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, Volume 29, Issue 1, pp 93–111 | Cite as

A splitting theorem for certain cohomology theories associated to BP* (−)

  • Urs Würgler


Let P(n)*(−) be Brown-Peterson cohomology modulo In and put B(n)*(−)=P(n)*(−)[1/vn]. In this note we construct a canonical multiplicative and idempotent operation Ωn in a suitable completion\(\bar B\) (n)*(−) of B(n)*(−) which has the property that its image is canonically isomorphic to the n-th Morava K-theory K(n)*(−). In particular, the ring theory K(n)*(−) is contained as a direct summand in the theory\(\bar B\) (n)*(−). A similar result is not true before completing. pleting. Because the completion map B (n)*(−) →\(\bar B\) (n)*(−) is injective, the above splitting theorem contains also information about B(n)*(−). The proof of the theorem depends on a result about the behaviour of formal groups of finite height over complete graded Fp.


Number Theory Formal Group Algebraic Geometry Topological Group Direct Summand 
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  1. [1]
    Adams, J. F.: Lectures on generalized cohomology. LN vol.99, Springer 1969Google Scholar
  2. [2]
    Adams, J. F.: Quillen's work on formal group laws and complex cobordism. University of Chicago lecture notes series, 1970Google Scholar
  3. [3]
    Araki,S.: Typical formal groups in complex cobordism and K-theory. Lecture notes in Mathematics, Kyoto University, 1973Google Scholar
  4. [4]
    Hazewinkel, M.: Formal groups and applications. Academic Press, 1978Google Scholar
  5. [5]
    Johnson, D. C., Wilson, S.: BP-operations and Morava's extraordinary K-theories. Math. Z.144, 55–75 (1975)Google Scholar
  6. [6]
    Landweber, P. S.: Annihilator ideals and primitive elements in complex bordism. Illinois J. Math.17, 273–284 (1973)Google Scholar
  7. [7]
    Landweber, P. S.: Homological properties of comodules over MU*(MU) and BP*(BP). Amer. J. Math.98, 591–610 (1976)Google Scholar
  8. [8]
    Landweber, P. S.: BP*(BP) and typical formal groups. Osaka J. Math.12, 357–363 (1975)Google Scholar
  9. [9]
    Miller, H. R., Ravenel, D.C.: Morava stabilizer algebras and localization of Novikov's E2-term. Duke Math. J.44, 433–448 (1977)Google Scholar
  10. [10]
    Morava, J.: Completions of complex cobordism. Springer LN of Math.658, 349–361 (1978)Google Scholar
  11. [11]
    Quillen, D.: On the formal group laws of unoriented and complex cobordism theory. Bull. Amer. Math. Soc.75, 1293–1298 (1969)Google Scholar
  12. [12]
    Ravenel, D. C.: The structure of BP*BP modulo an invariant prime ideal. Topology15, 149–153 (1976)Google Scholar
  13. [13]
    Ravenel, D.C..: The structure of the Morava stabilizer algebras. Inv. Math.37, 109–120 (1976)Google Scholar
  14. [14]
    Ravenel, D. C., Wilson, W. S.: The Hopf ring for complex cobordism. J. pure appl. algebra9, 241–280 (1977)Google Scholar
  15. [15]
    Würgler, U.: On the relation of Morava K-theories to Brown-Peterson homology. Monographie no.26 de l'Enseignement Math., 269–280 (1978)Google Scholar
  16. [16]
    Würgler, U.: On products in a family of cohomology theories associated to the invariant prime ideals of π*. Comment. Math. Helvetici52, 457–481 (1977)Google Scholar
  17. [17]
    Würgler, U.: Cobordism theories of unitary manifolds with singularities and formal group laws. Math. Z.150, 239–260 (1976)Google Scholar
  18. [18]
    Yagita, N.: On the algebraic structure of cobordism operations with singularities. J. London Math. Soc.16, 131–141 (1977)Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Urs Würgler
    • 1
  1. 1.Mathematisches InstitutUniversität BernBernSwitzerland

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