Separable functors in corings

Abstract

We develop some basic functorial techniques for the study of the categories of comodules over corings. In particular, we prove that the induction functor stemming from every morphism of corings has a left adjoint, called ad-induction functor. This construction generalizes the known adjunctions for the categories of Doi-Hopf modules and entwined modules. The separability of the induction and ad-induction functors are characterized, extending earlier results for coalgebra and ring homomorphisms, as well as for entwining structures.2000 Mathematics Subject Classification: 16W30.1. Introduction. The notion of separable functor was introduced by Nȃstȃsescu et al. [12], where some applications for group-graded rings were done. This notion fits satisfactorily to the classical notion of separable algebra over a commutative ring. Every separable functor between abelian categories encodes a Maschke's theorem, which explains the interest concentrated in this notion within the module-theoretical developments in recent years. Thus, separable functors have been investigated in the framework of coalgebras [8], graded homomorphisms of rings [9,10], Doi-Koppinen modules [6,7], or finally, entwined modules [4,5]. These situations are generalizations of the original study of the separability for the induction and restriction of scalars functors associated to a ring homomorphism done in [12]. It turns out that all the aforementioned categories of modules are instances of comodule categories over suitable corings [3]. In fact, the separability of some fundamental functors relating the category of comodules over a coring and the underlying category of modules has been studied in [3]. Thus, we can expect that the characterizations obtained in [4] of the separability of the induction functor associated to an admissible morphism of entwining structures and its adjoint generalize to the corresponding functors stemming from a homomorphism of corings. This is done in this paper.To state and prove the separability theorems, a basic theory of functors between categories of comodules has been developed in this paper, making the arguments independent from the Sweedler's sigma-notation. The plan here is to use purely categorical methods which could be easily adapted to more general developments of the theory. These methods had been sketched in [1,2] in the framework of coalgebras over commutative rings and are expounded in Sections 2, 3, and 4. In Section 5, a notion of homomorphism of corings is given, which leads to a pair of adjoint functors (the induction functor and its adjoint, called here ad-induction functor). The morphisms of entwining structures [4] are instances of homomorphisms of corings in our setting. Finally, the separability of these functors is characterized.We use essentially the categorical terminology of [16], with the exception of the

Quasi-Frobenius Functors. Applications

“…Following [10], a coring homomorphism from the A-coring C to the B-coring D is a pair (ϕ, ρ), where ρ : A → B is a homomorphism of k-algebras and ϕ : C → D is a homomorphism of A-bimodules such that C and f ′ , g ′ ∈ C * given by the formulas…”

Extended Distributive Law: Cowreath Over Corings

“…This in fact comes from a general setting. Namely, if we have any (A ′ , A)-corings morphism (φ, ϕ) : (C ′ : A ′ ) → (C : A) in the sense of [10]. That is ϕ : A ′ → A is a rings morphism and φ : C ′ → C is by scalar restriction an A ′ -bilinear map satisfying…”

“…In that case (−) φ admits − C D as right adjoint, see [10] or [6] for more details. Proposition 3.5 gives a right adjoint functor to (−) ξ without requiring any assumption on the cowreath product C ⊗ A M .…”

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“…Now, being Σ † = Σ ′ ⊗ R R firm as a right R-module, it has a structure of left Σ † ⊗ R Σ-comodule with coaction (see (21)) Proof. First, observe that f is nothing but the inverse of the isomorphism Σ ′ ⊗ R d Σ .…”