A LyndonHochschildSerre spectral sequence for certain homotopy fixed point spectra
Author:
Ethan S. Devinatz
Translated by:
Journal:
Trans. Amer. Math. Soc. 357 (2005), 129150
MSC (2000):
Primary 55N20; Secondary 55P43, 55T15
Published electronically:
January 23, 2004
MathSciNet review:
2098089
Fulltext PDF Free Access
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), 173199 (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. 4984. MR 2000m:55024
 [3]
A. K. Bousfield, The localization of spectra with respect to homology (correction in Comment. Math. Helv. 58 (1983), 599600), Topology 18 (1978), 257281. MR 80m:55006
 [4]
A. K. Bousfield and D. M. Kan, Homotopy Limits, Completions and Localizations, 2nd corrected edition, Lecture Notes in Mathematics 304, SpringerVerlag, 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. 93118. 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), 147.
 [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]
, AndréQuillen (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. 4185. 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), 149. 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. 397429. 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), 287312. MR 82f:55029
 [16]
C. Rezk, Notes on the HopkinsMiller 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. 313366. MR 2000i:55023
 [17]
J. P. Serre, Galois Cohomology, SpringerVerlag, 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. 349410. MR 2001k:22025
 [1]
 A. Baker and A. Lazerev, On the Adams spectral sequence for modules, Algebr. Geom. Topol. 1 (2001), 173199 (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. 4984. MR 2000m:55024
 [3]
 A. K. Bousfield, The localization of spectra with respect to homology (correction in Comment. Math. Helv. 58 (1983), 599600), Topology 18 (1978), 257281. MR 80m:55006
 [4]
 A. K. Bousfield and D. M. Kan, Homotopy Limits, Completions and Localizations, 2nd corrected edition, Lecture Notes in Mathematics 304, SpringerVerlag, 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. 93118. 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), 147.
 [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]
 , AndréQuillen (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. 4185. 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), 149. 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. 397429. 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), 287312. MR 82f:55029
 [16]
 C. Rezk, Notes on the HopkinsMiller 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. 313366. MR 2000i:55023
 [17]
 J. P. Serre, Galois Cohomology, SpringerVerlag, 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. 349410. MR 2001k:22025
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Additional Information
Ethan S. Devinatz
Affiliation:
Department of Mathematics, University of Washington, Seattle, Washington 98195
Email:
devinatz@math.washington.edu
DOI:
http://dx.doi.org/10.1090/S000299470403394X
PII:
S 00029947(04)03394X
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
