Monads and Comonads on Module Categories

Let A be a ring and MA the category of right A-modules. It is well known in module theory that any A-bimodule B is an A-ring if and only if the functor − ⊗A B : MA → MA is a monad (or triple). Similarly, an A-bimodule C is an A-coring provided the functor − ⊗A C : MA → MA is a comonad (or cotriple). The related categories of modules (or algebras) of −⊗A B and comodules (or coalgebras) of − ⊗A C are well studied in the literature. On the other hand, the right adjoint endofunctors HomA(B,−) and HomA(C,−) are a comonad and a monad, respectively, but the corresponding (co)module categories did not find much attention so far. The category of HomA(B,−)-comodules is isomorphic to the category of B-modules, while the category of HomA(C,−)-modules (called Ccontramodules by Eilenberg and Moore) need not be equivalent to the category of C-comodules. The purpose of this paper is to investigate these categories and their relationships based on some observations of the categorical background. This leads to a deeper understanding and characterisations of algebraic structures such as corings, bialgebras and Hopf algebras. For example, it turns out that the categories of C-comodules and HomA(C,−)-modules are equivalent provided C is a coseparable coring. Furthermore, we describe equivalences between categories of HomA(C,−)-modules and comodules over a coring D in terms of new Galois properties of bicomodules. Finally, we characterise Hopf algebras H over a commutative ring R by properties of the functor HomR(H,−) and the category of mixed HomR(H,−)-bimodules. This generalises in particular the fact that a finite dimensional vector space H is a Hopf algebra if and only if the dual space H∗ is a Hopf algebra.