Delta-discrete $G$-spectra and iterated homotopy fixed points

Abstract

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 X^{hH} 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 (X^{hH})^{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 (X^{hH})^{h_δK/H}, and it is just X^{hK}. Additionally, we show that for any delta-discrete G-spectrum Y, (Y^{h_δH})^{h_δK/H} \simeq Y^{h_δK}. Furthermore, if G is an arbitrary profinite group, there is a delta-discrete G-spectrum {X_δ} that is equivalent to X and, though X^{hH} is not even known in general to have a K/H-action, there is always an equivalence ((X_δ)^{h_δH})^{h_δK/H} \simeq (X_δ)^{h_δK}. 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.