The structure of corings: Induction functors, Maschke-type theorem, and Frobenius and Galois-type properties

Abstract

Given a ring $A$ and an $A$-coring we study when the forgetful functor from the category of right -comodules to the category of right $A$-modules and its right adjoint -⊗_A are separable. We then proceed to study when the induction functor -⊗_A is also the left adjoint of the forgetful functor. This question is closely related to the problem when A→ _A Hom(,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 is an equivalence. We also comment on possible dualisation of the notion of a coring.