A MASCHKE-TYPE THEOREM FOR THE GRADED SMASH COPRODUCT C⋊kG

  • Kim, Eun-Sup (Department of Mathematics, Kyungpook National University) ;
  • Park, Young-Soo (Department of Mathematics, Kyungpook National University) ;
  • Yoon, Suk-Bong (Department of Mathematics, Kyungpook National University)
  • Published : 1999.05.01

Abstract

M. Cohen and S. Montgomery showed that a Maschke-type theorem for the smash product, which unlike the corresponding result for group actions, does not require any assumptions about the characterstic of the algebra. Our purpose in this paper is a Maschke-type theorem for the graded smash coproduct C⋊kG: let V be a right C⋊kG-comodule and W a C⋊kG-subcomoduleof V which is a C-direct summand of V. Then W is a C⋊kG-direct summand of V. Also this result is equivalent to the following : let V be a graded right C-comodule and W a graded subcomodule of V which has a complement as a C-subcomodule of V. Then W has a graded complement.

References

  1. Hopf algebras E. Abe
  2. Trans. AMS v.282 Group graded rings, smash products and group actions M. Cohen;S. Montgomery
  3. Tsukuba J. Math. v.19 Graded coalgebras and Morita-Takeuchi contexts S. Dascalescu;C. Nastasescu;S. Raianu;F. Van Oystaeyen
  4. Comm. in Algebra v.25 Comodules graded by G-sets and applications S. Dascalescu;C. Nastasescu;B. Torrecillas;F. Van Oystaeyen
  5. J. Math. Soc. Japan v.33 Homological coalgebra Y. Doi
  6. Noncommutative Noetherian rings J. C. McConnell;J. C. Robson
  7. J. Algebra v.47 Semi-direct products of Hopf algebras R. K. Molnar
  8. CBMS Lecture Notes 82, American Math. Soc. Hopf algebras and their actions on rings S. Montgomery
  9. Tsukuba J. Math. v.17 Graded coalgebras C. Nastasescu;B. Torrecillas
  10. Benjamin Hopf algebras M. Sweedler