Separable functors in corings

We give characterizations of the separability of the induction and ad-induction functors associated to a coring morphism.


Introduction
The notion of separable functor was introduced by C. Nȃstȃsescu, M. Van den Bergh and F. Van Oystaeyen in [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 [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]. It turns out that that all the aforementioned categories of modules are instances of comodule categories over suitable corings [4]. 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 [4]. Thus, we can expect that the characterizations obtained in [3] 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, I have developed a basic theory of functors between categories of comodules, making the arguments independent from the Sweedler's 'sigma-notation'. My 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 [2] and [1] in the framework of coalgebras over commutative rings and are expounded in Sections 1, 2 and 3. In Section 4, I give a notion of homomorphism of corings, which leads to a pair of adjoint functors (the induction functor and its adjoint, called here ad-induction functor). The morphisms of entwining structures [3] 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 term K-linear category and functor, for K a commutative ring (see, e.g. [15, I.0.2]). There are, however, some minor differences: The notation X ∈ A for a category A means that X is an object of A, and the identity morphism attached to any object X will be represented by the object itself. The notation M K stands for the category of all unital K-modules. The fact that G is a right adjoint to some functor F will be denoted by F ⊣ G. For the notion of separable functor, the reader is referred to [12]. Finally, let f, g : X → Y be a pair of morphisms of right modules over a ring R, and let k : K → X be its equalizer (that is, the kernel of f − g). We will say that a left R-module Z preserves the equalizer of (f, g) if k ⊗ R Z : K ⊗ R Z → X ⊗ R Z is the equalizer of the pair (f ⊗ R Z, g ⊗ R Z). Of course, every flat module R Z preserves all equalizers.
1 Bicomodules and the cotensor product functor I first recall from [17] the notion of coring. The concepts of comodule and bicomodule over a coring are generalizations of the corresponding notions for coalgebras. We briefly state some basic properties of the cotensor product of bicomodules. Similar associativity properties were studied in [11] in the framework of coseparable corings and in [2] for coalgebras over commutative rings.
Throughout this paper, A, A ′ , A ′′ , . . . denote associative and unitary algebras over a commutative ring K.

1.1.
Corings. An A-coring is a three-tuple (C, ∆ C , ǫ C ) consisting of an A-bicomodule C and two A-bimodule maps such that the diagrams and commute.
commute. Left C-comodules are similarly defined; we use the notation λ M for their structure maps. A morphism of right C-comodules (M, ρ M ) and (N, ρ N ) is an A-linear map f : M → N such that the following diagram is commutative.
The K-module of all right C-comodule morphisms from M to N is denoted by Hom C (M, N). The K-linear category of all right C-comodules will be denoted by M C . When C = A, with the trivial coring structure, the category M A is just the category of all right A-modules, which is 'traditionally' denoted by M A . Coproducts and cokernels in M C do exist and can be already computed in M A . Therefore, M C has arbitrary inductive limits. If A C is a flat module, then M C is easily proved to be an abelian category.
1.3. Bicomodules. Let ρ M : M → M ⊗ A C be a comodule structure over an A ′ − Abimodule M, and assume that ρ M is A ′ -linear. For any right A ′ -module X, the right This leads to an additive functor − ⊗ A ′ M : M A ′ → M C . When A ′ = A and M = C, the functor − ⊗ A C is left adjoint to the forgetful functor U A : M C → M A (see [11,Proposition 3.1], [4,Lemma 3.1]). Now assume that the In this case, we say that M is a C ′ −C-bicomodule. The C ′ −C-bicomodules are the objects of a K-linear category C ′ M C whose morphisms are those A ′ − A-bimodule maps which are morphisms of C ′ -comodules and of C-comodules. Some particular cases are now of interest. For instance, when C ′ = A ′ , the objects of the category The map is then a C ′ − C ′′ -bicomodule map. Let M C N denote the kernel of (9). If C ′ A ′ and A ′′ C ′′ preserve the equalizer of (ρ M ⊗ A N, M ⊗ A λ N ), then M C N is both a C ′ and a C ′′ -subcomodule of M ⊗ A N and, hence, it is a C ′ − C ′′ -bicomodule.

Proposition. Assume that C ′
A ′ and A ′′ C ′′ preserve the equalizer of (ρ M ⊗ A N, M ⊗ A λ N ) for every M ∈ C ′ M C and N ∈ C M C ′′ . We have an additive bifunctor In particular, the cotensor product bifunctor (10) is defined when C ′ A ′ and A ′′ C ′′ are flat modules or when C is a coseparable A-coring in the sense of [11].
In the special case C ′ = A ′ and C ′′ = A ′′ , we have the bifunctor and, if we further assume A ′ = A ′′ = K, we have the bifunctor 1.5. Compatibility between tensor and cotensor. Let M ∈ C ′ M C and N ∈ C M C ′′ be bicomodules. For any right A ′ -module W , consider the commutative diagram where ψ is given by the universal property of the kernel in the second row. This leads to the following Next, we prove a basic fact concerning with the associativity of cotensor product.
and A ′′ C ′′ preserve the equalizer of (ρ M ⊗ A N, M ⊗ A λ N ), and that A C, A N, A C ⊗ A N and C ′′′ A ′′′ preserve the equalizer of (ρ L ⊗ A ′ M, L⊗ A ′ λ M ). Then we have a canonical isomorphism of C ′′′ − C ′′ -bicomodules The exactness of the first row is then deduced by using that A C ⊗ A N is assumed to preserve the equalizer of (ρ L ⊗ A ′ M, L ⊗ A ′ λ M ). Now, consider the commutative diagram with exact rows: y y Lemma 1.5 gives the isomorphisms ψ 2 and ψ 3 , which induce the isomorphism ψ 1 .

Functors between comodule categories
This section contains technical facts concerning with K-linear functors between categories of comodules over corings. Part of these tools were first developed for coalgebras over commutative rings in [2] and [1]. Roughly speaking, I prove an analogue to Watts theorem, which allow to represent good enough functors as cotensor product functors. I also include a result which states that, under mild conditions, a natural transformation gives a bicomodule morphism at any bicomodule. This will be used in the statement and proof of our separability theorems in Section 4. Let C, D be corings over K-algebras A and B, respectively, and consider a K-linear functor We shall construct a natural transformation Let Υ T,M be the unique isomorphism of D-comodules making commutative the diagram To prove that Υ T,M is natural at T , consider a homomorphism f : T → T of right Tmodules and define g : Moreover, g makes the following diagram commutative: the commutativity of the front rectangle, which gives that Υ T,M is natural, follows from the commutativity of the rest of the diagram. From Mitchell's theorem ([13, Theorem 3.6.5]) we obtain a natural transformation Moreover, if F preserves coproducts (resp. direct limits, resp. inductive limits) then Υ X,M is an isomorphism for X T projective (resp. flat, resp. any right T -module).

Proposition. If the functor F preserves coproducts, then
Proof. By 2.1, Υ −,M is natural for every M ∈ T M C . Thus, we have only to show that Υ X,− is natural for every X ∈ M T . We argue first for we get that Υ A,− is natural. Now, use a free presentation T (Ω) → X to obtain that Υ X,− is natural for a general X T . The rest of the statements are easily derived from this.
is commutative.
Proof. We need just to prove that (15) commutes for X = T . In this case, the diagram can be factored out as Since all trapezia and triangles commute, the back rectangle does, as desired.

Lemma.
Let T, S be K-algebras and assume that F : M C → M D preserves coproducts. Given X ∈ M S , Y ∈ S M T and M ∈ T M C , the following formula holds Proof. The equality will be first proved for X = S. Consider the diagram v v n n n n n n n n n n n n The back rectangle is commutative by definition of Υ S,Y ⊗ T M , while the other two parallelograms are commutative because Υ −,− is natural. Therefore, the right triangle is commutative. The equality is now easily extended for X = S (Ω) and, by using a free presentation S (Ω) → X → 0, for any X.

Natural transformations and bicomodule morphisms. Let
Proof. In order to prove that the coaction λ F (M ) is coassociative, let us consider the diagram: We want to see that the top side is commutative. Since F is assumed to be M-compatible, we have just to prove that the mentioned side is commutative after composing with the isomorphism Υ C ′ ⊗ A ′ C ′ ,M . This is deduced by using Lemma 2.4, in conjunction with the naturality of Υ −,M and the very definition of λ F (M ) . The counitary property is deduced from the commutative diagram To prove the second statement, let us consider the diagram Both triangles commute by definition of λ F 1 (M ) and λ F 2 (M ) , and the upper trapezium is commutative because η is natural. The bottom trapezium commutes by Proposition 2.3. Therefore, the back rectangle is commutative, which just says that η M is a morphism of left C ′ -comodules. This finishes the proof.
2.6. Theorem. Assume that C A is flat. If F : M C → M D is exact and preserves direct limits (e.g. if F is an equivalence of categories), then F is naturally isomorphic to − C F (C).
Proof. Let ρ N → N ⊗ A C be a right C-comodule. We have the following diagram with exact rows where the desired isomorphism is given by the universal property of the kernel.

Co-hom functors
This section contains a quick study of the left adjoint to a cotensor product functor, if it does exist. The presentation is inspired from the one given in [18, 1.8] for coalgebras over a field. Let C, D be corings over K-algebras A and B, respectively. The natural isomorphism which gives the adjunction will be denoted by for Y ∈ M A , X ∈ M D .

3.2.
Let θ : Id → h D (N, −) ⊗ A N be the unit of the adjunction (16). Therefore, the isomorphism Φ X,Y is given by the assignment f → (f ⊗ A N)θ X . In particular, the map 3.3. The following is a basic tool in our investigation.
Proposition. Let N be a C − D-bicomodule. Assume that B D preserves the equalizer of (ρ Y ⊗ A N, Y ⊗ A λ N ) for every right C-comodule Y (e.g., B D is flat or C is a coseparable A-coring in the sense of [11]).
1. If N is quasi-finite as a right D-comodule, then the natural isomorphism (16) restricts to an isomorphism

Separable homomorphisms of corings
I propose a notion of homomorphism of corings which generalizes both the concept of morphism of entwining structures [3] and the coring maps originally considered in [17]. An induction functor is constructed, which is shown to have a right adjoint, called ad-induction functor. The separability of these two functors is characterized in terms which generalize both the previous results on rings [12] and on coalgebras [8]. Our approach rests on the fundamental characterization of the separability of adjoint functors given in [14] and [10]. Consider an A-coring C and a B-coring D, where A and B are K-algebras.

Definition.
A coring homomorphism 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 and such that the following diagrams are commutative: Throughout this section, we consider a coring homomorphism (ϕ, ρ) : C → D. We will define the induction and ad-induction functors connecting the categories of comodules M C and M D .

4.2.
We start with some unavoidable technical work. For every B-bimodule X, let us denote by σ X : B ⊗ A X → X ⊗ B B the B-bimodule morphism given by b ⊗ A x → bx ⊗ B 1. Given B-bimodules X, Y , a straightforward computation shows that the diagram commutes, where ω X,Y : X ⊗ A Y → X ⊗ B Y is the obvious map. We have as well that for every homomorphism of B-bimodules f : X → Y the following diagram is commutative:

Proof. Consider the diagram
The pentagon labeled as (1) commutes since ϕ is a homomorphism of corings. The fouredged diagram (2) is commutative since M is a left comodule. The commutation of the quadrilateral (3) follows easily from the displayed decomposition of D ⊗ A λ M . Therefore, the pentagon with bold arrows commutes. Now consider the diagram We have proved before that (4) is commutative. Moreover, (5) is obviously commutative and (6) and (7) commute by (17) and (18). It follows that the outer curved diagram commutes, which gives the pseudo-coassocitative property for the coaction λ B⊗ A M . To check the counitary property, let ι M : M → A ⊗ A M be the canonical isomorphism. We get from the commutative diagram In order to show that the assignment M → B ⊗ A M is functorial, we will prove that B ⊗ A f is a homomorphism of left D-comodules for every morphism f : M → N in C M. So, we have to show that the outer rectangle in the following diagram is commutative From the definition of λ M , λ N and the fact that f is a morphism of C-comodules, it follows that the upper trapezium commutes. The lower trapezium is commutative by (18). Since the two triangles commute by definition, we get that the outer rectangle is commutative and B ⊗ A f is a morphism in D M.

Proposition 4.3 also implies by symmetry that for every right
We have as well a canonical structure of right C-comodule Proof. Since the comultiplication ∆ C is coassociative, we get that the diagram y y y y y y y y y y y y y is commutative. This implies that the left trapezium of the following diagram is commu-tative.
Since we know that the rest of inner diagrams are also commutative, we obtain that the outer diagram commutes, too. This proves that B ⊗ A C is a D − C-bicomodule.
We will see that B⊗ A C is a quasi-finite right C-comodule, i.e., that − ⊗ A B : G G M C . Now, these functors fit in the commutative diagrams y y y y y y y y y y y y y where U A denotes the forgetful functor. Since − ⊗ A B is left adjoint to − ⊗ B B and U A is left adjoint to − ⊗ A C we get the desired adjunction. The rest of the proposition follows from Proposition 3.3.
Remark. This proposition applies in the case that A C is flat or when D is a coseparable B-coring in the sense of [11].
This map is a homomorphism of Cbicomodules. The unit θ of the adjunction − ⊗ A B ⊣ − ⊗ B B ⊗ A C at a right C-comodule M is given by the composite map We see, in particular, that θ C = ∆ C . By Proposition 3.3, θ M factorizes throughout (M ⊗ A B) D (B ⊗ A C) and, therefore, it gives the unit θ M for the adjunction − ⊗ A B ⊣ − D (B ⊗ A C) at M. So, the multiplication ∆ C finally induces a map which is a homomorphism of C-bicomodules.
We are now ready to state the characterization of the separability of the induction functor.

Theorem. Assume that
Proof. Assume that − ⊗ A B is separable. By [10,Theorem 4.1] or [14,Theorem 1.2], the unit of the adjunction is split-mono, that is, there is a natural transformation ω : To prove the converse, we need to construct a natural transformation ω from the bicomodule map ω C . Given a right C-comodule M, consider the diagram where the vertical are equalizer diagrams (here, we are using the definition of the cotensor product and the fact that M A preserves the equalizer of (ρ C⊗ A B ⊗ B B ⊗ A C, C ⊗ A B ⊗ B λ B⊗ A C )). If we prove that the top rectangle commutes, then there is a unique dotted arrow κ M making the bottom rectangle commute. The identity is obvious; so we need just to prove that which is equivalent to (M ⊗ Proof. Obviously, the forgetful functor coincides with −⊗ A A, so that we get from Theorem 4.7 a characterization of the separability of this functor in terms of the existence of a Cbicomodule map ω C : C ⊗ A C → C such that ω C ∆ C = C. Now, notice that the adjointness isomorphism transfers faithfully the mentioned properties of ω C to the desired properties of γ = ǫ C ω C .
4.9. The counit of the adjunction. Letǫ C be the homomorphism of B-bimodules that makes commute the following diagram This natural transformation χ gives the counit of the adjunction − ⊗ A B ⊣ − ⊗ B (B ⊗ A C). By 3.3, the counit of the adjunction − ⊗ A B ⊣ − D (B ⊗ A C) is given by the restriction of χ to (− D (B ⊗ A C) ⊗ A B. We shall use the same notation for this counit. Now, defineφ as the B-bimodule map completing the diagram where m D : B ⊗ A D ⊗ A B → D is the obvious multiplication map given by the B-bimodule structure of D. We claim thatφ is a D-comodule map. To prove this, we first show that the diagram is commutative. This is done by the following computation, where m : B ⊗ B B ⊗ A B → B denotes the obvious multiplication map: We know that λ B⊗ A C ⊗ A B is a homomorphism of D-bicomodules and, by 2.5, χ D is D-bicolinear, too. This proves thatφ : We are now in a position to prove our separability theorem for the ad-induction functor. Therefore, we need just to check the equality

Theorem. Assume that
This is done by the following computation: Thus, we have proved that the natural transformation given in (20) factorizes throughout a natural transformation This means that we have a commutative diagram Finally, we will show that ν M splits off χ M by means of the following computation: By applying the stated theorem to ǫ C : C → A we obtain: 4.11. Corollary. [4,Theorem 3.3] Let C be an A-coring. Then the functor − ⊗ A C is separable if and only if there exists an invariant e ∈ C (that is, e ∈ C satisfying ae = ea for every a ∈ A) such that ǫ C (e) = 1.
Proof. This follows from Theorem 4.10 taking that the A-bimodule homomorphisms from A to C correspond bijectively with the invariants of C into account.