Separable Functors Research Articles

Heavily separable functors of the second kind and applications

We introduce heavily separable functors of the second kind and study them in three different situations. The first of these is with restrictions and extensions of scalars for modules over small preadditive categories. The second is with free functors taking values in Eilenberg-Moore categories associated to a monad or a comonad. Finally, we consider entwined modules and give if and only if conditions for heavy separability of the second kind for functors forgetting either the comodule action or the module action.

Semiseparable functors

In this paper we introduce and investigate the notion of semiseparable functor. One of its first features is that it allows a novel description of separable and naturally full functors in terms of faithful and full functors, respectively. To any semiseparable functor we attach an invariant, given by an idempotent natural transformation, which controls when the functor is separable and yields a characterization of separable functors in terms of (dual) Maschke and conservative functors. We prove that any semiseparable functor admits a canonical factorization as a naturally full functor followed by a separable functor. Here the main tool is the construction of the coidentifier category attached to the associated idempotent natural transformation. Then we move our attention to the semiseparability of functors that have an adjoint. First we obtain a Rafael-type Theorem. Next we characterize the semiseparability of adjoint functors in terms of the (co)separability of the associated (co)monads and the natural fullness of the corresponding (co)comparison functor. We also focus on functors that are part of an adjoint triple. In particular, we describe bireflections as semiseparable (co)reflections, or equivalently, as either Frobenius or naturally full (co)reflections. As an application of our results, we study the semiseparability of functors traditionally attached to ring homomorphisms, coalgebra maps, corings and bimodules, introducing the notions of semicosplit coring and semiseparability relative to a bimodule which extend those of cosplit coring and Sugano's separability relative to a bimodule, respectively.

On the monoidal invariance of the cohomological dimension of Hopf algebras

We discuss the question of whether the global dimension is a monoidal invariant for Hopf algebras, in the sense that if two Hopf algebras have equivalent monoidal categories of comodules, then their global dimensions should be equal. We provide several positive new answers to this question, under various assumptions of smoothness, cosemisimplicity or finite dimension. We also discuss the comparison between the global dimension and the Gerstenhaber-Schack cohomological dimension in the cosemisimple case, obtaining equality in the case the latter is finite. One of our main tools is the new concept of twisted separable functor.

Heavily separable cowreaths

Motivated by an example related to the tensor algebra, a stronger version of the notion of separable functor, called heavily separable (h-separable for short), was introduced and investigated in [1]. Here we study h-coseparable coalgebras in monoidal categories with special concern with the monoidal category TA♯ of right transfer morphisms through an algebra A in a monoidal category. We characterize the h-separability of the forgetful functor from the category of entwined modules associated to a cowreath to the base category using suitable Casimir morphisms. Even if there are non trivial examples of h-coseparable coalgebras over a field [1, Theorem 4.4], here we provide non trivial examples of h-coseparable coalgebras in the monoidal category TA⊗Hop# where H=H4 is the Sweedler 4-dimensional Hopf algebra over a field k and A=Cl(α,β,γ) the Clifford algebra.

Heavily separable functors

Prompted by an example related to the tensor algebra, we introduce and investigate a stronger version of the notion of separable functor that we call heavily separable. We test this notion on several functors traditionally connected to the study of separability.

Frobenius and Separable Functors for the Category of Entwined Modules over Cowreaths, I: General Theory

Entwined modules over cowreaths in a monoidal category are introduced. They can be identified to coalgebras in an appropriate monoidal category. It is investigated when such coalgebras are Frobenius (resp. separable), and when the forgetful functor from entwined modules to representations of the underlying algebra is Frobenius (resp. separable). These properties are equivalent when the unit object of the category is a ⊗-generator.

Frobenius and separable functors for the category of entwined modules over cowreaths, II: Applications

Let H be a quasi-Hopf algebra. We apply results obtained in [8] to give necessary and sufficient conditions for the forgetful functor from Doi–Hopf modules, two-sided Hopf modules or Yetter–Drinfeld modules over H to representations of the underlying algebra to be Frobenius (resp. separable). We show that in some situations these conditions reduce to the unimodularity and/or (co)semisimplicity of the quasi-Hopf algebra H.

Separable functors in group coring

Separable functors in group coring

A note on separable functors and monads with an application to equivariant derived categories

For an adjoint pair \((F, U)\) of functors, we prove that \(U\) is a separable functor if and only if the defined monad is separable and the associated comparison functor is an equivalence up to retracts. In this case, under an idempotent completeness condition, the adjoint pair \((F, U)\) is monadic. This applies to the comparison between the derived category of the category of equivariant objects in an abelian category and the category of equivariant objects in the derived category of the abelian category.

Separable Functors and Formal Smoothness

AbstractThe natural problem we approach in the present paper is to show how the notion of formally smooth (co)algebra inside monoidal categories can substitute that of (co)separable (co)algebra in the study of splitting bialgebra homomorphisms. This is performed investigating the relation between formal smoothness and separability of certain functors and led to other results related to Hopf algebra theory. Between them we prove that the existence ofad-(co)invariant integrals for a Hopf algebraHis equivalent to the separability of some forgetful functors. In the finite dimensional case, this is also equivalent to the separability of the Drinfeld DoubleD(H)overH. Hopf algebras which are formally smooth as (co)algebras are characterized. We prove that if π :E→His a bialgebra surjection with nilpotent kernel such thatHis a Hopf algebra which is formally smooth as aK-algebra, then π has a section which is a rightH-colinear algebra homomorphism. Moreover, ifHis also endowed with anad-invariant integral, then this section can be chosen to beH-bicolinear. We also deal with the dual case.

Naturally Full Functors in Nature

We introduce and discuss the notion of a naturally full functor. The definition is similar to the definition of a separable functor; a naturally full functor is a functorial version of a full functor, while a separable functor is a functorial version of a faithful functor. We study the general properties of naturally full functors. We also discuss when functors between module categories and between categories of comodules over a coring are naturally full.

Maschke functors, semisimple functors and separable functors of the second kind: applications

We introduce separable functors of the second kind (or H-separable functors) and H-Maschke functors. H-separable functors are generalizations of separable functors. Various necessary and sufficient conditions for a functor to be H-separable or H-Maschke, in terms of generalized (co)Casimir elements (integrals, in the case of Hopf algebras), are given. An H-separable functor is always H-Maschke, but the converse holds in particular situations. A special role will be played by Frobenius functors and their relations to H-separability. Our concepts are applied to modules, comodules, entwined modules, quantum Yetter–Drinfeld modules, relative Hopf modules.

Separable functors in corings

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.

Separable Functors for the Category of Doi–Hopf Modules, Applications

We prove a Maschke type theorem for the category of Doi–Hopf modules. In fact, we give necessary and sufficient conditions for the functor that forgets the C-coaction to be separable. This leads to a generalized notion of integrals. Our results can be applied to obtain Maschke type theorems for Yetter–Drinfel'd modules, Long dimodules, and modules graded by G-sets. Existing Maschke type theorems due to Doi and the authors are recovered as special cases.

Separable functors in graded rings

Separable functors were introduced by C. Năstăsescu et al. (J. Algebra 123 (1989) 397–413). We characterize separability of left or right adjoint functors defined on a Grothendieck category having a set of projective generators. This general results are particularized to the canonical functors arising from a graded homomorphism of group-graded rings (restriction of scalars, induction and coinduction functors). We relate the separability of these functors with that of their ungraded versions. In particular, we recover the characterizations given in loc. cited for the ungraded restriction of scalars and induction functors.

Separable functors in coalgebras. Applications

Separable functors in coalgebras. Applications

Inductions and coinductions for Hopf extensions

The induction and coinduction functors for the two types of Hopf extensions (Hopf Galois extensions and dual to Hopf Galois extension) and the symmetry between them are studied; by using the theory of separable functors further links between these two classes are provided.

Separable functors revisited

(1990). Separable functors revisited. Communications in Algebra: Vol. 18, No. 5, pp. 1445-1459.

Separable functors applied to graded rings

In this paper we introduce the notion of separability for functors. As special cases we mention the restriction (of scalars) functor, ‘p*, and the induction functor, cp *, associated to a ring homomorphism ‘p: R + S; separability of ‘p* relates to separability of S over R which is defined in terms of splitting of the canonical map II/: S 0 R S -+ S as an S-bimodule map. Separable functors and their properties have various applications, but here we restrict to applications in the theory of graded rings where the morphism cp is usually taken to be the map R, + R, where e is the neutral element of the group G. However, there are a few very natural functors around that allow interesting applications. After the general properties of separability studied in Section 1 we turn to separability of the restriction functor for graded rings in Section 2. In case R is strongly graded by G then R is separable over R, if and only if G is finite and the trace function is surjective. The forgetful functor U: R-gr + R-mod (U forgets the gradiation) is always a separable functor. The body of this paper deals with the properties of the right adjoint F of U in case G is finite of order n. Let Cal,(M) be the set of n x l-columns over it4 and H: R-mod + R-gr the functor obtained by putting H(M) = Cal.(M) viewing it as a graded R-module in some way, then the functors F and H are equivalent. In Theorem 3.6 we provide a separability criterion for Fusing smash-product constructions appearing in the duality for coac-