Galois theory and Lubin-Tate cochains on classifying spaces

Andrew Baker; Birgit Richter

Open Mathematics (2011)

  • Volume: 9, Issue: 5, page 1074-1087
  • ISSN: 2391-5455

Abstract

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We consider brave new cochain extensions F(BG +,R) → F(EG +,R), where R is either a Lubin-Tate spectrum E n or the related 2-periodic Morava K-theory K n, and G is a finite group. When R is an Eilenberg-Mac Lane spectrum, in some good cases such an extension is a G-Galois extension in the sense of John Rognes, but not always faithful. We prove that for E n and K n these extensions are always faithful in the K n local category. However, for a cyclic p-group C p r , the cochain extension F ( B C p r + , E n ) → F ( E C p r + , E n ) is not a Galois extension because it ramifies. As a consequence, it follows that the E n-theory Eilenberg-Moore spectral sequence for G and BGdoes not always converge to its expected target.

How to cite

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Andrew Baker, and Birgit Richter. "Galois theory and Lubin-Tate cochains on classifying spaces." Open Mathematics 9.5 (2011): 1074-1087. <http://eudml.org/doc/269295>.

@article{AndrewBaker2011,
abstract = {We consider brave new cochain extensions F(BG +,R) → F(EG +,R), where R is either a Lubin-Tate spectrum E n or the related 2-periodic Morava K-theory K n, and G is a finite group. When R is an Eilenberg-Mac Lane spectrum, in some good cases such an extension is a G-Galois extension in the sense of John Rognes, but not always faithful. We prove that for E n and K n these extensions are always faithful in the K n local category. However, for a cyclic p-group $C_\{p^r \} $, the cochain extension $F(BC_\{p^r + \} ,E_n ) \rightarrow F(EC_\{p^r + \} ,E_n )$ is not a Galois extension because it ramifies. As a consequence, it follows that the E n-theory Eilenberg-Moore spectral sequence for G and BGdoes not always converge to its expected target.},
author = {Andrew Baker, Birgit Richter},
journal = {Open Mathematics},
keywords = {Lubin-Tate spectrum; Morava K-theory; Classifying spaces of finite groups; Galois extensions; Morava -theory; classifying spaces of finite groups},
language = {eng},
number = {5},
pages = {1074-1087},
title = {Galois theory and Lubin-Tate cochains on classifying spaces},
url = {http://eudml.org/doc/269295},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Andrew Baker
AU - Birgit Richter
TI - Galois theory and Lubin-Tate cochains on classifying spaces
JO - Open Mathematics
PY - 2011
VL - 9
IS - 5
SP - 1074
EP - 1087
AB - We consider brave new cochain extensions F(BG +,R) → F(EG +,R), where R is either a Lubin-Tate spectrum E n or the related 2-periodic Morava K-theory K n, and G is a finite group. When R is an Eilenberg-Mac Lane spectrum, in some good cases such an extension is a G-Galois extension in the sense of John Rognes, but not always faithful. We prove that for E n and K n these extensions are always faithful in the K n local category. However, for a cyclic p-group $C_{p^r } $, the cochain extension $F(BC_{p^r + } ,E_n ) \rightarrow F(EC_{p^r + } ,E_n )$ is not a Galois extension because it ramifies. As a consequence, it follows that the E n-theory Eilenberg-Moore spectral sequence for G and BGdoes not always converge to its expected target.
LA - eng
KW - Lubin-Tate spectrum; Morava K-theory; Classifying spaces of finite groups; Galois extensions; Morava -theory; classifying spaces of finite groups
UR - http://eudml.org/doc/269295
ER -

References

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  1. [1] Alperin J.L., Local representation theory, Cambridge Stud. Adv. Math., 11, Cambridge University Press, Cambridge, 1986 http://dx.doi.org/10.1017/CBO9780511623592 
  2. [2] Baker A., Richter B., Galois extensions of Lubin-Tate spectra, Homology, Homotopy Appl., 2008, 10(3), 27–43 Zbl1175.55007
  3. [3] Bauer T., Convergence of the Eilenberg-Moore spectral sequence for generalized cohomology theories, preprint available at http://arxiv.org/abs/0803.3798 
  4. [4] Becker J.C., Gottlieb D.H., The transfer map and fiber bundles, Topology, 1975, 14(1), 1–12 http://dx.doi.org/10.1016/0040-9383(75)90029-4 
  5. [5] Chase S.U., Harrison D.K., Rosenberg A., Galois theory and Galois cohomology of commutative rings, In: Mem. Amer. Math. Soc., 52, American Mathematical Society, Providence, 1965, 15–33 Zbl0143.05902
  6. [6] Elmendorf A., Kriz I., Mandell M.A., May J.P., Rings, Modules, and Algebras in Stable Homotopy Theory, Math. Surveys Monogr., 47, American Mathematical Society, Providence, 1997 Zbl0894.55001
  7. [7] Hopkins M.J., Kuhn N.J., Ravenel D.C., Morava K-theories of classifying spaces and generalized characters for finite groups, In: Algebraic Topology, San Feliu de Guíxols, 1990, Lecture Notes in Math., 1509, Springer, Berlin, 1992, 186–209 
  8. [8] Hopkins M.J., Kuhn N.J., Ravenel D.C., Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc, 2000, 13(3), 553–594 http://dx.doi.org/10.1090/S0894-0347-00-00332-5 Zbl1007.55004
  9. [9] Hovey M., Strickland N.P., Morava K-Theories and Localisation, Mem. Amer. Math. Soc, 139(666), American Mathematical Society, Providence, 1999 
  10. [10] Kriz I., Lee K.P., Odd-degree elements in the Morava K(n) cohomology of finite groups, Topology Appl., 2000, 103(3), 229–241 http://dx.doi.org/10.1016/S0166-8641(99)00031-0 Zbl0963.55004
  11. [11] Lam T.Y., A First Course in Noncommutative Rings, 2nd ed., Grad. Texts in Math., 131, Springer, New York, 2001 http://dx.doi.org/10.1007/978-1-4419-8616-0 
  12. [12] Møller J.M., Frobenius categories for Chevalley groups, available at http://www.math.ku.dk/~moller/talks/frobenius.pdf 
  13. [13] Ravenel D.C., Morava K-theories and finite groups, In: Symposium on Algebraic Topology in honor of José Adem, Oaxtepec, 1981, Contemp. Math., 12, American Mathematical Society, Providence, 1982, 289–292 
  14. [14] Ravenel D.C., Wilson W.S., The Morava K-theories of Eilenberg-MacLane spaces and the Conner-Floyd conjecture, Amer. J. Math., 1980, 102(4), 691–748 http://dx.doi.org/10.2307/2374093 Zbl0466.55007
  15. [15] Robinson D.J.S., A Course in the Theory of Groups, 2nd ed., Grad. Texts in Math., 80, Springer, New York, 1996 http://dx.doi.org/10.1007/978-1-4419-8594-1 
  16. [16] Rognes J., Galois Extensions of Structured Ring Spectra, Mem. Amer. Math. Soc, 192(898), American Mathematical Society, Providence, 2008 Zbl1166.55001
  17. [17] Rognes J., A Galois extension that is not faithful, preprint available at http://folk.uio.no/rognes/papers/unfaithful.pdf 
  18. [18] Tate J., Nilpotent quotient groups, Topology, 1964, 3(suppl. 1), 109–111 http://dx.doi.org/10.1016/0040-9383(64)90008-4 

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