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I am reading Bousfield's paper entitled "The localization of spectra with respect to homology" (MSN). In that paper, Corollary 6.13 states that, if a ring spectrum $E$ has countable homotopy and satisfies some vanishing conditions in the associated Adams spectral sequence, then the localization is equivalent to nilpotent completion.

So, my question is the following:

What are the ring spectra $E$ (I need an updated list) satisfying the above conditions?

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    $\begingroup$ For reference, Bousfield's paper can be found here core.ac.uk/download/pdf/82464212.pdf . Corollary 6.13 is on page 24 (the page number is 280) $\endgroup$
    – Victoria M
    Commented Jun 4, 2019 at 17:36
  • $\begingroup$ @VictoriaM, thanks! I have edited in a link to the abstract (which I think is generally to be preferred to the PDF, especially since it's freely available). $\endgroup$
    – LSpice
    Commented Jun 4, 2019 at 17:43
  • $\begingroup$ Should your hypothesis "if a ring spectrum $E$ with countable homotopy and satisfies …" be "if a ring spectrum $E$ has countable homotopy and satisfies …"? $\endgroup$
    – LSpice
    Commented Jun 4, 2019 at 17:44
  • $\begingroup$ @LSpice: Thank you. $\endgroup$
    – Surojit Ghosh
    Commented Jun 5, 2019 at 8:03
  • $\begingroup$ Morava $E$ theories satisfy the condition I think. That's why they are smashing! $\endgroup$
    – Prasit
    Commented Jun 6, 2019 at 4:10

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