Profinite and discrete $G\hskip-2pt$–spectra and iterated homotopy fixed points

Abstract

For a profinite group $G$, let ${\left(-\right)}^{hG}$, ${\left(-\right)}^{h_{d}G}$ and ${\left(-\right)}^{h^{\prime }G}$ denote continuous homotopy fixed points for profinite $G$–spectra, discrete $G$–spectra and continuous $G$–spectra (coming from towers of discrete $G$–spectra), respectively. We establish some connections between the first two notions, and by using Postnikov towers, for $K{◃}_{c}G$ (a closed normal subgroup), we give various conditions for when the iterated homotopy fixed points ${\left(X^{hK}\right)}^{hG∕K}$ exist and are $X^{hG}$. For the Lubin–Tate spectrum $E_n$ and $G{<}_c{G}_n$, the extended Morava stabilizer group, our results show that $E_n^{hK}$ is a profinite $G∕K$–spectrum with ${\left(E_n^{hK}\right)}^{hG∕K}\simeq E_n^{hG}$; we achieve this by an argument that possesses a certain technical simplicity enjoyed by neither the proof that ${\left(E_n^{h^{\prime }K}\right)}^{h^{\prime }G∕K}\simeq E_n^{h^{\prime }G}$ nor the Devinatz–Hopkins proof (which requires $|G∕K|<\infty$) of ${\left(E_n^{dhK}\right)}^{h_{d}G∕K}\simeq E_n^{dhG}$, where $E_n^{dhK}$ is a construction that behaves like continuous homotopy fixed points. Also, we prove that (in general) the $G∕K$–homotopy fixed point spectral sequence for ${\pi }_{\ast }\left({\left(E_n^{hK}\right)}^{hG∕K}\right)$, with $E_2^{s,t}=H_c^s\left(G∕K;{\pi }_t\left(E_n^{hK}\right)\right)$ (continuous cohomology), is isomorphic to both the strongly convergent Lyndon–Hochschild–Serre spectral sequence of Devinatz for ${\pi }_{\ast }\left(E_n^{dhG}\right)$ and the descent spectral sequence for ${\pi }_{\ast }\left({\left(E_n^{h^{\prime }K}\right)}^{h^{\prime }G∕K}\right)$.