On the Adams spectral sequence for $R$–modules

Abstract

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 $E_2$–term involves the cohomology of certain ‘brave new Hopf algebroids’ $E_{\ast }^{R}E$. 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 $S_R$ based on a commutative localized regular quotient $R$ ring spectrum $E=R∕I\left[X^{-1}\right]$ converges to the homotopy of the $E$–nilpotent completion

$${\pi }_{\ast }{\widehat{L}}_E^R{S}_R=R_{\ast }\left[X^{-1}\right]\widehat{I_{\ast }}.$$

We also show that when the generating regular sequence of $I_{\ast }$ is finite, ${\widehat{L}}_E^R{S}_R$ is equivalent to $L_E^R{S}_R$, the Bousfield localization of $S_R$ with respect to $E$–theory. The spectral sequence here collapses at its $E_2$–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 tower

$$R∕I\leftarrow R∕I^2\leftarrow \cdots \leftarrow R∕I^s\leftarrow R∕I^{s+1}\leftarrow \cdots$$

whose homotopy limit is ${\widehat{L}}_E^R{S}_R$. We describe some examples for the motivating case $R=MU$.