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 non-abelian derived functors evaluated at H^*(X) as a module over the Steenrod algebra, and it converges to the cohomology of Ω^∞ X. We provide general methods for computing the E_2-term, including the construction of a multiplicative spectral sequence of Serre type

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... 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.