#### Volume 16, issue 4 (2016)

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

### Daniel G Davis and Gereon Quick

Algebraic & Geometric Topology 16 (2016) 2257–2303
##### 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 ${\pi }_{\ast }\left({\left({E}_{n}^{{h}^{\prime }K}\right)}^{{h}^{\prime }G∕K}\right)$.

##### Keywords
profinite $G$–spectrum, homotopy fixed point spectrum, iterated homotopy fixed point spectrum, Lubin–Tate spectrum, descent spectral sequence
##### Mathematical Subject Classification 2010
Primary: 55P42
Secondary: 55S45, 55T15, 55T99