Morita Equivalences Induced by Two-Sided Group Relative Hopf-Module

Abstract:

      Let $H$ be a Hopf $\pi$-coalgebra over a commutative ring $k$ with bijective antipode $S$, and $A$ and $B$ right $\pi$-$H$-comodulelike algebras. We show that the pair of adjoint functors $(F_3=A\o B^{op}\o_{A\square_H B^{op}} -, G_3=(-)^{co H})$ between the categories ${}_{A\square_H B^{op}}{\m{M}}$ and ${}_A{\m{M}}^{\pi-H}_B$ is a pair of inverse equivalences, when $A$ is a faithfully flat $\pi$-$H$-Galois extension. Furthermore, the categories $\underline{\mbox{\bf{M}orita}}^{\pi-H}(A, B)$ and $\underline{\mbox{\bf{M}orita}}^{\square_{\pi-H}}(A^{co H}, B^{co H})$ are equivalent, if $A$ and $B$ are faithfully flat $\pi$-$H$-Galois extensions.