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

Published electronically:
January 23, 2004

MathSciNet review:
2098089

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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]**Andrew Baker and Andrej Lazarev,*On the Adams spectral sequence for 𝑅-modules*, Algebr. Geom. Topol.**1**(2001), 173–199. MR**1823498**, 10.2140/agt.2001.1.173**[2]**J. Michael Boardman,*Conditionally convergent spectral sequences*, Homotopy invariant algebraic structures (Baltimore, MD, 1998) Contemp. Math., vol. 239, Amer. Math. Soc., Providence, RI, 1999, pp. 49–84. MR**1718076**, 10.1090/conm/239/03597**[3]**A. K. Bousfield,*The localization of spectra with respect to homology*, Topology**18**(1979), no. 4, 257–281. MR**551009**, 10.1016/0040-9383(79)90018-1**[4]**A. K. Bousfield and D. M. Kan,*Homotopy limits, completions and localizations*, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR**0365573****[5]**Ethan S. Devinatz,*Morava’s change of rings theorem*, The Čech centennial (Boston, MA, 1993) Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 83–118. MR**1320989**, 10.1090/conm/181/02031**[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. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal,*Analytic pro-𝑝 groups*, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 61, Cambridge University Press, Cambridge, 1999. MR**1720368****[8]**A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May,*Rings, modules, and algebras in stable homotopy theory*, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997. With an appendix by M. Cole. MR**1417719****[9]**P. G. Goerss and M. J. Hopkins, Resolutions in model categories, manuscript, 1997.**[10]**-, Simplicial structured ring spectra, manuscript, 1998.**[11]**Paul G. Goerss and Michael J. Hopkins,*André-Quillen (co)-homology for simplicial algebras over simplicial operads*, Une dégustation topologique [Topological morsels]: homotopy theory in the Swiss Alps (Arolla, 1999) Contemp. Math., vol. 265, Amer. Math. Soc., Providence, RI, 2000, pp. 41–85. MR**1803952**, 10.1090/conm/265/04243**[12]**-, Realizing commutative ring spectra as ring spectra, manuscript, 2000.**[13]**Michael J. Hopkins and Jeffrey H. Smith,*Nilpotence and stable homotopy theory. II*, Ann. of Math. (2)**148**(1998), no. 1, 1–49. MR**1652975**, 10.2307/120991**[14]**Dale Husemoller and John C. Moore,*Differential graded homological algebra of several variables*, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) Academic Press, London, 1970, pp. 397–429. MR**0310040****[15]**Haynes R. Miller,*On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space*, J. Pure Appl. Algebra**20**(1981), no. 3, 287–312. MR**604321**, 10.1016/0022-4049(81)90064-5**[16]**Charles Rezk,*Notes on the Hopkins-Miller theorem*, Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997) Contemp. Math., vol. 220, Amer. Math. Soc., Providence, RI, 1998, pp. 313–366. MR**1642902**, 10.1090/conm/220/03107**[17]**Jean-Pierre Serre,*Galois cohomology*, Springer-Verlag, Berlin, 1997. Translated from the French by Patrick Ion and revised by the author. MR**1466966****[18]**Stephen S. Shatz,*Profinite groups, arithmetic, and geometry*, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. Annals of Mathematics Studies, No. 67. MR**0347778****[19]**Peter Symonds and Thomas Weigel,*Cohomology of 𝑝-adic analytic groups*, New horizons in pro-𝑝 groups, Progr. Math., vol. 184, Birkhäuser Boston, Boston, MA, 2000, pp. 349–410. MR**1765127**

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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:
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