Separable functors in graded rings

Iglesias, Florencio Castaño; Torrecillas, José Gómez; Năstăsescu, C.

doi:10.1016/s0022-4049(96)00181-8

Cited by 13 publications

(5 citation statements)

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“…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 [10,9], Doi-Koppinen modules [7,6] or, finally, entwined modules [3,5]. 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].…”

Section: Introductionmentioning

confidence: 99%

Separable functors in corings

Gómez-Torrecillas

2002

International Journal of Mathematics and Mathematical Sciences

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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

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Section: Introductionmentioning

confidence: 99%

Separable functors in corings

Gómez-Torrecillas

2002

International Journal of Mathematics and Mathematical Sciences

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“…The terminology stems from the fact that, for a ring homomorphism i : R → S, the restriction of scalars functor is separable if and only if S/R is separable (see [NVO], [RAFAEL]); Separable functors satisfy the following version of Maschke's Theorem: if a morphism f in C is such that F (f ) has a left or right inverse in D, then f has a left or right inverse in C. Separable functors have been studied in several particular cases recently, see e.g. [CGN97], [CGN98], [CIMZ99]. The functor F is called Frobenius if F has a right adjoint G that is at the same time a right adjoint.…”

Section: Categorical Interpretationmentioning

confidence: 99%

Are Biseparable Extensions Frobenius?

Caenepeel¹,

Kadison²

2001

K-Theory

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In Secion 1 we describe what is known of the extent to which a separable extension of unital associative rings is a Frobenius extension. A problem of this kind is suggested by asking if three algebraic axioms for finite Jones index subfactors are dependent. In Section 2 the problem in the title is formulated in terms of separable bimodules. In Section 3 we specialize the problem to ring extensions, noting that a biseparable extension is a two-sided finitely generated projective, split, separable extension. Some reductions of the problem are discussed and solutions in special cases are provided. In Section 4 various examples are provided of projective separable extensions that are neither finitely generated nor Frobenius and which give obstructions to weakening the hypotheses of the question in the title. We show in Section 5 that characterizations of the separable extensions among the Frobenius extensions in [HS, K, K99] are special cases of a result for adjoint functors.

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“…In the first two examples we deal with graded rings. Various functors between categories of graded modules have been considered in [17]- [19], [15], [9] and [10].…”

Section: Applicationsmentioning

confidence: 99%