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

Abstract

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

22 pages, AMS-LaTeX, uses amscd. More examples of corings and references added

Cited by in corpus (5)

• Monads and comonads in module categories
• Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures
• The weak theory of monads
• Contramodules
• Idempotent splittings, colimit completion, and weak aspects of the theory of monads