The homotopy fixed point spectra of profinite Galois extensions

Abstract

LetEEbe akk-local profiniteGG-Galois extension of anE∞E_\infty-ring spectrumAA(in the sense of Rognes). We show thatEEmay be regarded as producing a discreteGG-spectrum. Also, we prove that ifEEis a profaithfulkk-local profinite extension which satisfies certain extra conditions, then the forward direction of Rognes’s Galois correspondence extends to the profinite setting. We show that the function spectrumFA((EhH)k,(EhK)k)F_A((E^{hH})_k, (E^{hK})_k)is equivalent to the localized homotopy fixed point spectrum((E[[G/H]])hK)k((E[[G/H]])^{hK})_k, whereHHandKKare closed subgroups ofGG. Applications to MoravaEE-theory are given, including showing that the homotopy fixed points defined by Devinatz and Hopkins for closed subgroups of the extended Morava stabilizer group agree with those defined with respect to a continuous action in terms of the derived functor of fixed points.