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Profinite and discrete $$G$$-spectra and iterated homotopy fixed points. (English) Zbl 1373.55010
The paper under review is a contribution to the study of relations between the notions of homotopy fixed point spectrum for the action of a profinite group, and the associated homotopy fixed point spectral sequence.
The authors investigate the following important question: Can one simplify the analysis of the homotopy fixed points under the group $$G$$ by reducing it to the study of those under proper closed normal subgroups $$K$$ and the quotients $$G/K$$? This would require two steps. For $$X$$ a fibrant profinite $$G$$-spectrum, determine whether $$X^{hK}$$ is a profinite $$G/K$$-spectrum. If this is the case, then determine whether the comparison map $X^{hG}\to \left(X^{hK}\right)^{hG/K}$ is an equivalence. The authors provide various sets of sufficient conditions on $$G$$ and $$X$$, namely $$X$$ is a $$K$$-Postnikov $$G$$-spectrum and $$G/K$$ has finite virtual cohomological dimension, that allow for obtaining the equivalence.
The main application of these results is the important example of the extended Morava stabilizer group $$G_{n}$$ on the Lubin-Tate spectrum $$E_{n}$$. For this purpose, the previous results are extended to homotopy inverse limits of diagrams of spectra as $$E_{n}^{hK}$$ can be identified with the homotopy inverse limit of a suitable diagram, where $$K$$ is a normal subgroup of some closed subgroup $$G$$ of $$G_{n}$$. Therefore, $$E_{n}^{hK}$$ is a profinite $$G/K$$-spectrum and the homotopy fixed point spectrum $$\left(E_{n}^{hK}\right)^{hG/K}$$ is defined and is equivalent to $$E_{n}^{hG}$$. The associated spectral sequence has the $$E_{2}$$-page of the form $$H^{s}_{c}\left(G/K;\pi_{t}\left(E_{n}^{hK}\right)\right)$$. Other constructions of the homotopy fixed point with respect to a continuous action of a closed subgroup of $$G_{n}$$ have also been studied by Devinatz, Hopkins and the first author. However, none of these constructions enjoys the fact that the equivalence of iterated homotopy fixed points $$E_{n}^{hG}\cong\left(E_{n}^{hK}\right)^{hG/K}$$ always holds. The authors expect that $$\left(E_{n}^{hK}\right)^{hG/K}$$ will be a useful tool in chromatic theory.
##### MSC:
 55P42 Stable homotopy theory, spectra 55S45 Postnikov systems, $$k$$-invariants 55T15 Adams spectral sequences 55T99 Spectral sequences in algebraic topology
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