Let be a profinite group with finite virtual cohomological dimension and let be a discrete –spectrum. If and are closed subgroups of , with , then, in general, the –spectrum is not known to be a continuous –spectrum, so that it is not known (in general) how to define the iterated homotopy fixed point spectrum . To address this situation, we define homotopy fixed points for delta-discrete –spectra and show that the setting of delta-discrete –spectra gives a good framework within which to work. In particular, we show that by using delta-discrete –spectra, there is always an iterated homotopy fixed point spectrum, denoted , and it is just .
Additionally, we show that for any delta-discrete –spectrum , there is an equivalence . Furthermore, if is an arbitrary profinite group, there is a delta-discrete –spectrum that is equivalent to and, though is not even known in general to have a –action, there is always an equivalence Therefore, delta-discrete –spectra, by letting equal and , give a way of resolving undesired deficiencies in our understanding of homotopy fixed points for discrete –spectra.
"Delta-discrete $G$–spectra and iterated homotopy fixed points." Algebr. Geom. Topol. 11 (5) 2775 - 2814, 2011. https://doi.org/10.2140/agt.2011.11.2775