morava stabilizer group on height $n$ Morava E-theory, so these are homotopy inverse limits. (You have to be careful here for general $n$.) Second, the reason that KO and TMF/Tmf are homotopy inverse limits is because they're global sections of a sheaf of $\mathbf{E}_\infty$-rings over a (Deligne-Mumford) stack. As in the classical setting, the way you take global sections over a nice stack $M$ in some xyz-topology is by taking the inverse limit of all rings $A$ equipped ... $\endgroup$
    – skd
    Commented Aug 22, 2019 at 17:07
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    $\begingroup$ ... with an xyz-map $\mathrm{Spec}(A) \to M$. In the case of TMF, such $A$ will be elliptic ring spectra; more generally, chromatically interesting spectra (which arise as the global sections of a sheaf of $\mathbf{E}_\infty$-rings over a spectral stack) will be expressible as a homotopy inverse limit over a diagram of complex orientable ring spectra. Complex cobordism, for instance, admits a trivial description as a homotopy inverse limit from this point of view. If you could realize the sphere spectrum as the global sections of some sheaf over the moduli stack $M_{fg}$ of formal groups ... $\endgroup$
    – skd
    Commented Aug 22, 2019 at 17:07
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    $\begingroup$ ... in some xyz-topology, then it'd similarly be an inverse limit over the ring spectra constructed from xyz-maps $\mathrm{Spec}(A) \to M_{fg}$. $\endgroup$
    – skd
    Commented Aug 22, 2019 at 17:09
  • $\begingroup$ @skd Thank you for interesting comments. Would it be correct to say that all of your cases are holims of spectra of "the same complexity" as the holim itself? Do you know of examples when the holim is "more complicated" than the constituents of the diagram? I realize that the diagrams you mention consist of ring spectra that are "local" in some sense while the holim is "global", but, other than that, I have impression that they are on the same "level of complicatedness" in certain sense. Am I right? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Aug 22, 2019 at 20:02

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