Given a profinite group G with finite virtual cohomological dimension, let {X_i} be a tower of discrete G-spectra, each of which is fibrant as a spectrum, so that X=holim_i X_i is a continuous G-spectrum, with homotopy fixed point spectrum X^hG. The E₂-term of the descent spectral sequence for \pi_∗(X^hG) cannot always be expressed as continuous cohomology. However, we show that the E₂-term is always built out of a certain complex of spectra, that, in the context of abelian groups, is used to compute the continuous cochain cohomology of G with coefficients in lim_i M_i, where {M_i} is a tower of discrete G-modules.

The author was partially supported by an NSF grant. Most of this paper was written during a visit to the Institut Mittag-Leffler (Djursholm, Sweden).