Separable functors for the category of Doi-Hopf modules. II (Q2762036)

[For part I see the authors, Adv. Math. 145, No. 2, 239-290 (1999; Zbl 0943.18007).]NEWLINENEWLINENEWLINEA Doi-Hopf module is a module with an action by an \(H\)-comodule algebra \(A\) and a coaction by an \(H\)-module coalgebra \(C\), with \(H\) a bialgebra or a Hopf algebra, and such that a certain compatibility relation holds. Such objects form the category of Doi-Hopf modules, and we can associate to every morphism in this category a pair of adjoint functors \((F,G)\). In the first part of the paper it is studied the separability of the (induction) functor \(F\) and its adjoint \(G\) and then two Maschke type theorems are obtained: for the induction functor and for the adjoint of the induction functor, respectively. These results are applied to the functor forgetting the \(A\)-action and also to Hopf-Galois extensions. In particular, a Maschke type theorem for the forgetful functor and a characterization for Hopf-Galois extensions are given.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00024].