Entwined comodules and contramodules over coalgebras with several objects: Frobenius, separability and Maschke theorems

Abstract

We study module like objects over categorical quotients of algebras by the action of coalgebras with several objects. These take the form of ``entwined comodules'' and ``entwined contramodules'' over a triple $(\mathcal{C},A,\ddot{I}ˆ)$, where $A$ is an algebra, $\mathcal{C}$ is a coalgebra with several objects and $\ddot{I}ˆ$ is a collection of maps that ``entwines'' $\mathcal{C}$ with $A$. Our objective is to prove Frobenius, separability and Maschke type theorems for functors between categories of entwined comodules and entwined contramodules.