## Algebraic & Geometric Topology

### 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 $E2$–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 $SR$ based on a commutative localized regular quotient $R$ ring spectrum $E=R∕I[X−1]$ 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

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

Dates
Revised: 4 April 2001
Accepted: 6 April 2001
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.agt/1513882589

Digital Object Identifier
doi:10.2140/agt.2001.1.173

Mathematical Reviews number (MathSciNet)
MR1823498

Zentralblatt MATH identifier
0970.55006

#### Citation

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. https://projecteuclid.org/euclid.agt/1513882589

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