The \(E_2\)-term of the descent spectral sequence for continuous \(G\)-spectra (Q851549)

Let \(G\) be a profinite group and \(\{ X_n\}\) a tower of discrete \(G\)-spectra. The author proposes a definition of the homotopy fixed spectrum \((\lim X_n)^{hG}\) and writes down a Bousfield-Kan spectral sequence for computing the homotopy groups of this fixed point spectrum. As explained in the last section, it is not possible to describe the \(E_2\)-term as a functor which factors through a category of \(G\)-modules; this is because of the possible non-vanishing of the derived functors of the inverse limit of the towers of homotopy groups \(\{ \pi_t X_n \}\). Instead the author gives a natural isomorphism from the standard \(E_2\)-term built from Godement resolutions to the cohomology of another resolution closer to the standard definition of continuous cohomology. This second resolution remains a functor on towers of spectra.