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.
Categories: math.AT