Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

HKR characters, $ p$-divisible groups and the generalized Chern character


Author: Takeshi Torii
Journal: Trans. Amer. Math. Soc. 362 (2010), 6159-6181
MSC (2010): Primary 55N22; Secondary 55R40, 14L05, 55N20
DOI: https://doi.org/10.1090/S0002-9947-2010-05194-3
Published electronically: June 11, 2010
MathSciNet review: 2661512
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

In this paper we describe the generalized Chern character of classifying spaces of finite groups in terms of Hopkins-Kuhn-Ravenel generalized group characters. For this purpose we study the $ p$-divisible group and its level structures associated with the $ K(n)$-localization of the $ (n+1)$st Morava $ E$-theory.


References [Enhancements On Off] (What's this?)

  • 1. M. Ando, Isogenies of formal group laws and power operations in the cohomology theories $ E\sb n$. Duke Math. J. 79 (1995), no. 2, 423-485. MR 1344767 (97a:55006)
  • 2. M. Ando and J. Morava, A renormalized Riemann-Roch formula and the Thom isomorphism for the free loop space. Topology, geometry, and algebra: interactions and new directions (Stanford, CA, 1999), 11-36, Contemp. Math., Vol. 279, Amer. Math. Soc., Providence, RI, 2001. MR 1850739 (2002h:57046)
  • 3. M. Ando, J. Morava and H. Sadofsky, Completions of $ {\mathbf Z}/(p)$-Tate cohomology of periodic spectra, Geom. Topol. 2 (1998), 145-174. MR 1638030 (99e:55016)
  • 4. M.F. Atiyah, Characters and cohomology of finite groups, Inst. Hautes Études Sci. Publ. Math. No. 9 (1961), 23-64. MR 0148722 (26:228)
  • 5. M. Demazure, Lectures on $ p$-divisible groups, Lecture Notes in Mathematics, Vol. 302, Springer-Verlag, Berlin-New York, 1972. MR 0344261 (49:9000)
  • 6. V.G. Drinfeld, Elliptic modules. Math. USSR-Sb. 23 (1974), no. 4, 561-592 (1976). MR 0384707 (52:5580)
  • 7. P. Goerss, H.-W. Henn, M. Mahowald and C. Rezk, A resolution of the $ K(2)$-local sphere at the prime 3. Ann. of Math. (2) 162 (2005), no. 2, 777-822. MR 2183282 (2006j:55016)
  • 8. P.G. Goerss and M.J. Hopkins, Moduli spaces of commutative ring spectra. Structured ring spectra, 151-200, London Math. Soc. Lecture Note Ser., 315, Cambridge Univ. Press, Cambridge, 2004. MR 2125040 (2006b:55010)
  • 9. B.H. Gross, Ramification in $ p$-adic Lie extensions. Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. III, pp. 81-102, Astérisque, 65, Soc. Math. France, Paris, 1979. MR 0563473 (81e:12018)
  • 10. J.P.C. Greenlees and N.P. Strickland, Varieties and local cohomology for chromatic group cohomology rings, Topology 38 (1999), no. 5, 1093-1139. MR 1688422 (2001c:55003)
  • 11. M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura varieties, With an appendix by Vladimir G. Berkovich. Annals of Mathematics Studies 151, Princeton University Press, Princeton, NJ, 2001. MR 1876802 (2002m:11050)
  • 12. M.J. Hopkins, Characters and elliptic cohomology, London Math. Soc. Lecture Note Ser., 139, 87-104, Cambridge Univ. Press, Cambridge, 1989. MR 1055870 (91c:55007)
  • 13. M.J. Hopkins, Topological modular forms, the Witten genus, and the theorem of the cube. Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Zürich, 1994), 554-565, Birkhäuser, Basel, 1995. MR 1403956 (97i:11043)
  • 14. M.J. Hopkins, Algebraic topology and modular forms. Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 291-317, Higher Ed. Press, Beijing, 2002. MR 1989190 (2004g:11032)
  • 15. M.J. Hopkins, N.J. Kuhn and D.C. Ravenel, Morava $ K$-theories of classifying spaces and generalized characters for finite groups, Algebraic topology, (San Feliu de Guíxols, 1990), Lecture Notes in Math. 1509, 186-209, Springer, Berlin, 1992. MR 1185970 (93k:55008)
  • 16. M.J. Hopkins, N.J. Kuhn and D.C. Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000), no. 3, 553-594. MR 1758754 (2001k:55015)
  • 17. M.J. Hopkins and B.H. Gross, The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory. Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 1, 76-86. MR 1217353 (94k:55009)
  • 18. M. Hovey and N.P. Strickland, Morava $ K$-theories and localisation. Mem. Amer. Math. Soc. 139 (1999), no. 666. MR 1601906 (99b:55017)
  • 19. K. Iwasawa, Local class field theory. Oxford Science Publications. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1986. MR 0863740 (88b:11080)
  • 20. N.M. Katz, $ p$-adic properties of modular schemes and modular forms. Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pp. 69-190. Lecture Notes in Mathematics, Vol. 350, Springer, Berlin, 1973. MR 0447119 (56:5434)
  • 21. N.M. Katz and B. Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies 108, Princeton University Press, Princeton, NJ, 1985. MR 0772569 (86i:11024)
  • 22. W. Messing, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics, Vol. 264. Springer-Verlag, Berlin-New York, 1972. MR 0347836 (50:337)
  • 23. J. Morava, Noetherian localisations of categories of cobordism comodules. Ann. of Math. (2) 121 (1985), no. 1, 1-39. MR 0782555 (86g:55004)
  • 24. J. Morava, HKR characters and higher twisted sectors, Gromov-Witten theory of spin curves and orbifolds, 143-152, Contemp. Math., Vol. 403, Amer. Math. Soc., Providence, RI, 2006. MR 2234888 (2007d:55007)
  • 25. C. Rezk, Notes on the Hopkins-Miller theorem. Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997), 313-366, Contemp. Math., Vol. 220, Amer. Math. Soc., Providence, RI, 1998. MR 1642902 (2000i:55023)
  • 26. J.P. Serre, Linear representations of finite groups. Translated from the second French edition by Leonard L. Scott. Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. MR 0450380 (56:675)
  • 27. N.P. Strickland, Finite subgroups of formal groups, J. Pure Appl. Algebra 121 (1997), no. 2, 161-208. MR 1473889 (98k:14065)
  • 28. N.P. Strickland and P.R. Turner, Rational Morava $ E$-theory and $ DS\sp 0$, Topology 36 (1997), no.1, 137-151. MR 1410468 (97g:55005)
  • 29. J.T. Tate, $ p$-divisible groups, Proc. Conf. Local Fields (Driebergen, 1966), 158-183, Springer, Berlin, 1967. MR 0231827 (38:155)
  • 30. T. Torii, The geometric fixed point spectrum of $ ({\mathbf Z}/p)\sp k$ Borel cohomology for $ E\sb n$ and its completion. Recent progress in homotopy theory (Baltimore, MD, 2000), 343-369, Contemp. Math., Vol. 293, Amer. Math. Soc., Providence, RI, 2002. MR 1890743 (2003b:55008)
  • 31. T. Torii, On degeneration of formal group laws and application to stable homotopy theory, Amer. J. Math. 125 (2003), 1037-1077. MR 2004428 (2004i:55006)
  • 32. T. Torii, Comparison of Morava $ E$-theories, preprint, arXiv:0901.3396, to appear in Math. Z.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 55N22, 55R40, 14L05, 55N20

Retrieve articles in all journals with MSC (2010): 55N22, 55R40, 14L05, 55N20


Additional Information

Takeshi Torii
Affiliation: Department of Mathematics, Okayama University, Okayama 700–8530, Japan
Email: torii@math.okayama-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-2010-05194-3
Keywords: HKR character, $p$-divisible group
Received by editor(s): February 21, 2009
Received by editor(s) in revised form: August 29, 2009
Published electronically: June 11, 2010
Additional Notes: The author was partially supported by Grant-in-Aid for Young Scientists (B) (No. 18740040) from JSPS
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society