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

Abstract

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).