The E_2-term of the descent spectral sequence for continuous G-spectra

TL;DR

This paper investigates the E_2-term of the descent spectral sequence for continuous G-spectra, revealing it is constructed from a complex related to continuous cochain cohomology, extending understanding beyond traditional continuous cohomology expressions.

Contribution

It introduces a new perspective on the E_2-term of the descent spectral sequence, connecting it to a complex that computes continuous cochain cohomology for towers of G-modules.

Findings

E_2-term is built from a complex of spectra.

The complex computes continuous cochain cohomology.

Provides a new framework for understanding the descent spectral sequence.

Abstract

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 E_2-term of the descent spectral sequence for \pi_*(X^{hG}) cannot always be expressed as continuous cohomology. However, we show that the E_2-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.