## Algebraic & Geometric Topology

### Cellular approximations and the Eilenberg–Moore spectral sequence

Shoham Shamir

#### Abstract

We set up machinery for recognizing $k$–cellular modules and $k$–cellular approximations, where $k$ is an $R$–module and $R$ is either a ring or a ring-spectrum. Using this machinery we can identify the target of the Eilenberg–Moore cohomology spectral sequence for a fibration in various cases. In this manner we get new proofs for known results concerning the Eilenberg–Moore spectral sequence and generalize another result.

#### Article information

Source
Algebr. Geom. Topol., Volume 9, Number 3 (2009), 1309-1340.

Dates
Revised: 17 April 2009
Accepted: 19 April 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513797031

Digital Object Identifier
doi:10.2140/agt.2009.9.1309

Mathematical Reviews number (MathSciNet)
MR2520402

Zentralblatt MATH identifier
1225.55004

#### Citation

Shamir, Shoham. Cellular approximations and the Eilenberg–Moore spectral sequence. Algebr. Geom. Topol. 9 (2009), no. 3, 1309--1340. doi:10.2140/agt.2009.9.1309. https://projecteuclid.org/euclid.agt/1513797031

#### References

• A K Bousfield, The localization of spectra with respect to homology, Topology 18 (1979) 257–281
• W Chachólski, On the functors $CW\sb A$ and $P\sb A$, Duke Math. J. 84 (1996) 599–631
• E Dror Farjoun, Cellular spaces, null spaces and homotopy localization, Lecture Notes in Math. 1622, Springer, Berlin (1996)
• W G Dwyer, Strong convergence of the Eilenberg–Moore spectral sequence, Topology 13 (1974) 255–265
• W G Dwyer, Exotic convergence of the Eilenberg–Moore spectral sequence, Illinois J. Math. 19 (1975) 607–617
• W G Dwyer, J P C Greenlees, Complete modules and torsion modules, Amer. J. Math. 124 (2002) 199–220
• W G Dwyer, J P C Greenlees, S Iyengar, Duality in algebra and topology, Adv. Math. 200 (2006) 357–402
• W G Dwyer, J P C Greenlees, S Iyengar, Finiteness in derived categories of local rings, Comment. Math. Helv. 81 (2006) 383–432
• W G Dwyer, C W Wilkerson, The fundamental group of a $p$–compact group, to appear in Bull. Lond. Math. Soc.
• A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Math. Surveys and Monogr. 47, Amer. Math. Soc. (1997) With an appendix by M Cole
• P Hilton, G Mislin, J Roitberg, Localization of nilpotent groups and spaces, North-Holland Math. Stud. 15, North-Holland, Amsterdam (1975)
• P S Hirschhorn, Model categories and their localizations, Math. Surveys and Monogr. 99, Amer. Math. Soc. (2003)
• I Kriz, Morava $K$–theory of classifying spaces: some calculations, Topology 36 (1997) 1247–1273
• M Rothenberg, N E Steenrod, The cohomology of classifying spaces of $H$–spaces, Bull. Amer. Math. Soc. 71 (1965) 872–875
• S Schwede, B Shipley, Stable model categories are categories of modules, Topology 42 (2003) 103–153