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This is basically a reference request. Does anyone know if the structure of the homotopy category of spectra (or maybe just the model, i.e. w/o the homotopy, category), localized at infinite wedges of Morava K-theories, or BP, is either well described somewhere or somehow stupid and uninteresting? This question is motivated by i) I believe Morava K-theories and the telescope spectra T(n) are the same BP-locally, i.e. a sort of telescope conjecture, and ii) wondering if localizing at BP or maybe the wedge of all the Morava K-theories would somehow pick out all the chromatic information in the category of spectra. That is (and I think my details will be off because I haven't thought about this in a little bit), there is some idea in the derived category of a Noetherian ring that we have these localization functors which correspond to prime ideals of the ring, and that maybe localizing at BP is somehow localizing the stable homotopy category at the $I_n$ ideals or something. Any commentary is appreciated.

Thanks!

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  • $\begingroup$ Jon, you are probably already familiar with these, but have you found any information in Hovey-Strickland's "Morava K-theories and localisation" or Hovey's "Bousfield localization functors and Hopkins' chromatic splitting conjecture"? $\endgroup$
    – Tyler Lawson
    Commented Aug 21, 2012 at 21:59
  • $\begingroup$ @Tyler Sort of. Hovey's paper on the chromatic splitting conjecture was where I found (i) above. I have not yet found more information regarding this question specifically, but I have not entirely read these papers, though they are two of my favorites! But thanks for mentioning them again, I will check thru them more carefully I think $\endgroup$
    – Jonathan Beardsley
    Commented Aug 21, 2012 at 22:48
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    $\begingroup$ In case anyone ever comes across this and is interested, I was able to prove that the Bousfield lattice of the p-local stable homotopy category localized at the infinite wedge of Morava K-theories is the Boolean algebra generated by the Morava K-theories. $\endgroup$
    – Jonathan Beardsley
    Commented Nov 16, 2012 at 4:12

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I mentioned that I proved this in the comment above but am "answering" just for closure. A link to the proof is here:

http://chromotopy.org/?p=1110

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  • $\begingroup$ sorry didn't mean for this to get bumped up to the top... not a very interesting question. $\endgroup$
    – Jonathan Beardsley
    Commented Apr 5, 2013 at 22:35

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