Algebraic & Geometric Topology 1 (2001) 173–199

DOI: 10.2140/agt.2001.1.173

arXiv: math.AT/0105079

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 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 S_R 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, L_E^RS_R is equivalent to L_E^RS_R, the Bousfield localization of S_R 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 tower

whose homotopy limit is L_E^RS_R. We describe some examples for the motivating case R = MU.

Keywords

S–algebra, R–module, R ring spectrum, Adams Spectral Sequence, regular quotient

Mathematical Subject Classification

Primary: 55N20, 55P42, 55P43, 55T15

References

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Publication

Received: 19 February 2001
Revised: 4 April 2001
Accepted: 6 April 2001
Published: 7 April 2001

Authors