Profinite and discrete G-spectra and iterated homotopy fixed points

Abstract

For a profinite group $G$, let $(\text{-})^{hG}$, $(\text{-})^{h_dG}$, and $(\text{-})^{h'G}$ 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 $K \vartriangleleft_c G$ (a closed normal subgroup), give various conditions for when the iterated homotopy fixed points $(X^{hK})^{hG/K}$ 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 $(E_n^{hK})^{hG/K} \simeq E_n^{hG}$, by an argument that possesses a certain technical simplicity not enjoyed by either the proof that $(E_n^{h'K})^{h'G/K} \simeq E_n^{h'G}$ or the Devinatz-Hopkins proof (which requires $|G/K| < \infty$) of $(E_n^{dhK})^{h_dG/K} \simeq E_n^{dhG}$, 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 $\pi_\ast((E_n^{hK})^{hG/K})$, with $E_2^{s,t} = H^s_c(G/K; \pi_t(E_n^{hK}))$ (continuous cohomology), is isomorphic to both the strongly convergent Lyndon-Hochschild-Serre spectral sequence of Devinatz for $\pi_\ast(E_n^{dhG})$, with $E_2^{s,t} = H^s_c(G/K; \pi_t(E_n^{dhK}))$, and the descent spectral sequence for $\pi_\ast((E_n^{h'K})^{h'G/K})$.