Function spectra and continuous G-spectra

TL;DR

This paper develops new methods for defining and analyzing homotopy fixed point spectra for profinite groups, showing their equivalences and applications to duality, spectral sequences, and Galois extensions in stable homotopy theory.

Contribution

It introduces a framework for homotopy fixed points of G-spectra, proving equivalences under finite vcd, and applies these to duality, spectral sequences, and Galois extensions.

Findings

Homotopy fixed point spectrum F(Z, holim_alpha X_alpha)^{hG} can be defined and related to fixed points.

K(n)-local Spanier-Whitehead dual is a homotopy fixed point spectrum.

Descent spectral sequence is equivalent to Adams-type spectral sequence.

Abstract

Let G be a profinite group, {X_alpha}_alpha a cofiltered diagram of discrete G-spectra, and Z a spectrum with trivial G-action. We show how to define the homotopy fixed point spectrum F(Z, holim_alpha X_alpha)^{hG} and that when G has finite virtual cohomological dimension (vcd), it is equivalent to F(Z, holim_alpha (X_alpha)^{hG}). With these tools, we show that the K(n)-local Spanier-Whitehead dual is always a homotopy fixed point spectrum, a well-known Adams-type spectral sequence is actually a descent spectral sequence, and, for a sufficiently nice k-local profinite G-Galois extension E, with K a closed normal subgroup of G, the equivalence (E^{h_kK})^{h_kG/K} \simeq E^{h_kG} (due to Behrens and the author), where (-)^{h_k(-)} denotes k-local homotopy fixed points, can be upgraded to an equivalence that just uses ordinary (non-local) homotopy fixed points, when G/K has finite vcd.