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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.
55P42 Stable homotopy theory, spectra
55S45 Postnikov systems, \(k\)-invariants
55T15 Adams spectral sequences
55T99 Spectral sequences in algebraic topology
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