Algebraic & Geometric Topology

On the Adams spectral sequence for $R$–modules

Andrew Baker and Andrey Lazarev

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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’ ERE. 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=RI[X1] converges to the homotopy of the E–nilpotent completion

π L ̂ E R S R = R [ X 1 ] I ̂ .

We also show that when the generating regular sequence of I is finite, L̂ERSR 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 tower

R I R I 2 R I s R I s + 1

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

Article information

Algebr. Geom. Topol., Volume 1, Number 1 (2001), 173-199.

Received: 19 February 2001
Revised: 4 April 2001
Accepted: 6 April 2001
First available in Project Euclid: 21 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P42: Stable homotopy theory, spectra 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.) 55T15: Adams spectral sequences 55N20: Generalized (extraordinary) homology and cohomology theories

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


Baker, Andrew; Lazarev, Andrey. On the Adams spectral sequence for $R$–modules. Algebr. Geom. Topol. 1 (2001), no. 1, 173--199. doi:10.2140/agt.2001.1.173.

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