Algebraic and Geometric Topology 1 (2001),
paper no. 9, pages 173-199.
On the Adams Spectral Sequence for R-modules
Andrew Baker, Andrey Lazarev
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.
Keywords.
S-algebra, R-module, R ring spectrum, Adams Spectral Sequence, regular quotient
AMS subject classification.
Primary: 55P42, 55P43, 55T15.
Secondary: 55N20.
DOI: 10.2140/agt.2001.1.173
E-print: arXiv:math.AT/0105079
Submitted: 19 February 2001.
(Revised: 4 April 2001.)
Accepted: 6 April 2001.
Published: 7 April 2001.
Erratum added 9 May 2001.
Notes on file formats
Andrew Baker, Andrey Lazarev
Mathematics Department, Glasgow University, Glasgow G12 8QW, UK.
Mathematics Department, Bristol University, Bristol BS8 1TW, UK.
Email: a.baker@maths.gla.ac.uk and a.lazarev@bris.ac.uk
URL: www.maths.gla.ac.uk/~ajb and
www.maths.bris.ac.uk/~pure/staff/maxal/maxal
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