A Lyndon-Hochschild-Serre spectral sequence for certain homotopy fixed point spectra

Author:
Ethan S. Devinatz

Translated by:

Journal:
Trans. Amer. Math. Soc. **357** (2005), 129-150

MSC (2000):
Primary 55N20; Secondary 55P43, 55T15

DOI:
https://doi.org/10.1090/S0002-9947-04-03394-X

Published electronically:
January 23, 2004

MathSciNet review:
2098089

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let and be closed subgroups of the extended Morava stabilizer group and suppose that is normal in . We construct a strongly convergent spectral sequence

where and are the continuous homotopy fixed point spectra of Devinatz and Hopkins. This spectral sequence turns out to be an Adams spectral sequence in the category of -local -modules.

**[1]**A. Baker and A. Lazerev, On the Adams spectral sequence for -modules,*Algebr. Geom. Topol.***1**(2001), 173-199 (electronic). MR**2002a:55011****[2]**J. M. Boardman, Conditionally convergent spectral sequences, Homotopy Invariant Algebraic Structures (J. P. Meyer, J. Morava, W. S. Wilson, eds.),*Contemp. Math.***239**, Amer. Math. Soc., Providence, RI, 1999, pp. 49-84. MR**2000m:55024****[3]**A. K. Bousfield, The localization of spectra with respect to homology (correction in*Comment. Math. Helv.***58**(1983), 599-600),*Topology***18**(1978), 257-281. MR**80m:55006****[4]**A. K. Bousfield and D. M. Kan, Homotopy Limits, Completions and Localizations, 2nd corrected edition,*Lecture Notes in Mathematics***304**, Springer-Verlag, New York, 1987. MR**51:1825****[5]**E. S. Devinatz, Morava's change of rings theorem, The Cech Centennial (M. Cenkl and H. Miller, eds.),*Contemp. Math.***181**, Amer. Math. Soc., Providence, RI, 1995, pp. 93-118. MR**96m:55007****[6]**E. S. Devinatz and M. J. Hopkins, Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups,*Topology***43**(2004), 1-47.**[7]**J. P. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic Pro- Groups, 2nd edition,*Cambridge Studies in Advanced Mathematics***61**, Cambridge University Press, Cambridge 1999. MR**2000m:20039****[8]**A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, Modules, and Algebras in Stable Homotopy Theory,*Amer. Math. Soc. Surveys and Monographs***47**, Amer. Math. Soc., Providence, RI, 1997. MR**97h:55006****[9]**P. G. Goerss and M. J. Hopkins, Resolutions in model categories, manuscript, 1997.**[10]**-, Simplicial structured ring spectra, manuscript, 1998.**[11]**-, André-Quillen (co-)homology for simplicial algebras over simplicial operads, Une Dégustation Topologique [Topological Morsels]: Homotopy Theory in the Swiss Alps (D. Arlettaz and K. Hess, eds.),*Contemp. Math.***265**, Amer. Math. Soc., Providence, RI, 2000, pp. 41-85. MR**2001m:18012****[12]**-, Realizing commutative ring spectra as ring spectra, manuscript, 2000.**[13]**M. J. Hopkins and J. H. Smith, Nilpotence and stable homotopy theory II,*Ann. of Math.*(2)**148**(1998), 1-49. MR**99h:55009****[14]**D. Husemoller and J. C. Moore, Differential graded homological algebra in several variables,*Symp. Math. IV, Inst. Naz. di Alta Mat.*, Academic Press, New York, 1970, pp. 397-429. MR**46:9143****[15]**H. R. Miller, On relations between Adams spectral sequences with an application to the stable homotopy of a Moore space,*J. of Pure and Applied Alg.***20**(1981), 287-312. MR**82f:55029****[16]**C. Rezk, Notes on the Hopkins-Miller theorem, Homotopy Theory via Algebraic Geometry and Group Representations (M. Mahowald and S. Priddy, eds.),*Contemp. Math.***220**, Amer. Math. Soc., Providence, RI, 1998, pp. 313-366. MR**2000i:55023****[17]**J. P. Serre,*Galois Cohomology*, Springer-Verlag, New York, 1997. MR**98g:12007****[18]**S. S. Shatz, Profinite Groups, Arithmetic, and Geometry,*Ann. of Math. Stud.***67**, Princeton University Press, Princeton, 1972. MR**50:279****[19]**P. Symonds and T. Weigel, Cohomology of -adic analytic groups, New Horizons in Pro- Groups,*Progr. Math.***184**, Birkhäuser Boston, Boston, 2000, pp. 349-410. MR**2001k:22025**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
55N20,
55P43,
55T15

Retrieve articles in all journals with MSC (2000): 55N20, 55P43, 55T15

Additional Information

**Ethan S. Devinatz**

Affiliation:
Department of Mathematics, University of Washington, Seattle, Washington 98195

Email:
devinatz@math.washington.edu

DOI:
https://doi.org/10.1090/S0002-9947-04-03394-X

Keywords:
Adams spectral sequence,
continuous homotopy fixed point spectra,
Morava stabilizer group

Received by editor(s):
September 13, 2002

Received by editor(s) in revised form:
May 21, 2003

Published electronically:
January 23, 2004

Additional Notes:
The author was partially supported by a grant from the National Science Foundation.

Article copyright:
© Copyright 2004
American Mathematical Society