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The homotopy fixed point spectra of profinite Galois extensions. (English) Zbl 1204.55007

There is a notion of Galois extension for commutative \(S\)-algebras (\(E_\infty\)-ring spectra), which was originally formulated by J. Rognes [Mem. Am. Math. Soc. 898, 137 p. (2008; Zbl 1166.55001)] as a kind of generalization of Galois extension for commutative rings. Let \(G\) be a profinite group, and let \(A\) be a \(k\)-local cofibrant commutative \(S\)-algebra where \(k\) is a fixed \(S\)-module. This paper is concerned with a \(k\)-local profinite \(G\)-Galois extension \(E\) of \(A\) defined using this notion. The main theme is to study its homotopy fixed point spectra \(E^{hH}\) with respect to closed subgroups \(H\) of \(G\), which were first studied by the second author [J. Pure Appl. Algebra 206, No. 3, 322–354 (2006; Zbl 1103.55005)].
The purpose of this paper is twofold: first, to prove that there holds the forward part of the profinite version of Rognes’ Galois correspondence for \(E\), and second, to identify the homotopy type of the function spectra of \(A\)-module maps between any two homotopy fixed point spectra of \(E\); the precise statements of the results are given as Theorems (7.2.1) and (7.3.1). But in order for the idea here to be brought to life, some further conditions must be imposed on the extension \(E\).
In fact, using newly introduced terms it is assumed here that in addition \(E\) satisfies the properties of being consistent, profaithful and of finite vcd. Then the results can be sketched as follows: (7.2.1) The homotopy fixed point spectra \(E^{hH}\) give rise to a \(k\)-local \(H\)-Galois extension of \((E^{hH})_k\) equivariantly equivalent to \(E\) for a closed subgroup \(H\) of \(G\), and to a \(k\)-local \(G/H\)-Galois extension of \(A\) equivariantly equivalent to \((E^{hH})_k\) when \(H\) is normal. (7.3.1) There is an equivalence \(F_A((E^{hH})_k, (E^{hK})_k)\simeq ((E[[G/H]])^{hK})_k\) for closed subgroups \(H\) and \(K\) of \(G\) where \(E[[G/H]]\) denotes a continuous permutation spectrum associated to the profinite \(G\)-set \(G/H\).
It was previously known that in the case of Morava \(E\)-theory the above equivalence formula has already been proved by P. Goerss, H.-W. Henn, M. Mahowald and C. Rezk [Ann. Math. (2) 162, No. 2, 777–822 (2005; Zbl 1108.55009)], but under the more restrictive hypothesis that \(K\) is finite. According to the authors, this seems to have been a major source of motivation for this work. Finally, in the last section (Section 8) the authors show that this Morava \(E\)-theory gives an important example of a profinite Galois extension, and prove that there is an equivalence between the homotopy fixed point spectra \(E_n^{hH}\) of the same type as those above and \(E_n^{dhH}\) of E. S. Devinatz and M. J. Hopkins [Topology 43, No. 1, 1–47 (2004; Zbl 1047.55004)].

MSC:

55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
55P91 Equivariant homotopy theory in algebraic topology
55Q51 \(v_n\)-periodicity
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References:

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