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Adjoint and Frobenius Pairs of Functorsfor CoringsMohssin Zarouali-Darkaoui aa Faculty of Sciences, Department of Algebra , University ofGranada , Granada, SpainPublished online: 02 Mar 2007.

To cite this article: Mohssin Zarouali-Darkaoui (2007) Adjoint and Frobenius Pairs of Functors forCorings, Communications in Algebra, 35:2, 689-724, DOI: 10.1080/00927870600876110

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Communications in Algebra®, 35: 689–724, 2007Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927870600876110

ADJOINT AND FROBENIUS PAIRS OF FUNCTORSFOR CORINGS

Mohssin Zarouali-DarkaouiFaculty of Sciences, Department of Algebra, University of Granada,Granada, Spain

We investigate adjoint and Frobenius pairs between categories of comodules over rathergeneral corings. We particularize to the case of the adjoint pair of functors associatedto a morphism of corings over different base rings, which leads to a reasonable notionof Frobenius coring extension. When applied to corings stemming from entwiningstructures, we obtain new results in this setting and in graded ring theory.

INTRODUCTION

Corings were introduced by Sweedler (1975) as a generalization of coalgebrasover commutative rings to the case of noncommutative rings, to give a formulationof a predual of the Jacobson–Bourbaki’s theorem for intermediate extensions ofdivision ring extensions. Thus, a coring over an associative ring with unit A isa comonoid in the monoidal category of all A-bimodules. Recently, motivatedby an observation of M. Takeuchi, namely that an entwining structure (resp.an entwined module) can be viewed as a suitable coring (resp. as a comoduleover a suitable coring), Brzezinski (2002) gave some new examples and generalproperties of corings. Among them, a study of Frobenius corings is developed,extending previous results on entwining structures (Brzezinski, 2000) and relativeHopf modules (Caenepeel et al., 1997).

A pair of functors �F�G� is said to be a Frobenius pair (Castaño Iglesias et al.,1999) if G is at the same time a left and right adjoint to F . That is a standardname which we use instead of Morita’s original “strongly adjoint pairs” (Morita,1965). The functors F and G are known as Frobenius functors (Caenepeel et al.,1997). The study of Frobenius functors was motivated by a article of K. Morita,where he proved (Morita, 1965, Theorem 5.1) that given a ring extension i � A →B, the induction functor −⊗ AB � �A → �B is a Frobenius functor if and only ifthe morphism i is Frobenius in the sense of Kasch (1961) (see also Nakayama

Received April 30, 2005; Revised May 2, 2006. Communicated by R. Wisbauer.Address correspondence to Mohssin Zarouali-Darkaoui, Department of Algebra, University of

Granada, E18071 Granada, Spain; E-mail: zaroual@correo.ugr.es

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and Tsuzuku, 1960). The dual result for coalgebras over fields was proved inCastaño Iglesias et al. (1999, Theorem 3.5) and it states that the corestriction functor�−�� � �C → �D associated to a morphism of coalgebras � � C → D is Frobeniusif and only if CD is quasi-finite and injective and there exists an isomorphismof bicomodules hD�CCD�D� � DCC (here, hD�C�−� denotes the “cohom” functor).Since corings generalize both rings and coalgebras over fields, one may expect thatMorita (1965, Theorem 5.1) and Castaño Iglesias et al. (1999, Theorem 3.5) arespecializations of a general statement on homomorphisms of corings. In this article,we find such a result (Theorem 4.1), and we introduce the notion of a (right)Frobenius extension of corings (see Definition 4.2). To prove such a result, westudy adjoint pairs and Frobenius pairs of functors between categories of comodulesover rather general corings. Precedents for categories of modules, and categoriesof comodules over coalgebras over fields are contained in Morita (1965), Takeuchi(1977b), Castaño Iglesias et al. (1999), and Caenepeel et al. (2002). More recently,Frobenius corings (i.e., corings for which the functor forgetting the coaction isFrobenius), have been intensively studied in Brzezinski (2002, 2003), Brzezinski andGómez-Torrecillas (2003), Brzezinski and Wisbauer (2003), and Caenepeel et al.(2002).

We think that our general approach produces results of independent interest,beyond the aforementioned extension to the coring setting of Morita (1965,Theorem 5.1) and Castaño Iglesias et al. (1999, Theorem 3.5), and contributes to theunderstanding of the behavior of the cotensor product functor for corings. In fact,our general results, although sometimes rather technical, have other applicationsand will probably find more. For instance, the author has used them to prove newresults on equivalences of comodule categories over corings (Zarouali-Darkaoui,2005). Moreover, when applied to corings stemming from different algebraictheories of current interest, they boil down to new (more concrete) results. As anillustration, we consider entwined modules over an entwining structure in Section 5,and graded modules over G-sets in Section 6.

The article is organized as follows. After Section 1, devoted to fix somebasic notations, Section 2 deals with adjoint and Frobenius pairs on categoriesof comodules. Some refinements of results from Gómez-Torrecillas (2002) andBrzezinski and Wisbauer (2003, §23) on the representation as cotensor productfunctors of certain functors between comodule categories are needed and are thusincluded in Section 2. From our general discussion on adjoint pairs of cotensorproduct functors we will derive our main general result on Frobenius pairs betweencomodule categories (Theorem 2.10) that extends the known characterizations inthe setting of modules over rings and of comodules over coalgebras. In the firstcase, the key property to derive the result on modules from Theorem 2.10 is theseparability of the trivial corings (see Remark 2.12). In the case of coalgebras,the fundamental additional property is the duality between finite left and rightcomodules. We already consider a much more general situation in Section 3, wherewe introduce the class of so called corings having a duality for which we provecharacterizations of Frobenius pairs that are similar to the coalgebra case.

Section 4 is one of the principal motivations of the article. After the technicaldevelopment of Sections 2 and 3, our main results follow without difficulty. Weprove in particular that the induction functor −⊗A B � �� → �� associated toa homomorphism ��� �� � � → � of corings � and � flat over their respective

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base rings A and B is Frobenius if and only if the �−�-bicomodule �⊗ AB isquasi-finite and injector as a right �-comodule and there exists an isomorphism of�− �-bicomodules h��⊗A B� � B ⊗A � (Theorem 4.1). We show as well how thistheorem unifies previous results for ring homomorphisms (Morita, 1965, Theorem5.1), coalgebra maps (Castaño Iglesias et al., 1999, Theorem 3.5), and Frobeniuscorings (Brzezinski and Wisbauer, 2003, 27.10, 28.8).

In Section 5, we specialize one of the general results on corings to entwiningstructures.

In Section 6, we particularize our results in the previous sections to the coringassociated to a G-graded algebra and a G-set, where G ia a group. Then we obtaina series of new results for graded modules by G-sets.

1. BASIC NOTATIONS

Throughout this article and unless otherwise stated, k denotes a commutativering (with unit), A� A′� A′′� and B denote associative and unitary algebras over k,and �� �′� �′′� and � denote corings over A� A′� A′′� and B, respectively.

We recall from Sweedler (1975) that an A-coring consists of an A-bimodule �with two A-bimodule maps

� � � −→ �⊗ A�� � � � −→ A

such that ��⊗ A�� � � = ��⊗ A�� � � and ��⊗ A�� � � = ��⊗ A�� � � = 1�.A right �-comodule is a pair �M� �M� consisting of a right A-module M andan A-linear map �M � M → M ⊗ A� (the coaction) satisfying �M ⊗ A�� � �M =��M ⊗ A�� � �M , �M ⊗ A�� � �M = 1M . A morphism of right �-comodules �M� �M�and �N� �N � is a right A-linear map f � M → N such that �f ⊗ A�� � �M = �N � f .The k-module of all such morphisms will be denoted by Hom��M�N�. Right �-comodules together with their morphisms form the k-category ��. Coproducts andcokernels (and then inductive limits) in �� exist and they coincide respectively withcoproducts and cokernels in the category of right A-modules �A. If A� is flat, then�� is a Grothendieck category. The converse is not true in general (see El Kaoutitet al., 2004, Example 1.1). When � = A with the trivial A-coring structure, then �A

coincide with the category of right A-modules �A.Now assume that the A′ − A-bimodule M is also a left comodule over an

A′-coring �′ with structure map M � M → �′ ⊗ A′M . Assume moreover that �M

is A′-linear, and M is A-linear. It is clear that �M � M → M ⊗ A� is a morphismof left �′-comodules if and only if M � M → �′ ⊗ A′M is a morphism of right�-comodules. In this case, we say that M is a �′ − �-bicomodule. The �′ − �-bicomodules are the objects of a k-category �′

��, whose morphisms are defined inthe obvious way.

We recall from Guzman (1989) that a coring � is said to be coseparable if thecomultiplication map �� is a section in the category of �-bicomodules. Obviously,the trivial A-coring � = A is coseparable.

Let Z be a left A-module and f � X → Y a morphism in �A. FollowingBrzezinski and Wisbauer (2003, 40.13) we say that f is Z-pure when the functor−⊗ AZ preserves the kernel of f . If f is Z-pure for every Z ∈ A� then we say simplythat f is pure in �A.

Finally, the notation ⊗ will stand for the tensor product over k.

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2. FROBENIUS FUNCTORS BETWEEN CATEGORIES OF COMODULES

Let T be a k-algebra, and M ∈ T�A. Let � � T → EndA�M� the morphism ofk-algebras given by the left T -module structure of the bimodule TMA. Now, supposemoreover that M ∈ ��. Then End��M� is a subalgebra of EndA�M� We have that��T� ⊂ End��M� if and only if �M is T -linear. Hence the left T -module structureof a T − �-bicomodule M can be described as a morphism of k-algebras � �T → End��M�. Given a k-linear functor F � �� → ��, and M ∈ T��, the algebra

morphism T�−→End��M�

F�−�−→End��F�M�� defines a T −�-bicomodule structureon F�M�. We have then two k-linear bifunctors

−⊗T F�−�� F�−⊗T −� � �T × T�� → ��

Let �T�M be the unique isomorphism of �-comodules making the following diagramcommutative

for every M ∈ T�� We have that �T�M is natural in T By the theorem of Mitchell(Popescu, 1973, Theorem 3.6.5), there exists a unique natural transformation

�−�M � −⊗T F�M� → F�−⊗T M�

extending the natural transformation �T�M

Remark 2.1. Mitchell’s (Popescu, 1973, Theorem 3.6.5) holds also if we onlysuppose that the target category �′ is pre-additive and has coproducts, or if thecategory �′ is pre-additive and the functor S preserves coproducts (notations asin Popescu, 1973, Theorem 3.6.5). This fact is used to show that the naturaltransformation � exists for every k-linear functor F � �� → �� even if the category�� is not Abelian. Note also that its corollary (Popescu, 1973, Corollary 3.6.6) alsoholds if we suppose only that the category �′ is pre-additive.

If moreover F preserves coproducts (resp. direct limits, resp. inductive limits),then �X�M is an isomorphism for every projective (resp. every flat, resp. every)right T -module X. By Gómez-Torrecillas (2002, Proposition 3.1), if F preservescoproducts, then � is a natural transformation of bifunctors. Moreover, �X�− is anatural isomorphism for all projective right T -module X. If F preserves direct limits,then �X�− is a natural isomorphism for all flat right T -module X. If F preservesinductive limits, then � is a natural isomorphism.

Let M ∈ �′�� be a bicomodule. We say that a k-linear functor F � �� → ��

is M-compatible if ��′�M and ��′⊗A′�′�M are isomorphisms. For example, F is M-compatible for every bicomodule M either if F preserves inductive limits, or �′

A′ is

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flat and F preserves direct limits, or �′A′ is projective and F preserves coproducts

(since ��′ ⊗ A′�′�A′ is projective).Now suppose that F is M-compatible and preserves coproducts. We define

F�M� �= �−1�′�M � F�M� � F�M� → �′ ⊗ A′F�M�

We have that F�M� is a morphism of A′ −�-bicomodules. By Gómez-Torrecillas(2002, Proposition 3.4), F�M� endowed with �F�M� and F�M�, is a �′ −�-bicomodule.Moreover, if F1� F2 � �

� → �� are M-compatible and preserve coproducts, and� � F1 → F2 is a natural transformation, then the map �M � F1�M� → F2�M� isa morphism of �′ −�-bicomodules. We refer to Gómez-Torrecillas (2002) orBrzezinski and Wisbauer (2003) for more details.

Now we will recall the definition of the cotensor product of two bicomodules.Let M ∈ �′

�� and N ∈ ���′′. The map

M�N = �M ⊗ AN −M ⊗ AN � M ⊗ AN → M ⊗ A�⊗ AN

is a �′ − �′′-bicomodule map. Its kernel in A′�A′′ is the cotensor product of M and N ,

and it is denoted by M��N . If M�N is �′A′ -pure and A′′�′′-pure, and the following

ker� M�N �⊗ A′′�′′ ⊗ A′′�′′� �′ ⊗ A′�′ ⊗ A′ker� M�N � and

�′ ⊗ A′ker� M�N �⊗ A′′�′′ (2)

are injective maps, then M��N is the kernel of M�N in �′��′′

. This is the case if M�N is ��′ ⊗ A′�′�A′ -pure, A′′��′′ ⊗ A′′�′′�-pure, and �′ ⊗ A′ M�N is A′′�′′-pure (e.g., if�′

A′ and A′′�′′ are flat, or if � is a coseparable A-coring).If for every M ∈ �′

�� and N ∈ ���′′� M�N is �′

A′ -pure and A′′�′′-pure, then wehave a k-linear bifunctor

−��− � �′�� × ���′′ −→ �′

��′′ (3)

If in particular �′A′ and A′′�′′ are flat, or if � is a coseparable A-coring, then the

bifunctor (3) is well defined. In the special case when � = A, −��− = −⊗ A−.By a proof similar to that of Al-Takhman (1999, II.1.3), we have, for every

M ∈ ��, that the functor M��− preserves direct limits.The following lemma was used implicitly in the proof of Gómez-Torrecillas

(2002, Proposition 3.4), and it will be useful for us in the proof of the next theorem.

Lemma 2.2. If M ∈ �′��, and F � �� → �� is a M-compatible k-linear functor,

which preserves coproducts, then for all X ∈ �A′�

�X⊗A′�′�M�X ⊗ A′F�M�� = F�X ⊗ A′M��X�M

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Proof. Let us consider the diagram

The commutativity of the top triangle follows from the definition of F�M�, whilethe right triangle commutes by Gómez-Torrecillas (2002, Lemma 3.3) (we takeS = T = A′, and Y = �′), and the left triangle is commutative since �X�− is natural.Therefore, the commutativity of the rectangle holds. �

From now on, every result for comodules over coseparable corings appliesin particular for modules over rings (since the trivial ring A-coring � = A iscoseparable).

Now we propose a generalization of Takeuchi (1977b, Proposition 2.1),Brzezinski and Wisbauer (2003, 23.1(1)), and Gómez-Torrecillas (2002,Theorem 3.5).

Theorem 2.3. Let F � �� → �� be a k-linear functor, such that:

(I) B� is flat and F preserves the kernel of �N ⊗ A�− N ⊗ A�� for every N ∈ ��, or(II) � is a coseparable A-coring and the categories �� and �� are Abelian.

Assume that at least one of the following statements holds:

(1) �A is projective, F preserves coproducts, and �N��, �N⊗A���are isomorphisms for

all N ∈ �� (e.g., if A is semisimple and F preserves coproducts); or(2) �A is flat, F preserves direct limits, and �N��, �N⊗A���

are isomorphisms for allN ∈ �� (e.g., if A is a von Neumann regular ring and F preserves direct limits);or

(3) F preserves inductive limits (e.g., if F has a right adjoint).

Then F is naturally equivalent to −��F���

Proof. In each case, we have F is �-compatible where � ∈ ��� Therefore,by Gómez-Torrecillas (2002, Proposition 3.4), F��� can be viewed as a �−�-bicomodule. From Lemma 2.2, and since �−�� is a natural transformation, wehave, for every N ∈ ��, the commutativity of the following diagram with exact

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rows in ��

The exactness of the bottom sequence is assumed in the case (I). For the case (II),it follows by factorizing the map N�� = �N ⊗ A�− N ⊗ A�� through its image,

and using the facts that the sequence 0 −→ N�N−→N ⊗ A�

N��−→N ⊗ A�⊗ A� is splitexact in �� in the sense of Brzezinski and Wisbauer (2003, 40.5), and that additivefunctors between Abelian categories preserve split exactness. By the universalproperty of the kernel, there exists a unique isomorphism �N � N��F��� → F�N� in�� making commutative the above diagram. It easy to show that � is natural. HenceF � −��F��� �

As an immediate consequence of the last theorem we have the followinggeneralization of the Eilenberg–Watts Theorem (Stenström, 1975, PropositionVI.10.1).

Corollary 2.4. Let F � �� → �� be a k-linear functor.

(1) If B� is flat and A is a semisimple ring (resp. a von Neumann regular ring), thenthe following statements are equivalent:

(a) F is left exact and preserves coproducts (resp. left exact and preserves directlimits);

(b) F � −��M for some bicomodule M ∈ ���.

(2) If A� and B� are flat, then the following statements are equivalent:

(a) F is exact and preserves inductive limits;(b) F � −��M for some bicomodule M ∈ ��� which is coflat in ��.

(3) If � is a coseparable A-coring and the categories �� and �� are Abelian, then thefollowing statements are equivalent:

(a) F preserves inductive limits;(b) F preserves cokernels and F � −��M for some bicomodule M ∈ ���.

(4) If � = A and the category �� is Abelian, then the following statements areequivalent

(a) F has a right adjoint;(b) F preserves inductive limits;(c) F � −⊗ AM for some bicomodule M ∈ A��

A bicomodule N ∈ ��� is said to be quasi-finite as a right �-comodule ifthe functor −⊗ AN � �A → �� has a left adjoint h��N�−� � �� → �A, the cohom

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functor. If Y�N is �⊗ B�-pure for every right �-comodule Y (e.g., B� is flat or �is coseparable) then N� is quasi-finite if and only if −��N � �� → �� has a leftadjoint, which we still to denote by h��N�−� (Gómez-Torrecillas, 2002, Proposition4.2). The particular case of the following statement when the cohom is exactgeneralizes Al-Takhman (2002, Corollary 3.12).

Corollary 2.5. Let N ∈ ��� be a bicomodule, quasi-finite as a right �-comodule,such that A� and B� are flat. If the cohom functor h��N�−� is exact or if � is acoseparable B-coring, then we have

h��N�−� � −��h��N��� � �� → ��

Proof. The functor h��N�−� is k-linear and preserves inductive limits, since it is aleft adjoint to the k-linear functor −��N � �� → �� (by Gómez-Torrecillas, 2002,Proposition 4.2). Hence Theorem 2.3 achieves the proof. �

Now we will use the following generalization of Takeuchi (1977b, Lemma 2.2).

Lemma 2.6. Let �, �′ be bicomodules in ��� and G = −���, G′ = −���′

Suppose moreover that A� is flat and B is a von Neumann regular ring, or A� is flatand G and G′ are cokernel preserving, or � is a coseparable coring. Then

Nat�G�G′� � Hom���������′�

Proof. Let � � G → G′ be a natural transformation. By Gómez-Torrecillas (2002,Lemma 3.2(1)), �� is left B-linear. For the rest of the proof it suffices to replace ⊗by ⊗B in the proof of Caenepeel et al. (2002, Lemma 4.1). �

The following proposition generalizes Morita (1965, Theorem 2.1) frombimodules over rings to bicomodules over corings.

Proposition 2.7. Suppose that A�, �A, B� and �B are flat. Let X ∈ ��� and� ∈ ���. Consider the following properties:

(1) −��X is left adjoint to −���;(2) � is quasi-finite as a right �-comodule and −��X � h����−�;(3) � is quasi-finite as a right �-comodule and X � h������ in ���;(4) There exist bicolinear maps

� � � → X��� and � ���X → �

in ��� and ��� respectively, such that

� ���� � ������ = � and �X�� � � ����X� = X� (4)

(5) ���− is left adjoint to X��−.

Then (1) and (2) are equivalent, and they imply (3). (3) implies (2) if � is acoseparable A-coring. If AX and B� are flat, and X�� = �X ⊗ B�− X ⊗ A�� is pure as

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an A-linear map and ��X = �� ⊗ AX −�⊗ B�X is pure as a B-linear map (e.g., if �Xand �� are coflat (Brzezinski and Wisbauer, 2003, 21.5) or A and B are von Neumannregular rings), or if � and � are coseparable, then (4) implies (1). The converse is trueif �X and �� are coflat, or if A and B are von Neumann regular rings, or if � and �are coseparable. Finally, if � and � are coseparable, or if X and � are coflat on bothsides, or if A�B are von Neumann regular rings, then (1), (4), and (5) are equivalent.

Proof. The equivalence between (1) and (2) follows from Gómez-Torrecillas (2002,Proposition 4.2). That (2) implies (3) is a consequence of Gómez-Torrecillas (2002,Proposition 3.4). If � is coseparable and we assume (3) then, by Corollary 2.5,h����−� � −��h������ � −��X. That (1) implies (4) follows from Lemma 2.6by evaluating the unit and the counit of the adjunction at � and �, respectively.Conversely, if we put F = −��X and G = −���, we have GF � −���X���� andFG � −������X� by Brzezinski and Wisbauer (2003, Proposition 22.6). Definenatural transformations

� � 1��

�−→−���−���−→GF

and

� � FG−�� −→ −���

�−→ 1���

which become the unit and the counit of an adjunction by (4). This gives theequivalence between (1) and (4). The equivalence between (4) and (5) followsby symmetry. �

Definition 2.8. Following Al-Takhman (2002) and Brzezinski and Wisbauer(2003), a bicomodule N ∈ ��� is called an injector as a right �-comodule if thefunctor −⊗ AN � �A → �� preserves injective objects.

Proposition 2.9. Suppose that A� and B� are flat. Let X ∈ ��� and � ∈ ��� Thefollowing statements are equivalent:

(i) −��X is left adjoint to −���, and −��X is left exact (or AX is flat or �X iscoflat);

(ii) � is quasi-finite as a right �-comodule, −��X � h����−�, and −��X is leftexact (or AX is flat or �X is coflat);

(iii) � is quasi-finite and injector as a right �-comodule and X � h������ in ���.

Proof. First, observe that if �X is coflat, then AX is flat Brzezinski and Wisbauer(2003, 21.6), and that if AX is flat, then the functor −��X is left exact. Thus,in view of Proposition 2.7, it suffices if we prove that the version of (ii) with−��X left exact implies (iii), and this last implies the version of (ii) with �X coflat.Assume that −��X � h����−� with −��X left exact. By Gómez-Torrecillas (2002,Proposition 3.4), X � h������ in ���. Being a left adjoint, h����−� is right exactand, henceforth, exact. By Popescu (1973, Theorem 3.2.8), �� is an injector andwe have proved (iii). Conversely, if �� is quasi-finite injector and X � h������as bicomodules, then −��X � −��h������ and, by Popescu (1973, Theorem

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3.2.8), we get that h����−� is an exact functor. By Corollary 2.5, h����−� �−��h������ � −��X, and �X is coflat. �

From the foregoing propositions, it is easy to deduce our characterization ofFrobenius functors between categories of comodules over corings.

Theorem 2.10. Suppose that A� and B� are flat. Let X ∈ ��� and � ∈ ���The following statements are equivalent:

(i) �−��X�−���� is a Frobenius pair;(ii) −��X is a Frobenius functor, and h��X��� � � as bicomodules;(iii) There is a Frobenius pair �F�G� for �� and �� such that F��� � � and G��� �

X as bicomodules;(iv) ��� X� are quasi-finite injectors, and X � h������ and � � h��X��� as

bicomodules;(v) ��� X� are quasi-finite, and −��X � h����−� and −��� � h��X�−�.

Proof. (i) ⇔ (ii) ⇔ (iii) This is obvious, after Theorem 2.3 and Gómez-Torrecillas(2002, Proposition 3.4).

(i) ⇔ (iv) Follows from Proposition 2.9.

(iv) ⇔ (v) If X� and �� are quasi-finite, then AX and B� are flat. Now, applyProposition 2.9. �

From Propositions 2.7 and 2.9 (or Theorem 2.10) we get the followingtheroem.

Theorem 2.11. Let X ∈ ��� and � ∈ ��� Suppose that A�, �A, B� and �B areflat. The following statements are equivalent:

(1) �−��X�−���� is a Frobenius pair, with X� and �� coflat;(2) ����−� X��−� is a Frobenius pair, with �X and �� coflat;(3) X and � are coflat quasi-finite injectors on both sides, and X � h������ in ���

and � � h��X��� in ���.

If moreover � and � are coseparable (resp. A and B are von Neumann regularrings), then the following statements are equivalent:

(1) �−��X�−���� is a Frobenius pair;(2) ����−� X��−� is a Frobenius pair;(3) X and � are quasi-finite (resp. quasi-finite injectors) on both sides, and X �

h������ in ��� and � � h��X��� in ���.

Remark 2.12. In the case of rings i.e., � = A and � = B, Theorem 2.10 and thesecond part of Theorem 2.11 give Castaño Iglesias et al. (1999, Theorem 2.1).To see this, observe that AXB is quasi-finite as a right B-module if and only if−⊗ AX � �A → �B has a left adjoint, that is, if and only if AX is finitely generatedand projective. In such a case, the left adjoint is −⊗ BHomA�X�A� � �B → �A.Of course, −�AX = −⊗ AX.

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The dual characterization in the framework of coalgebras over fields (CastañoIglesias et al., 1999, Theorem 3.3) will be deduced in Section 3.

3. FROBENIUS FUNCTORS BETWEEN CORINGS WITH A DUALITY

We will look to Frobenius functors for corings closer to coalgebras over fields,in the sense that the categories of comodules share a fundamental duality.

An object M of a Grothendieck category C is said to be finitely generated(Stenström, 1975, p. 121) if whenever M = ∑

i Mi is a direct union of subobjectsMi, then M = Mi0

for some index i0. Alternatively, M is finitely generated if thefunctor HomC�M�−� preserves direct unions (Stenström, 1975, Proposition V.3.2).The category C is locally finitely generated if it has a family of finitely generatedgenerators. Recall from Stenström (1975, p. 122) that a finitely generated objectM is finitely presented if every epimorphism L → M with L finitely generated hasfinitely generated kernel. By Stenström (1975, Proposition V.3.4), if C is locallyfinitely generated, then M is finitely presented if and only if HomC�M�−� preservesdirect limits. For the notion of a locally projective module we refer to Zimmermann-Huigsen (1976) and Brzezinski and Wisbauer (2003).

Lemma 3.1. Let � be a coring over a ring A such that A� is flat.

(1) A comodule M ∈ �� is finitely generated if and only if MA is finitely generated.(2) A comodule M ∈ �� is finitely presented if MA is finitely presented. The converse

is true whenever �� is locally finitely generated.(3) If A� is locally projective, then �� is locally finitely generated.

Proof. The forgetful functor U � �� → �A has an exact left adjoint −⊗ A� �

�A → �� which preserves direct limits. Thus, U preserves finitely generated objectsand, in case that �� is locally finitely generated, finitely presented objects. Now, ifM ∈ �� is finitely generated as a right A-module, and M = ∑

i Mi as a direct unionof subcomodules, then U�M� = U�

∑i Mi� =

∑i U�Mi�, since U is exact and preserves

coproducts. Therefore, U�M� = U�Mi0� for some index i0 which implies, U being a

faithfully exact functor, that M = Mi0. Thus, M is a finitely generated comodule.

We have thus proved (1), and the converse to (2). Now, if M ∈ �� is such thatMA is finitely presented, then for every exact sequence 0 → K → L → M → 0 in ��

with L finitely generated, we get an exact sequence 0 → KA → LA → MA → 0 withMA finitely presented. Thus, KA is finitely generated and, by (1), K ∈ �� is finitelygenerated. This proves that M is a finitely presented comodule. Finally, (3) is aconsequence of (1) and Brzezinski and Wisbauer (2003, 19.12(1)). �

The notation Cf stands for the full subcategory of a Grothendieck categoryC whose objects are the finitely generated objects. The category C is locallyNoetherian (Stenström, 1975, p. 123) if it has a family of Noetherian generators orequivalently, if C is locally finitely generated and every finitely generated object of Cis Noetherian. By Stenström (1975, Propositions V.4.2 and V.4.1, Lemma V.3.1(i)),in an arbitrary Grothendieck category, every finitely generated object is Noetherian

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if and only if every finitely generated object is finitely presented. The version forcategories of modules of the following result is well known.

Lemma 3.2. Let C be a locally finitely generated category.

(1) The category Cf is additive.(2) The category Cf has cokernels, and every monomorphism in Cf is a monomorphism

in C.(3) The following statements are equivalent:

(a) The category Cf has kernels;(b) C is locally Noetherian;(c) Cf is Abelian;(d) Cf is an Abelian subcategory of C.

Proof. (1) Straightforward.

(2) That Cf has cokernels is straightforward from Stenström (1975, LemmaV.3.1(i)). Now, let f � M → N be a monomorphism in Cf and � � X → M be amorphism in C such that f� = 0 Suppose that X = ⋃

i∈I Xi, where Xi ∈ Cf , and�i � Xi → X� i ∈ I the canonical injections. Then f��i = 0, and ��i = 0, for every i,and by the definition of the inductive limit, � = 0

(3) (b) ⇒ (a) Straightforward from Stenström (1975, Proposition V.4.1).(d) ⇒ (c) and (c) ⇒ (a) are trivial.

(a) ⇒ (b) Let M ∈ Cf , and K be a subobject of M Let � � L → M be the kernelof the canonical morphism f � M → M/K in Cf Suppose that K = ⋃

i∈I Ki, whereKi ∈ Cf , for every i ∈ I By the universal property of the kernel, there exist a uniquemorphism � � L → K� and a unique morphism �i � Ki → L, for every i ∈ I , makingcommutative the diagrams

By (2), � is a monomorphism in C, then for every Ki ⊂ Kj , the diagram

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commutes. Therefore we have the commutative diagram

Then K � L, and hence K ∈ Cf Finally, by Stenström (1975, Proposition V.4.1), Mis Noetherian in C.

(b) ⇒ (d) Straightforward from Freyd (1964, Theorem 3.41). �

The following generalization of Castaño Iglesias et al. (1999, Proposition3.1) will allow us to give an alternative proof to the equivalence “�1� ⇔ �4�” ofTheorem 3.11 (see Remark 3.12(1)).

Proposition 3.3. Let C and D be two locally Noetherian categories. Then:

(1) If F � C → D is a Frobenius functor, then its restriction Ff � Cf → Df is aFrobenius functor;

(2) If H � Cf → Df is a Frobenius functor, then H can be uniquely extended to aFrobenius functor H � C → D;

(3) The assignment F → Ff defines a bijective correspondence (up to naturalisomorphisms) between Frobenius functors from C to D and Frobenius functorsfrom Cf to Df ;

(4) In particular, if C = �� and D = �� are locally Noetherian such that A� and

B� are flat, then F � �� → �� is a Frobenius functor if and only if it preservesdirect limits and comodules which are finitely generated as right A-modules, and therestriction functor Ff � ��

f → ��f is a Frobenius functor.

Proof. The proofs of Castaño Iglesias et al. (1999, Proposition 3.1 and Remark3.2) remain valid for our situation, but with some minor modifications: To provethat H is well defined, we use Lemma 3.2. In the proof of the statements (1), (2),and (3) we use the Grothendieck AB 5 condition. �

In order to generalize Takeuchi (1977a, Proposition A.2.1) and its proof, weneed the following lemma.

Lemma 3.4. (1) Let C be a locally Noetherian category, let D be an arbitraryGrothendieck category, F � C → D be an arbitrary functor which preserves direct limits,and Ff � Cf → D be its restriction to Cf Then F is exact (faithfully exact, resp. left,right exact) if and only if Ff is exact (faithfully exact, resp. left, right exact).

In particular, an object M in Cf is projective (resp. projective generator) if andonly if it is projective (resp. projective generator) in C.

(2) Let C be a locally Noetherian category. For every object M of C, thefollowing conditions are equivalent:

(a) M is injective (resp. an injective cogenerator);

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(b) the contravariant functor HomC�−�M� � C → Ab is exact (resp. faithfully exact);(c) the contravariant functor HomC�−�M�f � Cf → Ab is exact (resp. faithfully exact).

In particular, an object M in Cf is injective (resp. injective cogenerator) if andonly if it is injective (resp. injective cogenerator) in C.

Proof. (1) The “only if” part is straightforward from the fact that the injectionfunctor Cf → C is faithfully exact.

For the “if” part, suppose that Ff is left exact. Let f � M → N be a morphismin C. Put M = ⋃

i∈I Mi and N = ⋃j∈J Nj� as direct union of directed families of

finitely generated subobjects. For �i� j� ∈ I × J� let Mi�j = Mi ∩ f−1�Nj�, and fi�j �

Mi�j → Nj be the restriction of f to Mi�j We have f = lim−−→I×J fi�j and then F�f� =

lim−−→I×J Ff �fi�j� Hence

ker F�f� = ker lim−−→I×J

Ff �fi�j� = lim−−→I×J

ker Ff �fi�j� = lim−−→I×J

Ff �ker fi�j�

= lim−−→I×J

F�ker fi�j� by Lemma 3.2 = F�lim−−→I×J

ker fi�j� = F�kerf�

Finally, F is left exact. Analogously, it can be proved that Ff is right exactimplies that F is also right exact. Now, suppose that Ff is faithfully exact. We havealready proved that F is exact. It remains to prove that F is faithful. For this, let0 �= M = ⋃

i∈I Mi be an object of C, where Mi is finitely generated for every i ∈ I

We have

F�M� = lim−→I

Ff �Mi� �∑i

Ff �Mi�

(since F is exact). Since M �= 0, there exists some i0 ∈ I such that Mi0�= 0

By Stenström (1975, Proposition IV.6.1), Ff �Mi0� �= 0� hence F�M� �= 0 Also by

Stenström (1975, Proposition IV.6.1), F is faithful.

(2) (a) ⇔ (b) Obvious.

(b) ⇒ (c) Analogous to that of the “only if” part of (1).

(c) ⇒ (a) That M is injective is a consequence of Stenström (1975,Proposition V.2.9, Proposition V.4.1). Now, suppose moreover that HomC�−�M�fis faithful. Let L be a nonzero object of C, and K be a nonzero finitely generatedsubobject of L By Stenström (1975, Proposition IV.6.1), there exists a nonzeromorphism K → M . Since M is injective, there exists a nonzero morphism L → M

making commutative the following diagram

From Stenström (1975, Proposition IV.6.5), it follows that M is a cogenerator. �

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If �A is flat and M ∈ �� is finitely presented as a right A-module, then(Brzezinski and Wisbauer, 2003, 19.19) the dual left A-module M∗ = HomA�M�A�has a left �-comodule structure

M∗ � Hom��M��� ⊆ HomA�M��� � �⊗ AM∗

(f → �f ⊗ A�� � �M .) Now, if AM∗ turns out to be finitely presented and A� is

flat, then ∗�M∗� = HomA�M∗� A� is a right �-comodule and the canonical map �M �

M → ∗�M∗� is a homomorphism in ��. This construction leads to a duality (i.e., acontravariant equivalence)

�−�∗��0 �

��0 �∗�−�

between the full subcategories ��0 and ��0 of �� and �� whose objects are the

comodules which are finitely generated and projective over A on the correspondingside (this holds even without flatness assumptions of �). Call it the basic duality(details may be found in Caenepeel et al., 2004). Of course, in the case that Ais semisimple (e.g., for coalgebras over fields) these categories are that of finitelygenerated comodules, and this basic duality plays a remarkable role in the studyof several notions in the coalgebra setting (e.g., Morita equivalence, Takeuchi,1977b; semiperfect coalgebras, Lin, 1977; Morita duality Gómez-Torrecillas andNastasescu, 1995, 1996, or Frobenius functors, Castaño Iglesias et al., 1999). Itwould be interesting to know, in the coring setting, to what extent the basic dualitycan be extended to the subcategories ��

f and ��f , since, as we will try to show inthis section, this allows to obtain better results. Of course, this is the underlyingidea when the ground ring A is assumed to be quasi-Frobenius (see El Kaoutit andGómez-Torrecillas, in press).

Consider contravariant functors between Grothendieck categories H � A �

A′ � H ′� together with natural transformations � � 1A →H ′ �H and �′ � 1A′ →H �H ′,satisfying the condition H��X� � �′H�X� = 1H�X� and H ′��′X′� � �H ′�X′� = 1H ′�X′� for X ∈ Aand X′ ∈ A′. Following Colby and Fuller (1983), this situation is called a rightadjoint pair.

Proposition 3.5. Let � be an A-coring such that A� and �A are flat. Assume that�� and �� are locally Noetherian categories. If AM

∗ and ∗NA are finitely generatedmodules for every M ∈ ��

f and N ∈ ��f , then the basic duality extends to a rightadjoint pair �−�∗ � ��

f �� �f � ∗�−�.

Proof. If M ∈ ��f then, since �� is locally Noetherian, M� is finitely presented.

By Lemma 3.1, MA is finitely presented and the left �-comodule M∗ makes sense.Now, the assumption AM

∗ finitely generated implies, by Lemma 3.1, that M∗ ∈��f . We have then the functor �−�∗ � ��

f → ��f . The functor ∗�−� is analogouslydefined, and the rest of the proof consists of straightforward verifications. �

Example 3.6. The hypotheses are fulfilled if A� and �A are locally projective andA is left and right Noetherian (in this case the right adjoint pair already appearsin El Kaoutit and Gómez-Torrecillas, in press). But there are situations in which

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no finiteness condition need to be required to A: This is the case, for instance, ofcosemisimple corings (see El Kaoutit et al., 2004, Theorem 3.1). In particular, if anarbitrary ring A contains a division ring B, then, by El Kaoutit et al. (2004, Theorem3.1) the canonical coring A⊗ BA satisfies all hypotheses in Proposition 3.5.

Definition 3.7. Let � be a coring over A satisfying the assumptions of Proposition3.5. We will say that � has a duality if the basic duality extends to a duality

�−�∗ � ��f �

��f � ∗�−�

We have the following examples of a coring which has a duality:

(i) � is a coring over a QF ring A such that A� and �A are flat (and henceprojective);

(ii) � is a cosemisimple coring, where ��f and ��f are equal to ��

0 and ��0,respectively;

(iii) � is a coring over A such that A� and �A are flat and semisimple, �� and�� are locally Noetherian categories, and the dual of every simple right (resp.left) A-module in the decomposition of �A (resp. A�) as a direct sum of simpleA-modules is finitely generated and A-reflexive (in fact, every right (resp. left) �-comodule M becomes a submodule of the semisimple right (resp. left) A-moduleM ⊗ A�, and hence MA (resp. AM) is also semisimple).

Proposition 3.8. Suppose that the coring � has a duality. Let M ∈ �� such that MA

is flat. The following are equivalent:

(1) M is coflat (resp. faithfully coflat);(2) Hom��−�M�f � ��

f → �k is exact (resp. faithfully exact);(3) M is injective (resp. an injective cogenerator).

Proof. Let M ∈ �� and N ∈ ��f We have the following commutative diagram(in �k):

0 −−−−→ M��N −−−−→ M ⊗A N�M⊗AN−M⊗AN−−−−−−−−−→ M ⊗A �⊗A N� �

0 −−−−→ Hom��N∗�M� −−−−→ HomA�N

∗�M�f →�Mf−�f⊗A���N∗−−−−−−−−−−−→ HomA�N

∗�M ⊗ A���

where the vertical maps are the canonical maps. By the universal property ofthe kernel, there is a unique morphism �M�N � M��N → Hom��N

∗�M� makingcommutative the above diagram. By the Cube Lemma (see Mitchell, 1965,Proposition II.1.1), � is a natural transformation of bifunctors. If MA is flat then�M�N is an isomorphism for every N ∈ ��f We have

M��− � Hom��−�M�f � �−�∗ � ��f → �k

Then, by Lemma 3.4, M� is coflat (resp. faithfully coflat) iff M��− � ��f → �k isexact (resp. faithfully exact) iff Hom��−�M�f � ��

f → �k is exact (resp. faithfullyexact) iff M� is injective (resp. an injective cogenerator). �

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The particular case of the second part of the following result for coalgebrasover a commutative ring is given in Al-Takhman (2002).

Corollary 3.9. Let N ∈ ��� be a bicomodule. Suppose that A is a QF ring.

(a) If N is an injector as a right �-comodule, then N is injective in ��.(b) If � has a duality, NB is flat, and N is injective in ��, then N is an injector as a

right �-comodule.

Proof. (a) Since A is a QF ring, then AA is injective. Hence N� � �A⊗ AN�� isinjective.

(b) Let XA be an injective module. Since A is a QF ring, XA is projective.We have then the natural isomorphism

�X ⊗ AN���− � X ⊗ A�N��−� � �� → �k

By Proposition 3.8, N� is coflat, and then X ⊗ AN is coflat. Now, since X ⊗ AN is aflat right B-module, and by Proposition 3.8, X ⊗ AN is injective in ��. �

The last two results allow to improve our general statements in Section 2 forcorings having a duality.

Proposition 3.10. Suppose that � and � have a duality. Consider the followingstatements:

(1) �−��X�−���� is an adjoint pair of functors, with AX and �A flat;(2) � is quasi-finite injective as a right �-comodule, with AX and �A flat and X �

h������ in ���.

We have (1) implies (2), and the converse is true if in particular B is a QF ring.

Proof. �1� ⇒ �2� From Proposition 3.8, � is injective in ��. Since the functor−��X is exact, � � ���� is injective in ��.

�2� ⇒ �1� Assume that B is a QF ring. From Corollary 3.9, � is quasi-finiteinjector as a right �-comodule, and Proposition 2.9 achieves the proof. �

We are now in a position to state and prove our main result of this section.

Theorem 3.11. Suppose that � and � have a duality. Let X ∈ ��� and � ∈ ���The following statements are equivalent:

(1) �−��X�−���� is a Frobenius pair, with XB and �A flat;(2) ����−� X��−� is a Frobenius pair, with AX and B� flat;(3) X and � are quasi-finite injector on both sides, and X � h������ in ��� and

� � h��X��� in ���.In particular, if A and B are QF rings, then the above statements are equivalent to:

(4) X and � are quasi-finite injective on both sides, and X � h������ in ��� and� � h��X��� in ���.

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Finally, suppose that � and � are cosemisimple corings. Let X ∈ ��� and � ∈ ���

The following statements are equivalent:

(1) �−��X�−���� is a Frobenius pair;(2) ����−� X��−� is a Frobenius pair;(3) X and � are quasi-finite on both sides, and X � h������ in ��� and � �

h��X��� in ���.

Proof. We start by proving the first part. In view of Theorems 2.11 and 2.10 itsuffices to show that if �−��X�−���� is a Frobenius pair, the condition “X� and�� are coflat” is equivalent to “XB and �A are flat”. Indeed, the first implicationis obvious, for the converse, assume that XB and �A are flat. By Proposition3.10, X and � are injective in �� and �� respectively, and they are coflat byProposition 3.8. The particular case is straightforward from Corollary 3.9 and theabove equivalences.

Now we will show the second part. We know that every comodule categoryover a cosemisimple coring is a spectral category (see Stenström, 1975, p. 128). FromProposition 3.8, the bicomodules �X� and ��� are coflat and injector on both sides(we can see this directly by using the fact that every additive functor between abeliancategories preserves split exactness). Finally, Theorem 2.11 achieves the proof. �

Remark 3.12. (1) The equivalence “�1� ⇔ �4�” of the last theorem is ageneralization of Castaño Iglesias et al. (1999, Theorem 3.3). The proof of CastañoIglesias et al. (1999, Theorem 3.3) gives an alternative proof of “�1� ⇔ �4�” ofTheorem 3.11, using Proposition 3.3.

(2) The adjunction of Propositions 2.7 and 3.10 generalizes the coalgebraversion of Morita’s theorem (Caenepeel et al., 2002, Theorem 4.2).

Example 3.13. Let A be a k-algebra. Put � = A and � = k The bicomodule A ∈��� is quasi-finite as a right �-comodule. A is an injector as a right �-comodule ifand only if the k-module A is flat. If we take A = k = �, the bicomodule A is quasi-finite and injector as a right �-comodule but it is not injective in �� Hence, theassertion “�−��X�−���� is an adjoint pair of functors” does not imply in generalthe assertion “� is quasi-finite injective as a right �-comodule and X � h������ in���”, and the following statements are not equivalent in general:

(1) �−��X�−���� is a Frobenius pair;(2) X and � are quasi-finite injective on both sides, and X � h������ in ��� and

� � h��X��� in ���.

On the other hand, there exists a commutative self-injective ring wich is notcoherent. By a theorem of S. U. Chase (see for example Anderson and Fuller, 1992,Theorem 19.20), there exists then a k-algebra A which is injective, but not flat ask-module. Hence, the bicomodule A ∈ ��� is quasi-finite and injective as a right�-comodule, but not an injector as a right �-comodule.

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4. APPLICATIONS TO INDUCTION FUNCTORS

We start this section by recalling from Gómez-Torrecillas (2002), that a coringhomomorphism from the coring � into the coring � is a pair ��� ��, where� � A → B is a homomorphism of k-algebras and � � � → � is a homomorphism ofA-bimodules such that

�� � � = � � �� and �� � � = ��� � ��⊗ A�� � ���

where ��� � �⊗ A� → �⊗ B� is the canonical map induced by � � A → B.Now we will characterize when the induction functor −⊗ AB � �� → ��

defined in Gómez-Torrecillas (2002, Proposition 5.3) is a Frobenius functor.The coaction of � over M ⊗ AB is given, when expressed in Sweedler’s sigmanotation, by

�M⊗AB�m⊗ Ab� =

∑m�0� ⊗ A1B ⊗ B��m�1��b�

where M is a right �-comodule with coaction �M�m� = ∑m�0� ⊗ Am�1�. If Y�B⊗A�

is A��⊗ A��-pure for every right �-comodule Y (i.e., ��� �� is a pure morphism ofcorings, see Brzezinski and Wisbauer, 2003, 24.8), we also define the coinductionfunctor −���B ⊗ A�� � �

� → ��, where the left coaction on the left B-module B ⊗A� is given by

B⊗A�� B ⊗ A� → �⊗ BB ⊗ A� � �⊗ A�� b ⊗ Ac → ∑

b��c�1��⊗ Ac�2��

where ���c� =∑

c�1� ⊗ Ac�2�.Moreover, we have an adjoint pair of functors �−⊗ AB�−���B ⊗ A��� (see

Brzezinski and Wisbauer, 2003, 24.11).

Theorem 4.1. Let ��� �� � � → � be a homomorphism of corings such that A� and

B� are flat. The following statements are equivalent:

(a) −⊗ AB � �� → �� is a Frobenius functor;(b) The �−�-bicomodule �⊗ AB is quasi-finite and injector as a right �-comodule

and there exists an isomorphism of �− �-bicomodules h���⊗ AB��� � B ⊗ A�

Moreover, if � and � are coseparable, then the condition “injector” in �b� can bedeleted.

Proof. First observe that −⊗ AB is a Frobenius functor if and only if �−⊗ AB�−���B ⊗ A��� is a Frobenius pair (by Gómez-Torrecillas, 2002, Proposition 5.4). Astraightforward computation shows that the map �M ⊗ AB � M ⊗ AB → M ⊗ A�⊗AB is a homomorphism of �-comodules. We have thus a commutative diagram in�� with exact row

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where �M is defined by the universal property of the kernel. Since B� is flat, to provethat �M is an isomorphism of �-comodules it is enough to check that it is bijective,as the forgetful functor U � �� → �B is faithfully exact. Some easy computationsshow that the map �M ⊗ A�� ⊗ AB� � � is the inverse in �B to �M . From this,we deduce a natural isomorphism � � −⊗ AB � −����⊗ AB�. The equivalencebetween (a) and (b) is then obvious from Propositions 2.9 and 2.7. �

Let us consider the particular case where � = A and � = B are thetrivial corings (which are separable). Then Theorem 4.1 gives functorial Morita’scharacterization of Frobenius ring extensions given in Morita (1965, Theorem 5.1).This follows from Gómez-Torrecillas (2002, Example 4.3): In the case � = A, � = Bwe have that A⊗ AB � B is quasi-finite as a right B-comodule if and only if AB isfinitely generated an projective, and, in this case, hB�B�−� � −⊗ BHomA�AB�A�.

Theorem 4.3 generalizes the characterization of Frobenius extension ofcoalgebras over fields (Castaño Iglesias et al., 1999, Theorem 3.5). It is thenreasonable to give the following definition.

Definition 4.2. Let ��� �� � � → � be a homomorphism of corings. It is said to bea right Frobenius morphism of corings if −⊗ AB � �� → �� is a Frobenius functor.

Theorem 4.3. Suppose that A and B are QF rings. Let ��� �� � � → � be ahomomorphism of corings such that the modules A�, �A and B� are projective. Thenthe following statements are equivalent:

(a) −⊗ AB � �� → �� is a Frobenius functor;(b) The �−�-bicomodule �⊗ AB is quasi-finite as a right �-comodule, ��⊗ AB�� is

injective and there exists an isomorphism of �− �-bicomodules h���⊗ AB��� �B ⊗ A�.

Proof. Obvious from Theorem 4.1 and Corollary 3.9. �

Now, suppose that the forgetful functor �� → �A is a Frobenius functor.Then the functor −⊗ A� � �A → �A is also a Frobenius functor (since it is acomposition of two Frobenius functors) and A� is finitely generated projective.On the other hand, since −⊗ A� � �A → �� is a left adjoint to Hom����−� ��� → �A Then Hom����−� is a Frobenius functor. Therefore, � is finitelygenerated projective in ��� and hence in �A.

Lemma 4.4. Let R be the opposite algebra of ∗�.

(1) � ∈ ��A is quasi-finite (resp. quasi-finite and injector) as a right A-comoduleif and only if A� is finitely generated projective (resp. A� is finitely generatedprojective and AR is flat). Let hA���−� = −⊗ AR � �A → �� be the cohomfunctor.

(2) If A� is finitely generated projective and AR is flat, then

AhA��� A�� � AR��

where the right �-comodule structure of R is defined as in Brzezinski (2002, Lemma4.3).

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Proof. (1) Straightforward from Gómez-Torrecillas (2002, Example 4.3).

(2) From Brzezinski (2002, Lemma 4.3), the forgetful functor �� → �A

is the composition of functors �� → �R → �A By Gómez-Torrecillas (2002,Proposition 4.2), hA���−� is a left adjoint to −��� � �� → �A which is isomorphicto the forgetful functor �� → �A Then hA���−� is isomorphic to the compositionof functors

�A

−⊗AR−−−→ �R −→ ��

In particular, AhA��� A�� � A�A⊗ AR�� � AR� �

Corollary 4.5 (Brzezinski and Wisbauer, 2003, 27.10). Let � be an A-coring and letR be the opposite algebra of ∗� Then the following statements are equivalent:

(a) The forgetful functor F � �� → �A is a Frobenius functor;(b) A� is finitely generated projective and � � R as �A�R�-bimodules, where � is a

right R-module via cr = c�1�r�c�2��, for all c ∈ � and r ∈ R

Proof. Straightforward from Theorem 4.1 and Lemma 4.4. �

The following proposition gives sufficient conditions to have that a morphismof corings is right Frobenius if and only if it is left Frobenius. Note that it saysin particular that the notion of Frobenius homomorphism of coalgebras over fields(by (b)) or of rings (by (d)) is independent on the side. Of course, the latter is wellknown.

Proposition 4.6. Let ��� �� � � → � be a homomorphism of corings such that A�,B�, �A and �B are flat. Assume that at least one of the following holds:

(a) � and � have a duality, and AB and BA are flat;(b) A and B are von Neumann regular rings;(c) B ⊗ A� is coflat in �� and �⊗ AB is coflat in �� and AB and BA are flat;(d) � and � are coseparable corings.

Then the following statements are equivalent:

(1) −⊗ AB � �� → �� is a Frobenius functor;(2) B ⊗ A− � �� → �� is a Frobenius functor.

Proof. Obvious from Theorems 2.11 and 3.11. �

Let us finally show how to derive from our results a remarkablecharacterization of the so called Frobenius corings.

Corollary 4.7 (Brzezinski and Wisbauer, 2003, 27.8). The following statements areequivalent:

(a) The forgetful functor �� → �A is a Frobenius functor;

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(b) The forgetful functor �� → A� is a Frobenius functor;(c) There exist an �A�A�-bimodule map � � A → � and a �����-bicomodule map � �

�⊗ A� → � such that ���⊗ A�� = � = ���⊗ A��.

Proof. The proof of “�1� ⇔ �4�” in Proposition 2.7 for X = � ∈ A�� and � = � ∈��A remains valid for our situation. Finally, notice that the condition (4) in thiscase is exactly the condition (c). �

5. APPLICATIONS TO ENTWINED MODULES

In this section we particularize some of our results in Section 4 to the categoryof entwined modules. We adopt the notations of Caenepeel et al. (2002). We startwith some remarks.

(1) Consider a right-right entwining structure �A�C��� ∈ �••�k� and a left-

left entwining structure �B�D� �� ∈ ••��k�. The category of two-sided entwined

modules DB���� ��CA defined in Caenepeel et al. (2002, pp. 68–69) is isomorphic to

the category of bicomodules D⊗B�A⊗C over the associated corings.

(2) If �A�C��� and �A′� C ′� �′� belong to �••�k� and are such that � is an

isomorphism, then � is an isomorphism of corings (see Caenepeel et al., 2002,Proposition 34), and consequently if the coalgebra C is flat as a k-module, then themodules A�A⊗ C� and �A⊗ C�A are flat, and

A⊗C�A′⊗C′ � CA���−1� �′�C

′A′

(3) Let ��� �� � �A�C��� → �A′� C ′� �′� be a morphism in �••�k�. We know

that��⊗ �� �� � A⊗ C → A′ ⊗ C ′ is a morphism of corings. The functor F defined inCaenepeel et al. (2002, Lemma 8) satisfies the commutativity of the diagram

where −⊗ AA′ � �A⊗C → �A′⊗C′

is the induction functor defined in Gómez-Torrecillas (2002, Proposition 5.3).

We obtain the following result concerning the category of entwined modules.

Theorem 5.1. Let ��� �� � �A�C��� → �A′� C ′� �′� be a morphism in �••�k�, such that

kC and kD are flat.

(1) The following statements are equivalent:

(a) The functor F = −⊗ AA′ � ����CA → ���′�C

′A′ defined in Caenepeel et al.

(2002, Lemma 8) is a Frobenius functor;

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(b) The A⊗ C − A′ ⊗ C ′-bicomodule �A⊗ C�⊗ AA′ is quasi-finite injector as a

right A′ ⊗ C ′-comodule and there exists an isomorphism of A′ ⊗ C ′ − A⊗ C-bicomodules hA′⊗C′��A⊗ C�⊗ AA

′� A′ ⊗ C ′� � A′ ⊗ A�A⊗ C�

Moreover, if A⊗ C and A′ ⊗ C ′ are coseparable corings, then the condition“injector” in (b) can be deleted.

(2) If A and A′ are QF rings and the module �A′ ⊗ C ′�A′ is projective, then thefollowing are equivalent:

(a) The functor F = −⊗ AA′ � ����CA → ���′�C

′A′ defined in Caenepeel et al.

(2002, Lemma 8) is a Frobenius functor;(b) The A⊗ C − A′ ⊗ C ′-bicomodule �A⊗ C�⊗ AA

′ is quasi-finite and injective asa right A′ ⊗ C ′-comodule and there exists an isomorphism of A′ ⊗ C ′ − A⊗ C-bicomodules hA′⊗C′��A⊗ C�⊗ AA

′� A′ ⊗ C ′� � A′ ⊗ A�A⊗ C�.

Proof. Follows from Theorems 4.1 and 4.3. �

Remark 5.2. Let a right-right entwining structure �A�C��� ∈ �••�k�. The cosepa-

rability of the coring A⊗ C is characterized in Caenepeel et al. (2002, Theorem38(1)) (see also Brzezinski, 2002, Corollary 3.6).

6. APPLICATIONS TO GRADED RING THEORY

In this section we apply our results in the previous sections to the categoryof graded modules by a G-set. Let G be a group, A be a G-graded k-algebra, andlet X be a left G-set. The category �G�X�A�− gr of graded left modules by X isintroduced and studied in Nastasescu et al. (1990). A study of the graded ring theorycan be found in the recent book Nastasescu and Van Oystaeyen (2004). We adoptthe notations of Del Río (1992) and Caenepeel et al. (2002), and we begin by givingsome useful lemmas.

6.1. Some Useful Lemmas

Let C = kX and C ′ = kX′ be two grouplike coalgebras, where X and X′ arearbitrary sets. We know (see Caenepeel et al., 2002, Example 4) that the category�C is isomorphic to the category of X-graded modules. Moreover, we have thefollowing lemma.

Lemma 6.1. For a k-module M which is both a X-graded and a X′-graded module,the following are equivalent:

(a) M ∈ C′�C;

(b) For every m ∈ M�x ∈ X� x′ ∈ X′� x′�mx� ∈ Mx;(c) For every m ∈ M�x ∈ X� x′ ∈ X′� �x′m�x ∈ x′M;(d) For every m ∈ M�x ∈ X� x′ ∈ X′� x′�mx� = �x′m�x.

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Proof. Let M = ⊕x∈X Mx =

⊕x′∈X′ x′M . At first, observe that the condition �a� is

equivalent to the fact that the diagram

is commutative.

(a) ⇒ (d) Let m ∈ M . We have, �M�m� = ∑x∈X mx ⊗ x and M�m� =∑

x′∈X′ x′ ⊗ x′m. From the commutativity of the above diagram,

∑x∈X

∑x′∈X′

x′ ⊗ x′�mx�⊗ x = ∑x′∈X′

∑x∈X

x′ ⊗ �x′m�x ⊗ x

Hence, x′�mx� = �x′m�x for all x ∈ X, and x′ ∈ X′.

(d) ⇒ (b) Trivial.

(b) ⇒ (a) Let m ∈ M . We have,

�M ⊗ kX��M�m� = ∑x∈X

∑x′∈X′

x′ ⊗ x′�mx�⊗ x�

and

�kX′ ⊗ �M�M�m� = �kX′ ⊗ �M�

(∑x∈X

∑x′∈X′

x′ ⊗ x′�mx�

)= ∑

x∈X

∑x′∈X′

x′ ⊗ x′�mx�⊗ x

Hence (a) follows.

(a) ⇔ (d) ⇔ (c) Follows by symmetry. �

Now let G and G′ be two groups, A a G-graded k-algebra, A′ a G′-gradedk-algebra, X a right G-set, and X′ a left G′-set. Let kG and kG′ be the canonicalHopf algebras.

Let � � kX ⊗ A → A⊗ kX� be the map defined by x⊗ ag → ag ⊗ xg, and�′ � A′ ⊗ kX′ → kX′ ⊗ A′, be the map defined by a′

g′ ⊗ x′ → g′x′ ⊗ a′g′ .

From Caenepeel et al. (2002, §4.6), we have �kG�A� kX� ∈ �K ••�k�� �kG

′� A′�kX′� ∈ •

•�K�k�� �A� kX��� ∈ �••�k�� �A′� kX′� �′� ∈ •

•��k�� and ��kG�kXA � gr −�A�X�G�� kX′

A′ ��kG′� � �G′� X′� A′�− gr.

Lemma 6.2. (1) Let M be a k-module having the structure of an X-graded right A-module and an X′-graded left A′-module. The following are equivalent:

(a) M ∈ kX′A′ ���′� ��kXA ;

(b) The following conditions hold:

(i) M is a �A′� A�-bimodule;

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(ii) For every m ∈ M�x ∈ X� x′ ∈ X′� x′�mx� ∈ Mx (or �x′m�x ∈ x′M , or x′�mx� =�x′m�x);

(iii) For every x ∈ X� Mx is a submodule of A′M;(iv) For every x′ ∈ X′� x′M is a submodule of MA.

(2) M ∈ kX′A′ ���′� ��kXA if and only if M is an X′ × X-graded A′ − A-bimodule

(see Del Río, 1992, pp. 492–493).

Proof. (1) (a) ⇔ (b) We will use the definition of an object in the category oftwo-sided entwined modules kX′

A′ ���′� ��kXA (see Caenepeel et al., 2002, pp. 68–69). ByLemma 6.1, the condition “M ∈ kX′

�kX” is equivalent to the condition (ii). We havemoreover that the left A′-action on M is kX-colinear if and only if for every x ∈ X�m ∈ Mx, �M�a

′m� = �a′m�⊗ x, if and only if for every x ∈ X�m ∈ Mx, a′m ∈ Mx, if

and only if (iii) holds. By symmetry, the condition “the right A-action on M is kX′-colinear” is equivalent to the condition (iv).

(2) The “if” part is clear (see Del Río, 1992, p. 493). For the “only if” part, putM = ⊕

�x′�x�∈X′×X M�x′�x�, where M�x′�x� = x′�Mx� = �x′M�x. Let a′ ∈ A′g′ �m ∈ M . Since

a′x′�mx� ∈ Mx and a′x′�mx� ∈ g′x′�M�, a′x′�mx� ∈ a′x′�Mx�. Therefore, A′g′ x′�Mx� ⊂

a′x′�Mx�. By symmetry, we obtain A′g′M�x′�x�Ag ⊂ Mg′�x′�x�g = M�g′x′�xg� (g ∈ G� g′ ∈

G′� x ∈ X� x′ ∈ X′). �

6.2. Adjoint Pairs and Frobenius Pairs of Functors BetweenCategories of Graded Modules Over G-sets

Throughout this subsection, G and G′ will be two groups, A a G-gradedk-algebra, A′ a G′-graded k-algebra, X a right G-set, and X′ a right G′-set. LetkG and kG′ be the canonical Hopf algebras. Let � � kX ⊗ A → A⊗ kX be the mapdefined by x⊗ ag → ag ⊗ xg. Analogously, we define the map �′ � kX′ ⊗ A′ → A′ ⊗kX′. The comultiplication and the counit maps of the coring A⊗ kX are defined by

�A⊗kX�a⊗ x� = �a⊗ x�⊗ A�1A ⊗ x�� �A⊗kX�a⊗ x� = a �a ∈ A� x ∈ X�

An important result is that the corings A⊗ kX and A′ ⊗ kX′ are coseparable.A proof is clear by using Brzezinski (2002, Corollary 3.6) and Caenepeel et al. (2002,Proposition 101). A direct proof in the setting of corings is deduced from Brzezinski(2002, Theorem 3.5) by using the cointegral in the coring A⊗ kX given by ��a⊗x⊗ y� = a�x�y (Kronecker’s delta) for a ∈ A� x� y ∈ X (see Brzezinski and Wisbauer,2003, 26.2 for the definition of a cointegral in a coseparable coring).

Lemma 6.3.

(1) � is bijective, �A� kX��−1� ∈ ••��k�, and

kXA ��kG� �= kX

A ���−1� � �G�X�A�− gr�

where the structure of left G-set on X is given by gx = xg−1 �g ∈ G� x ∈ X�.(2) Every object of the category A′⊗kX′

�A⊗kX � kX′A′ ����′�−1� ��kXA can be identified with

an X′ × X-graded A′ − A-bimodule.

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Proof. (1) From Caenepeel et al. (2002, Proposition 2), and since the antipode ofH = kG is S�g� = g−1, for all g ∈ G, then S � S = 1H and S = S−1 = S is a twistedantipode of H Hence � is bijective and �A� kX��−1� ∈ •

•��k� (see Caenepeel et al.,2002, p. 49), and kX

A ��kG� �= kXA ���−1� � �G�X�A�− gr, since �−1 � A⊗ kX →

kX ⊗ A� ag ⊗ x → gx⊗ ag, where gx = xg−1

(2) It follows from �1� and Lemma 6.2. �

Lemma 6.4. (1) Let M ∈ gr − �A�X�G�, and N ∈ �G�X�A�− gr. We know thatM ∈ �A⊗kX by the coaction: �M � M → M ⊗ A�A⊗ kX�, where �M�mx� = mx ⊗A�1A ⊗ x�, and N ∈ A⊗kX� by the coaction: N � N → �A⊗ kX�⊗ AN , where N �xn� =�1A ⊗ x�⊗ Axn. We have M��A⊗kX�N =M ⊗AN , where M ⊗AN is the additive subgroupof M ⊗ AN generated by the elements m⊗ An where x ∈ X�m∈Mx, and n∈ xN(see Del Río, 1992, p. 492).

(2) Let P be an X × X′-graded A− A′-bimodule. We have the commutativediagram:

where −⊗AP � gr − �A�X�G� → gr − �A′� X′�G′� is the functor defined in Del Río(1992, p. 493).

Proof. (1) At first we will show that the right A-module A⊗ kX is free. Moreprecisely that the family �1A ⊗ x � x ∈ X� is a right basis of it. It is clear thatag ⊗ x = �1A ⊗ xg−1�ag for ag ∈ Ag and x ∈ X. Now suppose that

∑i�1A ⊗ xi�ai = 0,

where xi ∈ X� ai ∈ A. Then∑

i ��xi ⊗ ai� = 0. Since � is bijective, we obtain∑

i xi ⊗ai = 0. Therefore ai = 0 for all i. Hence the above mentioned family is a basis of theright A-module A⊗ kX. (There is a shorter and indirect proof of this fact by usingthat � is an isomorphism of A-bimodules, and �x⊗ 1A�x ∈ X� is a basis of the rightA-module kX ⊗ A.)

Finally, suppose that∑

i mxi⊗ Ayi

n ∈ M��A⊗kX�N . Then

∑i

mxi⊗ A1A ⊗ xi ⊗ Ayi

n = ∑i

mxi⊗ A1A ⊗ yi ⊗ Ayi

n

Hence (the right A-module A⊗ kX is free), xi = yi for all i, and M��A⊗kX�N ⊂M⊗AN . The other inclusion is obvious.

(2) It suffices to show that the map M⊗AP → M⊗AP ⊗ A′�A′ ⊗ kX′� definedby

∑x∈F

mx ⊗ Axp −→ ∑x′∈X′

(∑x∈F

mx ⊗ Axp

)x′⊗ A′�1A′ ⊗ x′��

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where F is a finite subset of X, makes commutative the following diagram

That is clear since �∑

x∈F mx ⊗ Axp�x′ =∑

x∈F mx ⊗ A�xp�x′ =∑

x∈F mx ⊗ Ap�x�x′�,where p = ∑

x∈F xp. �

Let A = A⊗ kX be the X × X-graded A− A-bimodule associated to the �A⊗kX�− �A⊗ kX�- bicomodule A⊗ kX. It is clear that −⊗AA = −��A⊗kX��A⊗ kX� �1gr−�A�G�X�. The gradings are Ax = A⊗ kx, and xA = �

∑i ai ⊗ xi � xig−1 = x�∀i�∀g ∈

G � �ai�g �= 0� (x ∈ X).Recall from Menini (1993, Proposition 1.2) that for every X × X′-

graded A− A′-bimodule P, we have an adjunction �−⊗ AP�H�PA′−�. Theunit and the counit of this adjunction are given respectively by �M � M →H�PA′ �M⊗AP�� �M�m��p� = ∑

x∈X mx ⊗ Axp�m = ∑x∈X mx ∈ M�p = ∑

x∈X xp ∈ P�,and �N � H�PA′ � N�⊗AP → N� �N �f⊗AP� = f�p��f ∈ H�PA′ � N�x� p ∈ xP� x ∈ X�.

Proposition 6.5 (Menini, 1993, Proposition 1.3, Corollary 1.4).

(1) The following statements are equivalent for a k-linear functor F � gr − �A�X�G� →gr − �A′� X′�G′�:

(a) F has a right adjoint;(b) F is right exact and preserves coproducts;(c) F � −⊗AP for some X × X′-graded A− A′-bimodule P.

(2) A k-linear functor G � gr − �A′� X′�G′� → gr − �A�X�G� has a left adjoint if andonly if G � H�PA′ �−� for some X × X′-graded A− A′-bimodule P.

Proof. (1) (a) ⇒ (b) Clear. �b� ⇒ �c� It follows from Theorem 2.3 andLemma 6.4. (c) ⇒ (a) Obvious from the above mentioned result.

(2) It follows from �1� and Menini (1993, Proposition 1.2). �

Lemma 6.6. Let P be an X × X′-graded A− A′-bimodule.

(1) H�PA′ �−� is right exact and preserves direct limits if and only if xP is finitelygenerated projective in �A′ for every x ∈ X.

(2) Suppose that xP is finitely generated projective in �A′ for every x ∈ X.

(a) For every k-algebra T ,

�Z�M � Z ⊗ TH�PA′ �M��−→H�PA′ � Z ⊗ TM�

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defined by

�Z�M�z⊗ T f��p� =∑x∈X

z⊗ T fx�xp�

�z∈Z� f = ∑x∈X fx ∈H�PA′ �M�� p = ∑

x∈X xp ∈ P�, is the natural isomorphismassociated to the functor (see Section 2)

H�PA′ �−� � gr − �A′� X′�G′� → gr − �A�X�G�

(b) Moreover we have the natural isomorphism

�N � N ⊗A′H�PA′ � A′��−→H�PA′ � N�

defined by

�N �nx′⊗A′ x′f��p� =∑x∈X

nx′�(�1A′ ⊗ x′�⊗A′f�x′�x��xp�

)

�nx′ ∈ Nx′ � x′f ∈ x′H�PA′ � A′�� p = ∑x∈X xp ∈ P�, where � is the cointegral in the

coring A′ ⊗ kX′ defined in the beginning of this subsection. The left grading onH�PA′ � A′� is given by

x′H�PA′ � A′�

={f ∈ H�PA′ � A′� � ��A′⊗kX′�

(f�p�

) = �1A′ ⊗ x′�⊗A′∑x∈X

fx�xp�� ∀p ∈ P

}

�x′ ∈ X′�

Proof. (1) We have

H�PA′ �−� � ⊕x∈X

Homgr−�A′�X′�G′��xP�−� � gr − �A′� X′�G′� −→ Ab

Hence, H�PA′ �−� is right exact and preserves direct limits if and only ifHomgr−�A′�X′�G′��xP�−� is right exact and preserves direct limits for every x ∈ X ifand only if xP is finitely generated projective in �A′ for every x ∈ X (by Stenström,1975, Proposition V.3.4 and the fact that every finitely generated projective objectof a Grothendieck category is finitely presented).

(2) (a) Let T be a k-algebra. Let M be an X0 × X′-graded T − A′-bimodule,where X0 is a singleton, Z be a right T -module, and x ∈ X. We have a sequence ofT -submodules

H�PA′ �M�x ≤ H�PA′ �M� ≤ Homgr−�A′�X′�G′��P�M� ≤ THomA′�PA′ �M��

and the induced structure of left T -module on H�PA′ �M� is the same structure (seeSection 2) of left T -module associated to the functor H�PA′ �−� on it. Moreover,we have the isomorphism of left T -modules H�PA′ �M�x � Homgr−�A′�X′�G′��xP�M�.

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From Anderson and Fuller (1992, Proposition 20.10), for every x ∈ X, there is anisomorphism

�x � Z⊗THomA′�xP�M��−→HomA′�xP� Z⊗TM� (5)

defined by �x�z⊗T �x� � xp → z⊗T �x�xp� �z ∈ Z� �x ∈ HomA′�xP�M��. For everyx ∈ X, �x induces an isomorphism

�′x � Z ⊗ THomgr−�A′�X′�G′��xP�M��−→Homgr−�A′�X′�G′��xP� Z ⊗ TM� (6)

Now let us consider the isomorphism

�′Z�M �= ⊕x∈X

�′x � Z⊗TH�PA′ �M��−→H�PA′ � Z⊗TM�

We have �′Z�M�z⊗T f��p� =∑

x∈X z⊗T fx�xp� �z ∈ Z� f = ∑x∈X fx ∈ H�PA′ �M�� p =∑

x∈X xp ∈ P�. We can verify easily that �′Z�M is a morphism of right A-modules.Hence it is a morphism in gr − �A�X�G�. It is clear that �′Z�M is natural in Z,and �′T�M makes commutative the diagram (1) (see Section 2). Finally, by Mitchell’sTheorem (Mitchell, 1965, Theorem 3.6.5), �′Z�M = �Z�M .

(b) To prove the first statement it suffices to use the proof of Theorem 2.3,the property �a�, and the fact that every comodule over an A-coseparable coring isA-relative injective comodule (for the last see Brzezinski and Wisbauer, 2003, 26.1).

Finally, we will prove the last statement. We know that H�PA′ �A′� is defined(Gómez-Torrecillas, 2002) to be the unique A′-linear map making commutative thefollowing diagram

On the other hand, for each x′ ∈ X′, x′H�PA′ � A′� = �f ∈ H�PA′ � A′� � H�PA′ �A′��f� =�1A′ ⊗ x′�⊗A′f�. Since �A′�A′ is an isomorphism, x′H�PA′ � A′� = �f ∈ H�PA′ � A′� ���A′⊗kX′��f�p�� = �1A′ ⊗ x′�⊗A′

∑x∈X fx�xp� for all p ∈ P�. �

Remark 6.7. Let � be a finitely generated projective right A-module, and M bea right A-module. It easy to verify that the map ���M � HomA���M� → M⊗A�

∗

defined by ���M��� =∑

i ��ei�⊗Ae∗i , where �ei� e

∗i �i is a dual basis of �, is an

isomorphism, with �−1��M�m⊗Af��u� = mf�u��m ∈ M� f ∈ �∗� u ∈ ��.

Now we assume that the condition of Lemma 6.6(2) holds. It is easyto verify that for every x ∈ X, �x = �−1

xP�Z⊗TM� �Z⊗T�xP�M

�, where �x is the

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isomorphism (5). Therefore �−1x = �Z⊗T�

−1xP�M

� � �xP�Z⊗TM

Hence �−1Z�M�g� =∑

x∈X∑

i∈Ix �Z⊗T�−1xP�M

��gx�ex�i�⊗A′e∗x�i�, where �ex�i� e∗x�i�i∈Ix is a dual basis of xP

�x ∈ X�, and g = ∑x∈X gx ∈ H�PA′ � Z⊗TM�. Finally, for each N ∈ gr − �A′� X′�G′�,

�−1N � H�PA′ � N�

�−→N ⊗A′H�PA′ � A′�

is �−1N = �−1

N�A′ �H�PA′ � �N �, and then

�−1N ��� = ∑

x∈X

∑i∈Ix

∑x′∈X′

��x�ex�i��x′⊗A′�′�x′ ⊗ e∗x�i�−���

where � = ∑x∈X �x ∈ H�PA′ � N�.

In particular, if X′ = G′ = �e′�, then

�−1N � H�PA′ � N�

�−→N ⊗ A′H�PA′ � A′�

is defined by

�−1N ��� = ∑

x∈X

∑i∈Ix

��ex�i�⊗A′e∗x�i �� ∈ H�PA′ � N��

Lemma 6.8. (1) Let N ∈ A′�A⊗kX Then N is quasi-finite as a right A⊗ kX-comodule if and only if Nx is finitely generated projective in A′�, for every x ∈ X. Inthis case, the cohom functor hA⊗kX�N�−� is the composite

�A⊗kX �−→ gr − �A�X�G�−⊗AP−−−→ �A′ �

where P is the X × X′0-graded A− A′-bimodule H�A′N�A′� with X′

0 is a singleton.

(2) Now suppose that N ∈ A′⊗kX′�A⊗kX , and Nx is finitely generated projective

in �A′ , for every x ∈ X. Let �ex�i� e∗x�i�i∈Ix be a dual basis of Nx �x ∈ X�. Let P be

the bigraded bimodule defined as above, � � 1�A⊗kX → −⊗AP ⊗ A′N be the unit of theadjunction �−⊗AP�−⊗A′N�, and let M ∈ gr − �A�X�G�.

We can endow P with a structure of X × X′-graded A− A′-bimodule such that

hA⊗kX�N�−� � −⊗AP � gr − �A�X�G� → gr − �A′� X′�G′�

The A′ ⊗ kX′-comodule structure on P,

�P � P → P⊗A′�A′ ⊗ kX′��

is defined to be the unique A′-linear map satisfying the condition:

∑i∈Ix

�P�ae∗x�i�⊗A′ex�i =

∑i∈Ix

∑x′∈X′

ae∗x�i⊗A′�1A′ ⊗ x′�⊗A′�ex�i�x′ � (7)

for every a ∈ A� x ∈ X.

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ADJOINT AND FROBENIUS PAIRS 719

Proof. It follows from Lemma 6.4(2), that the functor F �= −⊗A′N � �A′ = gr −�A′� X′

0�G′� → gr − �A�X�G�, where X′

0 is a singleton, is the composite

�A′ = gr − �A′� X′0�G

′�−⊗A′N−−−→ �A⊗kX �−→ gr − �A�X�G�

Then, N is quasi-finite as a right A⊗ kX-comodule if and only if F has a left adjoint,if and only if (by Corollary 6.5(2)) there exists an X × X′

0-graded A− A′-bimoduleP such that F � H�PA′ �−�, if and only if (by Lemma 6.6, Gómez-Torrecillas, 2002,Lemma 3.2(1) and Theorem 2.3) there exists an X × X′

0-graded A− A′-bimoduleP such that xP is finitely generated projective in �A′ for every x ∈ X� and N �H�PA′ � A′� in A′

�A⊗kX .Now let us consider the X × X′

0-graded A− A′-bimodule P �= H�A′N�A′�,and the X′ × X-graded A′ − A-bimodule M �= H�PA′ � A′�, where X′

0 is a singleton.We have

M = �f ∈ P∗�f�xP� = 0 for almost all x ∈ X�

Mx ={f ∈ P∗�f�yP� = 0 for all y ∈ X − �x�

}�x ∈ X��

where P∗ = HomA′�PA′ � A′A′�. The structure of A′ − A-bimodule on P∗ is given by

�fa��p� = f�ap�� �a′f��p� = a′f�p� �f ∈ P∗� a ∈ A� a′ ∈ A′� p ∈ P�. We have

M ≤ �P∗�A� Mx ≤ M ≤ A′�P∗�� and Mx � �xP�∗ in �A′ �x ∈ X�

Analogously,

P = �f ∈ ∗N � f�Nx� = 0 for almost all x ∈ X�

xP = �f ∈ ∗N � f�Ny� = 0 for all y ∈ X − �x�� �x ∈ X��

where ∗N = HomA′�A′N� A′A′�. The structure of A− A′-bimodule on ∗N is given by�af��n� = f�na�� �fa′��n� = a′f�n� �f ∈ ∗N� a ∈ A� a′ ∈ A′� n ∈ N�. We have

P ≤ A�∗N�� xP ≤ P ≤ �∗N�A′ � and xP � ∗�Nx� in �A′ �x ∈ X�

Hence, Nx is finitely generated projective in A′�, for every x ∈ X implies that xP isfinitely generated projective in �A′ , for every x ∈ X, which implies that Mx is finitelygenerated projective in A′�, for every x ∈ X.

Finally, let us consider for every x ∈ X the isomorphism of left A′-modules

Hx � Nx

�−→ �∗�Nx��∗ �−→ �xP�

∗ �−→Mx

We have Hx�nx���� = ��nx� if � ∈ xP, and Hx�nx���� = 0 if � ∈ yP and y ∈ X − �x�.Set

H = ⊕x∈X

Hx � N�−→M

It can be proved easily that H is a morphism of right A-modules. Hence it is anisomorphism of graded bimodules.

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(2) We have �e∗x�i� �x�ex�i��i∈Ix is a dual basis of ∗�Nx� � xP, where �x isthe evaluation map �x ∈ X�. From the proof of Lemma 6.5 and Remark 6.7,the unit of the adjunction �−⊗AP�−⊗ A′N� is �M � M → M⊗AP ⊗ A′N , �M�m� =∑

x∈X∑

i∈Ix mx ⊗ Ae∗x�i ⊗ A′ex�i�m ∈ M�. By Gómez-Torrecillas (2002, 4.1), the coaction

on hA⊗kX�N�M� = M⊗AP � �M⊗AP� M⊗AP → M⊗AP ⊗ A′�A′ ⊗ kX′� is the unique

A′-linear map satisfying the commutativity of the diagram

Since �A⊗ kX�⊗AP � P as �A⊗ kX�− A′-bicomodules, P can be endowed witha structure of �A⊗ kX�− �A′ ⊗ kX′�-bicomodule such that �A⊗ kX�⊗AP � P as�A⊗ kX�− �A′ ⊗ kX′�-bicomodules. Moreover, the right coaction on P, �P � P →P⊗A′�A′ ⊗ kX′�, is the unique A′-linear map satisfying the commutativity of thediagram

The commutativity of the above diagram is equivalent to the condition (7). Finally,By Theorem 2.3, hA⊗kX�N�−� � −⊗A��A⊗ kX�⊗AP� � −⊗AP �

Now we are in a position to state and prove the main results of this sectionwhich characterize adjoint pairs and Frobenius pairs of functors between categoriesof graded modules over G-sets.

Theorem 6.9. Let M be an X × X′-graded A− A′-bimodule, and N be an X′ × X-graded A′ − A-bimodule. Then the following are equivalent:

(1) �−⊗AM�−⊗A′N� is an adjoint pair;(2) �N ⊗A−�M⊗A′−� is an adjoint pair;(3) Nx is finitely generated projective in A′�, for every x ∈ X, and H�A′N�A′� � M as

X × X′-graded A− A′-bimodules;(4) there exist bigraded maps

� � A → M⊗A′N and � N ⊗AM → A′�

such that

� ⊗A′N� � �N ⊗A�� = � and �M⊗A′ � � ��⊗AM� = M (8)

In particular, �−⊗AM�−⊗A′N� is a Frobenius pair if and only if �M⊗A′−� N ⊗A−� isa Frobenius pair.

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ADJOINT AND FROBENIUS PAIRS 721

Proof. We know that the corings A⊗ kX and A′ ⊗ kX′ are coseparable.Proposition 2.7 then achieves the proof. �

Theorem 6.10. Let G and G′ be two groups, A a G-graded k-algebra, A′ a G′-graded k-algebra, X a right G-set, and X′ a right G′-set. For a pair of k-linear functorsF � gr − �A�X�G� → gr − �A′� X′�G′� and G � gr − �A′� X′�G′� → gr − �A�X�G�,the following statements are equivalent:

(a) �F�G� is a Frobenius pair;(b) there exist an X × X′-graded A− A′-bimodule M , and an X′ × X-graded A′ − A-

bimodule N , with the following properties:

(1) xM and Nx is finitely generated projective in �A′ and �A′ respectively, forevery x ∈ X, and Mx′ and x′N is finitely generated projective in A� and �A

respectively, for every x′ ∈ X′;(2) H�AM�A� � N as X′ × X-graded A′ − A-bimodules, and H�A′N�A′� � M as

X × X′-graded A− A′-bimodules;(3) F � −⊗AM and G � −⊗A′N .

Proof. Straightforward from Theorem 2.11. �

6.3. When Is the Induction Functor T ∗ Frobenius?

Finally, let f � G → G′ be a morphism of groups, X a right G-set, X′ a rightG′-set, � � X → X′ a map such that ��xg� = ��x�f�g� for every g ∈ G� x ∈ X. Let Abe a G-graded k-algebra, A′ a G′-graded k-algebra, and � � A → A′ a morphism ofalgebras such that ��Ag� ⊂ A′

f�g� for every g ∈ G.We have, � � kX → kX′ such that ��x� = ��x� for each x ∈ X, is a morphism

of coalgebras, and ��� �� � �A� kX��� → �A′� kX′� �′� is a morphism in �••�k�

Let −⊗ AA′ � gr − �A�X�G� → gr − �A′� X′�G′� be the functor making

commutative the following diagram:

Let M ∈ gr − �A�X�G� We have M ⊗ AA′ is a right A′-module, and

�r�mx ⊗ Aa′g′� = �mx ⊗ Aa

′g′�⊗ ��x�g′

Therefore, �M ⊗ AA′�x′ is the subgroup of M ⊗ AA

′ spanned by the elements of theform mx ⊗ Aa

′g′ where x∈X� g′ ∈G′� ��x�g′ = x′� mx ∈Mx� a

′g′ ∈A′

g′ � for every x′ ∈X′.Therefore, the functor −⊗ AA

′ � gr − �A�X�G� → gr − �A′� X′�G′� is exactlythe induction functor T ∗ defined in Menini and Nastasescu (1994, p. 531). HenceT ∗ is a Frobenius functor if and only if the induction functor −⊗ AA

′ � �A⊗kX →�A′⊗kX′

is a Frobenius functor (see Section 5).

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Moreover, we have the commutativity of the following diagram:

The following consequence of Theorem 5.1 and Menini and Nastasescu (1994,Theorem 2.27) give two different characterizations when the induction functor T ∗ isa Frobenius functor.

Theorem 6.11. The following statements are equivalent:

(a) The functor T ∗ � gr − �A�X�G� → gr − �A′� X′�G′� is a Frobenius functor;(b) �T ∗�A��x′ is finitely generated projective in �A, for every x′ ∈ X′, and there exists

an isomorphism of X′ × X graded A′ − A-bimodules

H�AT∗�A�� A� � �T ∗�′�A�

As an immediate consequence of Proposition 4.6, we obtain the followingproposition.

Proposition 6.12. The following are equivalent:

(a) The functor T ∗ � gr − �A�X�G� → gr − �A′� X′�G′� is a Frobenius functor;(b) The functor �T ∗�′ � �G�X�A�− gr → �G′� X′� A′�− gr is a Frobenius functor.

ACKNOWLEDGMENTS

I would like to thank my advisor, Professor José Gómez-Torrecillas, for hisassistance in the writing of this article, and for interesting discussions.

I would also like to express my deep gratitude to Professor Edgar Enochsfor communicating the example of a commutative self-injective ring which is notcoherent, and to Professor Tomasz Brzezinski for his assistance, for his carefulreading of this article, and for his useful comments.

Finally, I wish to thank Editor Robert Wisbauer and the referee for somehelpful suggestions.

REFERENCES

Al-Takhman, K. (1999). Äquivalenzen zwischen Komodulkategorien von Koalgebren überRingen. Ph.D. thesis, Universität Düsseldorf, Düsseldorf, Germany.

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ADJOINT AND FROBENIUS PAIRS 723

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