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For a finite group $G$, how is a rational $G$-spectrum $X$ detected by the geometric fixed point functor $\phi^H$ where we consider the conjugacy class of $H\leq G$? I tried finding a reference for the same but could not. The best explanation I could find so far has been Corollary 3.4.28 from Schwede's 'Global homotopy theory': For every finite group $G$, every orthogonal $G$-spectrum $X$ and every integer $k$ the map $$ \left(\phi^H\circ\text{res}^G_H\right)_H:\pi^G_k(X)\longrightarrow \prod\limits_{(H)\leq G} \left(\phi^H_k(X)\right)^{W_GH} $$ becomes an isomorphism after inverting the order of $G$.

Is it true in the category of rational $G$-spectra, there is an equivalence described by the functor? $$ X\longmapsto\prod\limits_{(H)\leq G} \phi^H(X) $$

Any reference and/or explanation would be appreciated. Thank you!

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1 Answer 1

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This is implied by Theorem 3.10 in Wimmer's "A model for genuine equivariant commutative ring spectra away from the group order". Taking $R = \Bbb Q$ and $\mathcal F$ to be the full family of subgroups of a finite group $G$, it says that the geometric fixed-point functors induce an equivalence $$ Sp^G_{\Bbb Q} \to \prod_{[H] < G} Fun(BW_H, Sp_{\Bbb Q}). $$ So a rational genuine $G$-spectrum $X$ is equivalent data to its collection of rational spectra $\Phi^H(X)$ equipped with their $W_H$-actions.

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