Delta-discrete G-spectra and iterated homotopy fixed points

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Let G be a profinite group with finite virtual cohomological dimension and let X be a discrete G-spectrum. If H and K are closed subgroups of G, with H normal in K, then, in general, the K/H-spectrum XhH is not known to be a continuous K/H-spectrum, so that it is not known (in general) how to define the iterated homotopy fixed point spectrum (XhH)hK/H. To address this situation, we define homotopy fixed points for delta-discrete G-spectra and show that the setting of delta-discrete G-spectra gives a good framework within which to work. In particular, we show that by using delta-discrete K/H-spectra, there is always an iterated homotopy fixed point spectrum, denoted (XhH)ʰδᵏ/ʰ, and it is just XhK. Additionally, we show that for any delta-discrete G-spectrum Y, (Yʰδʰ)ʰδᵏ/ʰ ≃ Yʰδᵏ. Furthermore, if G is an arbitrary profinite group, there is a delta-discrete G-spectrum Xδ that is equivalent to X and, though XhH is not even known in general to have a K/H-action, there is always an equivalence ((Xδ)ʰδʰ)ʰδᵏ/ʰ ≃ (Xδ)ʰδᵏ. Therefore, delta-discrete L-spectra, by letting L equal H, K, and K/H, give a way of resolving undesired deficiencies in our understanding of homotopy fixed points for discrete G-spectra.