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There are a couple different models for spectra, or constructions of the categories of spectra that have the desired properties (homotopically and otherwise). The construction of the Categories of $S$-algebras in EKMM (Rings, Modules, and Algebras in stable homotopy theory by Elmendorf-Mandell-May-Kriz) is one such model. In EKMM, the authors develop a lot of "technology" for their category of spectra. In particular, they construct several spectral sequences and talk about Algebraic K-theory of S-algebras. I am curious to know if such developments have been pursued in the various other models, such as Symmetric Spectra.

All of these models yield the same homotopy category (I believe the paper of Mandell-May-Schwede-Shipley does this, or does a lot of it), but the impression I have gotten is that $S$-algebras have a lot of the technology already developed that one might want in stable homotopy theory. I believe Schwede has worked on computing homotopy groups of Symmetric Spectra, is this a sign that more technology is on its way?

I suppose the above is all background in some sense, so let me clearly state my question(s): Are there developed notions of spectral sequences for other models of spectra? Are there simple reasons why one would or would not expect such developments to occur (other than we already have them for $S$-algebras)?

Also, I don't mean for this to spark a debate about different models. I am just curious.

Thanks

EDIT: Clearly, I was unclear. I am wondering if people have used, written down, proved results regarding convergence etc, spectral sequences based on other models of spectra. Is it as Tyler points out that all constructions of spectral sequences essentially occur in the homotopy category? That would certainly answer my question. I thought that since computing homotopy groups of Symmetric Spectra (non-naively) was subtle that such constructions might be subtle as well.

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  • $\begingroup$ I'm unclear what you mean by "developed notions of spectral sequences". What kinds of spectral sequences are you interested in that aren't lifted from some notion in the homotopy category of spectra (which is, by definition of "equivalent", independent of model)? $\endgroup$
    – Tyler Lawson
    Dec 15, 2010 at 2:33
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    $\begingroup$ In EKMM they go through the construction of some basic hypercohomology spectral sequences. Can I just mimic such constructions in with the other models of spectra? My initial reaction is no, there seem to be a lot of hypothesis as to things being cellular and the like. Maybe such notions in the other models are simple. Certainly I would accept an answer saying "You can mimic pretty much all of the homological algebra chapter in EKMM in settings X,Y, and Z." I was under the impression that people dont compute with the other models, and I suppose I am looking for others "opinions." $\endgroup$
    – Sean Tilson
    Dec 15, 2010 at 2:54
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    $\begingroup$ I think that lots of computations occur in different models, in fact. You might try looking at the Goerss-Hopkins paper "Moduli spaces of commutative ring spectra," where the first chapter gives a lengthy description of what conditions they really need in order to carry out their work and how various different models satisfy them. $\endgroup$
    – Tyler Lawson
    Dec 15, 2010 at 13:27
  • $\begingroup$ Thanks for the suggestion. I spent some time talking with Bob, and he said roughly what you did above about it not making too much of a difference. $\endgroup$
    – Sean Tilson
    Dec 15, 2010 at 19:35
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    $\begingroup$ There's much technology on symmetric spectra in Schwede's math.uni-bonn.de/~schwede/SymSpec.pdf . OK, there are no spectral sequences in it, but there is a whole bunch of constructions. $\endgroup$
    – Lennart Meier
    Dec 17, 2010 at 23:03

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I don't think I noticed this question before. One point is that now that we have multiplicatively well-behaved Quillen equivalences between all reasonable models for the stable category, hence between reasonable models for categories of ring and module spectra, it is formal to transport constructions like spectral sequences from one model to any other. Another point is just historical: EKMM got there first and skimmed off the easy applications. In view of the first point, there is no reason for anyone to want to reinvent the wheel.

A technical point is that, for spectral sequences especially, it is convenient to work with CW spectra (or CW R-modules for a ring spectrum R), and these are nowhere written down in any context other than EKMM (harking back to LMS). They are only natural objects when all spectra are fibrant; more precisely, unless all spectra are fibrant, CW spectra will not be nicely related to the cell spectra relevant to the model structures of interest. (Section 24.2 of Parametrized Homotopy Theory http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf has some discussion of this.) This advantage of EKMM has directly related offsetting disadvantages; in particular, the sphere spectrum is not cofibrant. It is an old theorem of Gaunce Lewis that you can't have everything.

Repeating myself from other answers, it is best to be eclectic.

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