We discuss the Adams Spectral Sequence for R–modules based on commutative
localized regular quotient ring spectra over a commutative S–algebra R in the sense
of Elmendorf, Kriz, Mandell, May and Strickland. The formulation of this spectral
sequence is similar to the classical case and the calculation of its E2–term involves
the cohomology of certain ‘brave new Hopf algebroids’ E*RE. In working out the
details we resurrect Adams’ original approach to Universal Coefficient Spectral
Sequences for modules over an R ring spectrum.
We show that the Adams Spectral Sequence for SR based on a commutative
localized regular quotient R ring spectrum E = R ∕ IX−1 converges to the
homotopy of the E–nilpotent completion
We also show that when the generating regular sequence of I* is finite, LERSR is
equivalent to LERSR, the Bousfield localization of SR with respect to E–theory. The
spectral sequence here collapses at its E2–term but it does not have a vanishing line
because of the presence of polynomial generators of positive cohomological degree.
Thus only one of Bousfield’s two standard convergence criteria applies here even
though we have this equivalence. The details involve the construction of an I–adic
whose homotopy limit is LERSR. We describe some examples for the motivating
case R = MU.