On a spectral sequence for the cohomology of infinite loop spaces

Abstract

We study the mod-$2$ cohomology spectral sequence arising from delooping the Bousfield–Kan cosimplicial space giving the $2$–nilpotent completion of a connective spectrum $X$. Under good conditions its $E_2$–term is computable as certain nonabelian derived functors evaluated at $H^{\ast }\left(X\right)$ as a module over the Steenrod algebra, and it converges to the cohomology of ${\Omega }^{\infty }X$. We provide general methods for computing the $E_2$–term, including the construction of a multiplicative spectral sequence of Serre type for cofibration sequences of simplicial commutative algebras. Some simple examples are also considered; in particular, we show that the spectral sequence collapses at $E_2$ when $X$ is a suspension spectrum.