The homotopy fixed point spectra of profinite Galois extensions

Abstract

Let E E be a k k -local profinite G G -Galois extension of an E ∞ E_\infty -ring spectrum A A (in the sense of Rognes). We show that E E may be regarded as producing a discrete G G -spectrum. Also, we prove that if E E is a profaithful k k -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 spectrum F A ( ( E h H ) k , ( E h K ) k ) F_A((E^{hH})_k, (E^{hK})_k) is equivalent to the localized homotopy fixed point spectrum ( ( E [ [ G / H ] ] ) h K ) k ((E[[G/H]])^{hK})_k , where H H and K K are closed subgroups of G G . Applications to Morava E E -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.