Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 1, Number 1 (2001), 173-199.
On the Adams spectral sequence for $R$–modules
We discuss the Adams Spectral Sequence for –modules based on commutative localized regular quotient ring spectra over a commutative –algebra 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 –term involves the cohomology of certain ‘brave new Hopf algebroids’ . In working out the details we resurrect Adams’ original approach to Universal Coefficient Spectral Sequences for modules over an ring spectrum.
We show that the Adams Spectral Sequence for based on a commutative localized regular quotient ring spectrum converges to the homotopy of the –nilpotent completion
We also show that when the generating regular sequence of is finite, is equivalent to , the Bousfield localization of with respect to –theory. The spectral sequence here collapses at its –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 –adic tower
whose homotopy limit is . We describe some examples for the motivating case .
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
Permanent link to this document
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
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