Skip to main content
SpringerLink
Log in

On the Cartan Map for Crossed Products and Hopf–Galois Extensions

  • Published:
Algebras and Representation Theory Aims and scope Submit manuscript

Abstract

We study certain aspects of the algebraic K-theory of Hopf–Galois extensions. We show that the Cartan map from K-theory to G-theory of such an extension is a rational isomorphism, provided the ring of coinvariants is regular, the Hopf algebra is finite dimensional and its Cartan map is injective in degree zero. This covers the case of a crossed product of a regular ring with a finite group and has an application to the study of Iwasawa modules.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ardakov, K., Wadsley, S.J.: K 0 and the dimension filtration for p-torsion Iwasawa modules. In: Proceedings of the London Mathematical Society (2008, in press)

  2. Bass, H.: Algebraic K-Theory. Benjamin, New York (1968)

    MATH  Google Scholar 

  3. Cohen, M., Fischman, D., Montgomery, S.: Hopf Galois extensions, smash products, and Morita equivalence. J. Algebra 133(2), 351–372 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chase, S.U., Sweedler, M.E.: Hopf algebras and Galois theory. Lecture Notes in Mathematics, vol. 97. Springer (1969)

  5. Curtis, C.W., Reiner, I.: Methods of Representation Theory I. Wiley-Interscience, New York (1981)

    MATH  Google Scholar 

  6. Lorenz, M.: Representations of finite dimensional Hopf algebras. J. Algebra 188, 476–505 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  7. Kreimer, H.F., Takeuchi, M.: Hopf algebras and Galois extensions of an algebra. Indiana Univ. Math. J. 30(5), 675–692 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  8. McConnell, J.C., Robson, J.C.: Noncommutative Noetherian rings. Revised Edition, AMS Graduate Studies in Mathematics, vol. 30 (2001)

  9. Montgomery, S.: Hopf algebras and their actions on rings. CBMS Conference proceedings, AMS (1993)

  10. Quillen, D.: Higher algebraic K-theory I. Lecture Notes in Mathematics, vol. 341, pp. 85–147. Springer (1973)

  11. Rumynin, D.: Hopf–Galois extensions with central invariants and their geometric properties. Algebra Represent. Theory 1(4), 353–381 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Serre, J.-P.: Linear representations of finite groups. Graduate Texts in Mathematics, vol. 42. Springer (1977)

  13. Schneider, H.-J.: Representation theory of Hopf Galois extensions. Hopf algebras, Israel J. Math. 72(1–2), 196–231 (1990)

    Article  MATH  Google Scholar 

  14. Waldhausen, F.: Algebraic K-Theory of generalized free products, Part 1. The Ann. Math. 2nd Series 108(2), 135–204 (1978)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK

    Konstantin Ardakov

  2. Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, UK

    Simon J. Wadsley

Authors
  1. Konstantin Ardakov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Simon J. Wadsley
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Konstantin Ardakov.

Additional information

The first author thanks Christ’s College, Cambridge for financial support. The second author was supported by EPSRC research grant EP/C527348/1.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ardakov, K., Wadsley, S.J. On the Cartan Map for Crossed Products and Hopf–Galois Extensions. Algebr Represent Theor 13, 33–41 (2010). https://doi.org/10.1007/s10468-008-9095-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-008-9095-4

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation