Profinite and discrete G–spectra and iterated homotopy fixed points

Davis, Daniel¹ ; Quick, Gereon²

¹Department of Mathematics, University of Louisiana at Lafayette, 1403 Johnston Street, Maxim Doucet Hall, Room 217, Lafayette, LA 70504-3568, USA
²Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway

Algebraic and Geometric Topology, Tome 16 (2016) no. 4, pp. 2257-2303

Résumé

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) the G∕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 ).

Détail

DOI : 10.2140/agt.2016.16.2257

Classification : 55P42, 55S45, 55T15, 55T99
Keywords: profinite $G$–spectrum, homotopy fixed point spectrum, iterated homotopy fixed point spectrum, Lubin–Tate spectrum, descent spectral sequence

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