Skip to main content
Log in

On Products in Algebraic K-Theory of Crossed Hopf Algebras

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

Some multiplication is constructed for algebraic K-functors of the crossed products of a commutative algebra and a Hopf cocommutative algebra; the question on these functors to be Frobenius functors with respect to the constructed multiplication is studied.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. A. Bartels and W. Luck, “Induction theorems and isomorphism conjectures for K- and L-theory,” Forum Math., 19, 1–28 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  2. S. M. Gersten, “On the functor K2,” J. Algebra, 17, 212–237 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  3. R. G. Heyneman and D. E. Radford, “Reflexivity and coalgebras of finite type,” J. Algebra, 28, 215–246 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  4. K. Kawakubo, “Induction theorems for equivariant K-theory and J-theory,” J. Math. Soc. Jpn., 38, 173–198 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  5. T. Y. Lam, “Induction theorems for Grothendieck groups and Whitehead groups of finite groups,” Ann. Sci. Ec. Norm. Super., 1, 91–148 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  6. J. G. McConnel and M. E. Sweedler, “Simplicity of smash products,” Proc. London Math. Soc., 23, 251–266 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. I. Nemytov, “Kn() functors as Frobenius modules on the functor GR0(π),” Usp. Mat. Nauk, 28, 187–188 (1973).

    MathSciNet  Google Scholar 

  8. B. Pachuashvili, “Cohomologies in monoidal categories,” Bull. Georgian Acad. Sci., 106, 485—488 (1982).

    MathSciNet  MATH  Google Scholar 

  9. D. Quillen, “Higher algebraic K-Theory, I,” Lect. Notes Math., 341, 85–147 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  10. D. E. Radford, “Pointed Hopf algebras are free over Hopf subalgebras,” J. Algebra, 45, 266–273 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Rakviashvili, “Generalization of the Artin theorem for semisimple algebras and inductive theorems for orders and crossed group rings,” Bull. Georgian Acad. Sci., 96, 25—28 (1979).

    MathSciNet  MATH  Google Scholar 

  12. G. Rakviashvili, “Inductive theorems and projective modules over crossed group rings,” Proc. A. Razmadze Math. Inst., 3, 92–107 (1982).

    MathSciNet  MATH  Google Scholar 

  13. G. Rakviashvili, “On the crossed enveloping algebra of Lie p-algebra,” Bull. Georgian Acad. Sci., 96, No. 1, 265–268 (1979).

    MathSciNet  MATH  Google Scholar 

  14. G. Rakviashvili, “On the K-theory of the crossed product of a commutative algebra and a Hopf algebra,” Proc. A. Razmadze Math. Inst., 5, 79–95 (1986).

    MathSciNet  MATH  Google Scholar 

  15. G. Rakviashvili, “On algebraic K-functors of crossed group rings and its applications,” Tbilisi Math. J., 11, 1–15 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  16. R. G. Swan, “Induced representations and projective modules,” Ann. Math., 71, 552–578 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  17. R. G. Swan, “Nonabelian homological algebra and K-theory,” Proc. Symp. Pure Math., 17, 88–123 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Sweedler, “Cohomology of algebras over Hopf algebras,” Trans. Am. Math. Soc., 133, 205–239 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  19. S. M. J. Wilson, “K-Theory for twisted group rings,” Proc. London Math. Soc., 29, 257–271 (1974).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to G. Rakviashvili.

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 177, Algebra, 2020.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rakviashvili, G. On Products in Algebraic K-Theory of Crossed Hopf Algebras. J Math Sci 275, 758–766 (2023). https://doi.org/10.1007/s10958-023-06718-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-023-06718-1

Keywords and phrases

AMS Subject Classification

Navigation