Algebraic & Geometric Topology

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

Daniel Davis and Gereon Quick

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For a profinite group G, let ()hG, ()hdG and ()hG 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)hGK 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 GK–spectrum with (EnhK)hGK EnhG; we achieve this by an argument that possesses a certain technical simplicity enjoyed by neither the proof that (EnhK )hGK EnhG nor the Devinatz–Hopkins proof (which requires |GK| < ) of (EndhK)hdGK E ndhG, where EndhK is a construction that behaves like continuous homotopy fixed points. Also, we prove that (in general) the GK–homotopy fixed point spectral sequence for π((EnhK)hGK), with E2s,t = Hcs(GK;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((EnhK )hGK ).

Article information

Algebr. Geom. Topol., Volume 16, Number 4 (2016), 2257-2303.

Received: 29 July 2015
Revised: 13 October 2015
Accepted: 4 November 2015
First available in Project Euclid: 28 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P42: Stable homotopy theory, spectra
Secondary: 55S45: Postnikov systems, $k$-invariants 55T15: Adams spectral sequences 55T99: None of the above, but in this section

profinite $G$–spectrum homotopy fixed point spectrum iterated homotopy fixed point spectrum Lubin–Tate spectrum descent spectral sequence


Davis, Daniel; Quick, Gereon. Profinite and discrete $G\hskip-2pt$–spectra and iterated homotopy fixed points. Algebr. Geom. Topol. 16 (2016), no. 4, 2257--2303. doi:10.2140/agt.2016.16.2257.

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