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Let $R$ be an $L_n$-local ring spectrum. Then one knows that the Tate construction $R^{tC_p}$ (with respect to the trivial $C_p$-action on $R$) is $L_{n-1}$-local; this "blueshift" result is due to Hovey-Sadofsky and Greenlees-Sadofsky, and has been generalized to the telescopic setting by Kuhn.

I am curious to what extent there is an analog of this result when $R$ is instead connective. For example, suppose $R$ has the property that its $K(i)$-localization vanishes for $i \geq n$ (not including $\infty$, so for instance the connective cover of an $L_n$-local ring spectrum is allowed). Does this imply that the $K(i)$-localization of $R^{tC_p}$ vanishes for $i \geq n-1$?

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  • $\begingroup$ Super-ignorant question -- for the Hovey-Sadofsky result you mention, does it matter whether $R^{tC_p}$ is computed in the $L_n$-local category vs. being computed in the category of all spectra? $\endgroup$ Commented Sep 5, 2020 at 14:51
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    $\begingroup$ @TimCampion: It doesn't affect the statement, since the subcategory of $L_n$-local spectra is closed under all limits and colimits (the latter by the smash product theorem). $\endgroup$ Commented Sep 5, 2020 at 17:47
  • $\begingroup$ It seem to me the question for all ring spectra and connective ring spectra are equivalent. It's a matter of commuting localization and fixed points, and either composition kill coconective spectra since Ki localization kills cocobective stuff and fixed points preserve them. Do I miss something here? $\endgroup$
    – S. carmeli
    Commented Sep 6, 2020 at 6:51
  • $\begingroup$ @S.carmeli: Yes, I agree with you. $\endgroup$ Commented Sep 6, 2020 at 21:26

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