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Let  G  be a profinite group, { X  α } 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 , holim α   X  α )  hG   and that when  G  has finite virtual cohomological dimension (vcd), it is equivalent to  F ( Z , holim α  ( X  α )  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.

Function spectra and continuous G ‐spectra