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^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[X^{-1}] converges to the homotopy of the E-nilpotent completion pi_*hat{L}^R_ES_R=R_*[X^{-1}]^hat_{I_*}. We also show that when the generating regular sequence of I_* is finite, hatL^R_ES_R is equivalent to L^R_ES_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 <-- R/I^2 <-- ... <-- R/I^s <-- R/I^{s+1} <-- ... whose homotopy limit is hatL^R_ES_R. We describe some examples for the motivating case R=MU.

Published 7 April 2001 by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-9.abs.html . Erratum added 9 May 2001

Cited by in corpus (23)

• Galois extensions of structured ring spectra
• Topological Hochschild homology and cohomology of A_\infty ring spectra
• Commutative ring objects in pro-categories and generalized Moore spectra
• On the ring of cooperations for 2-primary connective topological modular forms
• Topological Hochschild cohomology and generalized Morita equivalence
• Bordism for the 2-group symmetries of the heterotic and CHL strings
• Power operations in the Kunneth spectral sequence and commutative HFp-algebras
• Completed representation ring spectra of nilpotent groups
• On the Brun spectral sequence for topological Hochschild homology
• An introduction to Bousfield localization
• I-adic towers in topology
• The topological Hochschild homology of algebraic $K$-theory of finite fields
• The topological modular forms of $\mathbb{R}{P}^2$ and $\mathbb{R}{P}^2\wedge \mathbb{C}{P}^2$
• Chromatic characteristic classes in ordinary group cohomology
• An Orientation Map for Height p-1 Real E Theory
• On the homology of regular quotients
• On the $\mathrm{EO}$-orientability of vector bundles
• Infinitesimal thickenings of Morava K-theories
• The 3-local tmf homology of BSigma_3
• $BP$: Close encounters of the $E_{\infty }$ kind
• Homotopy theory of modules over a commutative $S$-algebra: some tools and examples
• Clifford algebras from quotient ring spectra
• The String bordism of $B{E}_8$ and $B{E}_8\times B{E}_8$ through dimension 14