Timeline for Which ring spectra are homotopy limits of simpler ones?
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| Aug 22, 2019 at 20:02 | comment | added | მამუკა ჯიბლაძე | @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? | |
| Aug 22, 2019 at 17:09 | comment | added | skd | ... in some xyz-topology, then it'd similarly be an inverse limit over the ring spectra constructed from xyz-maps $\mathrm{Spec}(A) \to M_{fg}$. | |
| Aug 22, 2019 at 17:07 | comment | added | skd | ... 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 ... | |
| Aug 22, 2019 at 17:07 | comment | added | skd | 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 ... | |
| Aug 22, 2019 at 14:33 | history | asked | მამუკა ჯიბლაძე | CC BY-SA 4.0 |