Galois theory and Lubin-Tate cochains on classifying spaces

• Autorzy: Andrew Baker , Birgit Richter
• Języki publikacji: EN

Abstrakty

EN

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 $F(BC_{p^r + } ,E_n ) \to F(EC_{p^r + } ,E_n )$ 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.

Słowa kluczowe

EN

Lubin-Tate spectrum; Morava K-theory; Classifying spaces of finite groups; Galois extensions

Kategorie tematyczne

• 55P43: Spectra with additional structure ( E ∞ , A ∞ , ring spectra, etc.)
• 55P60: Localization and completion
• 13B05: Galois theory
• 55N22: Bordism and cobordism theories, formal group laws

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2011
• Tom: 9
• Numer: 5
• Strony: 1074-1087

Daty

wydano

2011-10-01

online

2011-07-26

Twórcy

autor

Andrew Baker

• University of Glasgow

autor

Birgit Richter

• Fachbereich Mathematik der Universität Hamburg

Bibliografia

• [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] Baker A., Richter B., Galois extensions of Lubin-Tate spectra, Homology, Homotopy Appl., 2008, 10(3), 27–43
• [3] Bauer T., Convergence of the Eilenberg-Moore spectral sequence for generalized cohomology theories, preprint available at http://arxiv.org/abs/0803.3798
• [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] 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
• [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
• [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] 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
• [9] Hovey M., Strickland N.P., Morava K-Theories and Localisation, Mem. Amer. Math. Soc, 139(666), American Mathematical Society, Providence, 1999
• [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
• [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] Møller J.M., Frobenius categories for Chevalley groups, available at http://www.math.ku.dk/~moller/talks/frobenius.pdf
• [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] 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
• [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] Rognes J., Galois Extensions of Structured Ring Spectra, Mem. Amer. Math. Soc, 192(898), American Mathematical Society, Providence, 2008
• [17] Rognes J., A Galois extension that is not faithful, preprint available at http://folk.uio.no/rognes/papers/unfaithful.pdf
• [18] Tate J., Nilpotent quotient groups, Topology, 1964, 3(suppl. 1), 109–111 http://dx.doi.org/10.1016/0040-9383(64)90008-4

Identyfikatory

DOI

10.2478/s11533-011-0058-3