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Is there a known relation between the space of units of a ring spectrum, from stable homotopy theory in the sense of Ando-Blumberg-Gepner https://arxiv.org/pdf/1002.3004v2.pdf and completions of spectra, in the sense of Bousfield -Kan, et al?

In particular, does completion commute with p- completion at a prime p? Since the first is a homotopy colimit like tipe of construction, and the second one is a homotopy inverse limit, this seems to be unlikely. Is there a spectral sequence relating them ?

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    $\begingroup$ I don't understand the relationship between the second paragraph and the first. $\endgroup$
    – Qiaochu Yuan
    Commented Nov 18, 2016 at 1:30
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    $\begingroup$ Are you asking if taking units commutes with p-completion? If so, then the answer is no because it is also false for ordinary rings: $(R^\wedge_p)^\times \neq (R^\times)^\wedge_p$ as you can see from the example of $\Bbb Z$. However, the issue is almost entirely a $\pi_0$-issue. $\endgroup$
    – Tyler Lawson
    Commented Nov 18, 2016 at 5:11

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