The Structure of Corings: Induction Functors, Maschke-Type Theorem, and Frobenius and Galois-Type Properties

Given a ring A and an A-coring C, we study when the forgetful functor from the category of right C-comodules to the category of right A-modules and its right adjoint −⊗AC are separable. We then proceed to study when the induction functor −⊗AC is also the left adjoint of the forgetful functor. This question is closely related to the problem when A→ AHom(C,A) is a Frobenius extension. We introduce the notion of a Galois coring and analyse when the tensor functor over the subring of A fixed under the coaction of C is an equivalence. We also comment on possible dualisation of the notion of a coring.