The homotopy fixed point spectra of profinite Galois extensions

TL;DR

This paper explores the structure of homotopy fixed point spectra in profinite Galois extensions, extending Rognes's Galois correspondence to the profinite setting and applying results to Morava E-theory.

Contribution

It extends Rognes's Galois correspondence to profinite Galois extensions and clarifies the structure of homotopy fixed point spectra in this context.

Findings

E can be regarded as a discrete G-spectrum.

The Galois correspondence extends to profinite extensions under certain conditions.

Homotopy fixed points for Morava E-theory agree with previous definitions.

Abstract

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