Advertisement

Manuscripta Mathematica

, Volume 136, Issue 1–2, pp 33–63 | Cite as

Clifford algebras from quotient ring spectra

  • A. JeanneretEmail author
  • S. Wüthrich
Article

Abstract

We give natural descriptions of the homology and cohomology algebras of regular quotient ring spectra of even E -ring spectra. We show that the homology is a Clifford algebra with respect to a certain bilinear form naturally associated to the quotient ring spectrum F. To identify the cohomology algebra, we first determine the derivations of F and then prove that the cohomology is isomorphic to the exterior algebra on the module of derivations. We treat the example of the Morava K-theories in detail.

Mathematics Subject Classification (2000)

55P42 55P43 55U20 18E30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Angeltveit V.: Topological Hochschild homology and cohomology of A ring spectra. Geom. Topol. 12(2), 987–1032 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Baker, A., Jeanneret, A.: Brave New Bockstein Operations (preprint)Google Scholar
  3. 3.
    Baker A., Lazarev A.: On the Adams spectral sequence for R-modules. Algebr. Geom. Topol. 1, 173–199 (2001) (electronic)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Boardman, J.M.: Stable operations in generalized cohomology. In: Handbook of Algebraic Topology, pp. 585–686. North-Holland publishing Co., Amsterdam (1995)Google Scholar
  5. 5.
    Bourbaki, N.: Éléments de mathématique. Première partie: Les structures fondamentales de l’analyse. Livre II: Algèbre. Chapitre 9: Formes sesquilinéaires et formes quadratiques, Actualités Sci. Ind. no. 1272, Hermann, Paris (1959) (French)Google Scholar
  6. 6.
    Eisenbud, D.: Commutative algebra: with a view toward algebraic geometry. In: Graduate Texts in Mathematics, vol. 150. Springer-Verlag, New York, (1995).Google Scholar
  7. 7.
    Elmendorf, A.D., Kriz, I., Mandell, M.A., May, J.P.: Rings, modules, and algebras in stable homotopy theory. In: Mathematical Surveys and Monographs, vol. 47. American Mathematical Society, Providence, RI, (1997)Google Scholar
  8. 8.
    Jeanneret, A., Wüthrich, S.: Quadratic forms classify products on quotient ring spectra (preprint) (2010)Google Scholar
  9. 9.
    Goerss, P.G., Hopkins, M.J.: Moduli spaces of commutative ring spectra. In: Structured Ring Spectra, London Mathematical Society. Lecture Note Series, vol. 315, pp. 151–200. Cambridge University Press, Cambridge (2004)Google Scholar
  10. 10.
    Lazarev A.: Towers of MU-algebras and the generalized Hopkins–Miller theorem. Proc. London Math. Soc. (3) 87(2), 498–522 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Mac Lane, S.: Categories for the working mathematician. In: Graduate Texts in Mathematics, vol. 5, 2nd edn. Springer-Verlag, New York (1998)Google Scholar
  12. 12.
    Matsumura, H.: Commutative ring theory. In: Cambridge Studies in Advanced Mathematics, vol. 8, 2nd edn. Cambridge University Press, Cambridge (1989)Google Scholar
  13. 13.
    Rognes, J.: Galois extensions of structured ring spectra. Stably dualizable groups. Mem. Am. Math. Soc., 192(898), viii+137 (2008)Google Scholar
  14. 14.
    Strickland N.P.: Products on MU-modules. Trans. Am. Math. Soc. 351(7), 2569–2606 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Wüthrich S.: I-adic towers in topology. Algebr. Geom. Topol. 5, 1589–1635 (2005) (electronic)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Mathematisches InstitutBernSwitzerland
  2. 2.SBBBernSwitzerland

Personalised recommendations