Profinite and discrete G-spectra and iterated homotopy fixed points

Abstract

For a profinite group G, let (−)hG, (−)hdG and (−)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◃cG (a closed normal subgroup), we give various conditions for when the iterated homotopy fixed points (XhK)hG∕K exist and are XhG. For the Lubin–Tate spectrum En and G < cGn, the extended Morava stabilizer group, our results show that EnhK is a profinite G∕K–spectrum with (EnhK)hG∕K ≃ EnhG; we achieve this by an argument that possesses a certain technical simplicity enjoyed by neither the proof that (Enh′K )h′G∕K ≃ Enh′G nor the Devinatz–Hopkins proof (which requires|G∕K| < ∞) of (EndhK)hdG∕K ≃ E ndhG, where EndhK is a construction that behaves like continuous homotopy fixed points. Also, we prove that (in general) theG∕K–homotopy fixed point spectral sequence for π∗((EnhK)hG∕K), with E2s,t =Hcs(G∕K;πt(EnhK)) (continuous cohomology), is isomorphic to both the strongly convergent Lyndon–Hochschild–Serre spectral sequence of Devinatz for π∗(EndhG)and the descent spectral sequence for π∗((Enh′K )h′G∕K ).