On the Adams spectral sequence for R–modules

Baker, Andrew¹ ; Lazarev, Andrey²

¹Mathematics Department, Glasgow Universit, Glasgow G12 8QW, UK
²Mathematics Department, Bristol University, Bristol BS8 1TW, UK

Algebraic and Geometric Topology, Tome 1 (2001) no. 1, pp. 173-199

Résumé

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

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

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

Détail

DOI : 10.2140/agt.2001.1.173

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

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