Morita Contexts, π-Galois Extensions for Hopf π-Coalgebras

ABSTRACT

Let H be a co-Frobenius Hopf π-coalgebra over a field k, and A a π-H-comodule algebra. We study the notions of a relative π-(H, A)-Hopf module and a smash product A ⊟ H ^⋇. We show that such a smash product is connected to the ring of coinvariants A ⁰ by constructing a Morita context. Furthermore, we study the notion of a π-Galois extension, and use the Morita context to find some equivalent conditions for A/A ⁰ to be a π-Galois extension, generalizing the main results of Galois extensions for co-Frobenius Hopf algebras in Beattie et al. (1997).

Key Words:

• π-Galois extension
• Hopf π-coalgebra
• Morita context
• Relative π-(H, A)-Hopf module

2000 Mathematics Subject Classification:

• 16W30

ACKNOWLEDGMENTS

The author would like to thank the referee for the helpful comments on this article. This work was finished when the author stayed in the K. U. Lenven, supported by the Research Council of the K. U. Leuven. He is most thankful to the staff of the Department of Mathematics of this university for their most generous hospitality. In particular, he is grateful to Prof. A.‘Van Daele, Prof. J. Quaegebeur, and Prof. L. Delvaux for their help. This work was partially supported by the FNS of China (10571026), Jiangsu Province (BK2005207).

Notes

Communicated by M. Cohen.