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A DESCENT SPECTRAL SEQUENCE FOR ARBITRARY K(n)-LOCAL SPECTRA WITH EXPLICIT E2-TERM

Published online by Cambridge University Press:  13 August 2013

DANIEL G. DAVIS  and
TYLER LAWSON
DANIEL G. DAVIS
Affiliation:
Department of Mathematics, University of Louisiana at Lafayette, 217 Maxim Doucet Hall, P.O. Box 41010, Lafayette, LA, 70504USA e-mail: dgdavis@louisiana.edu
TYLER LAWSON
Affiliation:
Department of Mathematics, University of Minnesota, 206 Church Street S.E., Minneapolis, MN, 55455USA e-mail: tlawson@math.umn.edu
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Abstract

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Let n be any positive integer and p be any prime. Also, let X be any spectrum and let K(n) denote the nth Morava K-theory spectrum. Then we construct a descent spectral sequence with abutment π∗(LK(n)(X)) and E2-term equal to the continuous cohomology of Gn, the extended Morava stabilizer group, with coefficients in a certain discrete Gn-module that is built from various homotopy fixed point spectra of the Morava module of X. This spectral sequence can be contrasted with the K(n)-local En-Adams spectral sequence for π∗(LK(n)(X)), whose E2-term is not known to always be equal to a continuous cohomology group.

Keywords

Primary 55P4255T1555Q51
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 2 , May 2014 , pp. 369 - 380
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Behrens, M. and Davis, D. G., The homotopy fixed point spectra of profinite Galois extensions, Trans. Amer. Math. Soc. 362 (9) (2010), 4983–5042.Google Scholar
2.Davis, D. G., The E 2-term of the descent spectral sequence for continuous G-spectra, New York J. Math. 12 (2006), 183–191.Google Scholar
3.Davis, D. G., Homotopy fixed points for LK(n)(En ∧ X) using the continuous action, J. Pure Appl. Algebra 206 (3) (2006), 322–354.Google Scholar
4.Davis, D. G., Explicit fibrant replacement for discrete G-spectra, Homology, Homotopy Appl. 10 (3) (2008), 137–150.Google Scholar
5.Davis, D. G., Iterated homotopy fixed points for the Lubin-Tate spectrum, Topol. Appl. 156 (17) (2009), 2881–2898. (With an appendix by D. G. Davis and B. Wieland.)Google Scholar
6.Davis, D. G., Obtaining intermediate rings of a local profinite Galois extension without localization, J. Homotopy Relat. Struct. 5 (1) (2010), 253–268.Google Scholar
7.Davis, D. G., Delta-discrete G-spectra and iterated homotopy fixed points, Algebr. Geom. Topol. 11 (5) (2011), 2775–2814.Google Scholar
8.Davis, D. G. and Torii, T., Every K(n)-local spectrum is the homotopy fixed points of its Morava module, Proc. Amer. Math. Soc. 140 (3) (2012), 1097–1103.Google Scholar
9.Devinatz, E. S., Morava modules and Brown-Comenetz duality, Amer. J. Math. 119 (4) (1997), 741–770.Google Scholar
10.Devinatz, E. S. and Hopkins, M. J., Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups, Topology 43 (1) (2004), 1–47.Google Scholar
11.Goerss, P. G. and Hopkins, M. J., Moduli spaces of commutative ring spectra, in Structured ring spectra, vol. 315 of London Mathematical Society Lecture Note Series (Cambridge University Press, Cambridge, UK 2004), 151–200.Google Scholar
12.Hirschhorn, P. S., Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99 (American Mathematical Society, Providence, RI, 2003).Google Scholar
13.Hopkins, M. J., Mahowald, M. and Sadofsky, H., Constructions of elements in Picard groups, in Topology and representation theory (Evanston, IL, 1992), Contemporary Mathematics, vol. 158 (American Mathematical Society, Providence, RI, 1994), 89–126.CrossRefGoogle Scholar
14.Hovey, M. and Strickland, N. P., Morava K-theories and localisation, Mem. Amer. Math. Soc. 139 (666) (1999), viii + 100.Google Scholar
15.Lawson, T. and Naumann, N., Commutativity conditions for truncated Brown–Peterson spectra of height 2, J. Topol. 5 (1) (2012), 137–168.Google Scholar
16.Mahowald, M. and Sadofsky, H., vn telescopes and the Adams spectral sequence, Duke Math. J. 78 (1) (1995), 101–129.Google Scholar
17.Mitchell, S. A., Hypercohomology spectra and Thomason's descent theorem, in Algebraic K-theory (Toronto, ON, 1996), Fields Institute Communications, vol. 16 (American Mathematical Society, Providence, RI, 1997), 221–277.Google Scholar
18.Serre, J.-P., Galois cohomology, English edition, translated from the French by P. Ion and revised by the author, Springer Monographs in Mathematics (Springer-Verlag, Berlin, Germany, 2002).Google Scholar