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Let $G$ be a finite group and $X$ an orthogonal $\mathbb{Z}\left[\frac{1}{\lvert G\rvert}\right]$-local spectrum with an $G$-action that is trivial on $\pi_*X$.

I want to show that then the map $X^{hG} \to X$ from the homotopy fixed points is a $\mathbb{Z}\left[\frac{1}{\lvert G\rvert}\right]$-local equivalence.

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    $\begingroup$ Maybe there is some subtlety I have missed (for example, I am assuming that by $BK$ you mean $G$), but the classical Bousfield-Kan spectral sequence of the form $H^*(G; \pi_*(X)) \Rightarrow \pi_*(X^{hG})$ is natural in the choice of group $G$. By Maschke's theorem, your localness hypothesis is enough to ensure that $H^*(G; \pi_*(X))$ agrees with $H^*(1; \pi_*(X))$, so the map from the BKSS for the $G$-action on $X$ to the action of the trivial group on $X$ is an isomorphism on $E_2$-terms. So it's an isomorphism on $E_{\infty}$-terms, so $\pi_*(X^{hG}) \rightarrow \pi_*(X)$ is an isomorphism. $\endgroup$
    – user164898
    Commented May 8, 2022 at 15:17
  • $\begingroup$ Thanks, yes you are right, I mean $G$ and corrected it. $\endgroup$
    – Urs
    Commented May 8, 2022 at 16:35

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