Function spectra and continuous G -spectra

Abstract

Let G be a profinite group, {Xα}α be a cofiltered diagram of discrete G-spectra, and Z be a spectrum with trivial G-action. We show how to define the homotopy fixed point spectrum F(Z, holimα Xα)hG and that when G has finite virtual cohomological dimension (vcd), it is equivalent to F(Z, holimα (Xα)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 ◃ G and closed, the equivalence ( E h k K ) h k G / K ≃ E h k G (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.