We discuss the Adams Spectral Sequence for
–modules
based on commutative localized regular quotient ring spectra over a commutative
–algebra
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
–term
involves the cohomology of certain ‘brave new Hopf algebroids’
.
In working out the details we resurrect Adams’ original approach
to Universal Coefficient Spectral Sequences for modules over an
ring
spectrum.
We show that the Adams Spectral Sequence for
based on a commutative localized regular quotient
ring spectrum
converges to the
homotopy of the –nilpotent
completion
We also show that when the generating regular sequence of
is finite,
is equivalent to
, the Bousfield localization
of with respect to
–theory. The spectral sequence
here collapses at its –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
–adic
tower
whose homotopy limit is .
We describe some examples for the motivating case
.
Keywords
$S$–algebra, $R$–module, $R$ ring spectrum, Adams Spectral
Sequence, regular quotient
