Frobenius and Maschke type theorems for Doi-Hopf modules and entwined modules revisited: a unified approach (Q2738441)

This paper presents a new approach to study under what conditions induction functors (and their adjoints) between categories of Doi-Hopf modules and, more generally, entwined modules are separable resp. Frobenius.NEWLINENEWLINENEWLINEFirst, the authors give a new approach to the old result based on explicit descriptions of \(V\) and \(W\). Then, let \((A,C,\psi)\) be a right-right entwining structure, \(F\colon M(\psi)^C_A\to M_A\) be a functor forgetting the coaction, \(G=\bullet\otimes C\) its adjoint; \(G'\colon M(\psi)^C_A\to M^C\) be a functor forgetting the \(A\)-action, \(F'\) its left adjoint; the authors show necessary and sufficient conditions for \(F\) and \(G\) to be separable, \((F,G)\) to be a Frobenius pair. With the additional condition that \(A\) is finitely generated and projective as a \(k\)-module, they also give necessary and sufficient conditions for \(F'\) and \(G'\) to be separable, \((F',G')\) to be a Frobenius pair. On the other hand, it is studied when the smash product of two algebras \(A\) and \(B\) is a Frobenius extension of \(A\) and \(B\). Finally, the authors present a direct application of the old results on factorization structures \((B,A,R)\).NEWLINENEWLINEFor the entire collection see [Zbl 0963.00025].