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Page 1: Representable Functors for Corings

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Representable Functors for CoringsG. Militaru aa Faculty of Mathematics and Computer Science , University of Bucharest , Bucharest ,RomaniaPublished online: 15 May 2012.

To cite this article: G. Militaru (2012) Representable Functors for Corings, Communications in Algebra, 40:5, 1766-1796, DOI:10.1080/00927872.2011.556694

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Communications in Algebra®, 40: 1766–1796, 2012Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927872.2011.556694

REPRESENTABLE FUNCTORS FOR CORINGS

G. MilitaruFaculty of Mathematics and Computer Science, University of Bucharest,Bucharest, Romania

We address four problems regarding representable functors and give answers to themfor functors connecting the category of comodules over a coring to the categoryof modules over a ring. A functor’s property of being Frobenius is restated as aparticular case of its representability by imposing the predefinition of the object ofrepresentability. Let R, S be two rings, C an R-coring and C

R� the category of leftC-comodules. The category Rep�CR�� S�� of all representable functors C

R� → S� isshown to be equivalent to the opposite of the category C

R�S . For U an �S�R�-bimodulewe give necessary and sufficient conditions for the induction functor U ⊗R − � C

R� →S� to be: a representable functor, an equivalence of categories, a separable or aFrobenius functor. The latter results generalize and unify the classical theorems ofMorita for categories of modules over rings and the more recent theorems obtained byBrezinski, Caenepeel et al. for categories of comodules over corings.

Key Words: Corings; Representable; Separable and Frobenius functors.

2010 Mathematics Subject Classification: 16T15; 18A22.

INTRODUCTION

The representable functor is a central concept in category theory for atleast two reasons. On one hand, fundamental mathematical constructions likefree groups, free modules, tensor products of modules, tensor algebras, symmetricalgebras, and algebras of noncommutative differential forms of an algebra are betterexplained using the language of representable functors. They are all answers to thesame question: Is a given functor F representable? On the other hand, representablefunctors are bridges from classical geometry to noncommutative geometry andquantum groups. Based on the fact that an affine scheme is a representable functoron the category of commutative k-algebras a quantum space is defined as a naturalgeneralization. More precisely, a quantum space is a representable functor on thecategory of k-algebras (not neccesarily commutative) while a quantum group is aquantum space for which the object of representability has a structure of Hopfalgebra [19].

Received September 28, 2010; Revised January 12, 2011. Communicated by E. Puczylowski.Address correspondence to Prof. G. Militaru, Faculty of Mathematics and Computer

Science, University of Bucharest, Str. Academiei 14, RO-010014 Bucharest 1, Romania; E-mail:[email protected] and [email protected]

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The classical definition of representable functors corresponding to �et-valuated functors was generalized [1] replacing the category of sets with a variety ofalgebras � in the sense of universal algebras (for example, � can be the category ofsemigroups, monoids, groups, abelian groups, rings, algebras over commutative ringor modules over a ring). At this level, a functor F � � → � is called representableif � � F � � → �et is representable in the classical sense, where � � � → �et is theforgetful functor. At first sight, this generalisation seems trivial. In reality, however,it poses great difficulties in the theory of representable functors. Let us take � acategory and � a variety of algebras. There are four general problems concerningrepresentable functors.

Problem A: Describe, whenever possible, the category Rep���� � of allrepresentable functors F � � → � .

Problem B: Give a necessary and sufficient condition for a given functor F �� → � to be representable (possibly predefining the object of representability).

Problem C: When is a composition of two representable functors a representablefunctor?

Problem D: Give a necessary and sufficient condition for a representable functorF � � → � and for its left adjoint1 to be separable or Frobenius.

All universal constructions mentioned in the beginning are in fact answers toProblem B in the trivial case � = �et. An excellent book dedicated exclusively toProblem A is [1] where the category Rep���� � is described for different categoriesof varieties of algebras � and � . In the case that � has finite coproducts theProblem A is equivalent to description of the opposite of the category of so calledco-� objects of � [1, Corollary 8.13]. In general, the problem is difficult and theresults can be very interesting and surprising: moreover, as explained in [1], ProblemA has relevant applications in the study of fundamental groups of topologicalspaces, operator algebras, connected graded algebras, probability distributions,Hopf algebras, etc. The pioneer of studying Problem A was Kan, who described allrepresentable functors from semigroups to semigroups [16]. A crucial step relatedto problem A was made by Freyd in his seminal article [13]: if � is a cocompletecategory and � a variety of algebras then a functor F � � → � is representableif and only if F is a right adjoint ([1, Theorem 8.14]). Abstract properties of thecategory Rep���� � are proven in the recent article [2] and in Section 1 we shallgive more examples and motivations for each of the problems above.

The study of corings and their corepresentations has generated an explosionof interest and has become a distinctive research area after [5] was published, for atleast two reasons. On one hand, the category of comodules over a coring is a verygeneral one. Different types of categories like modules over a ring, comodules overa coalgebra, descent data associated to a ring extension, Hopf modules and relativeHopf modules, Doi–Koppinen modules, entwined modules, or Yetter–Drinfel’dmodules are all special cases of comodules over various corings. Thus any resultproven for the category of comodules over a coring is also applicable for all

1A representable functor F has a left adjoint if the object of representability of F has copowers.

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these particular types of categories. On the other hand, working with comodulesover corings is simple and most proofs are natural and transparent. For moredetails about the importance of coring theory and its applications we refer to themonograph [6].

In this article, we shall give answers to all the above problems in case � = CR�,

the category of left C-comodules over an R-coring C and � = S�, the categoryof left S-modules over a ring S. The article is organized as follows. In Section 1,we recall the basic concepts that will be used throughout the article and givemore examples and motivations of the above problems. Theorem 1.5 is a structuretheorem that gives a first answer to Problem D: it describes all functors that areFrobenius and separable from an abelian category to the category of abelian groups.Representable functors having an object of representability with special propertiesplay the key role in this picture. Section 2 collects all technical results that we shalluse to prove the main theorems of the article. Let R, S be rings and C an R-coring.We are focusing on the large categories Functors

(S�� CR�

)and Functors

(CR�� S�

)of all covariant functors that connect the category of comodules over C and thecategory of modules over S. Three Yoneda type embeddings are constructed. Theclasses of all natural transformations between an induction functor and the identityfunctor on the category C

R� are explicitly computed. Section 3 contains the mainresults of the article. Theorem 3.1 gives an answer for Problem A: the categoryRep�CR�� S�� is equivalent to the opposite of the category C

R�S . Corollary 3.4 offersan answer for Problem C.

Let U be a �S� R�-bimodule and the induction functor U ⊗R − � CR� →

S�. Theorem 3.9 gives necessary and sufficient conditions for U ⊗R − to be arepresentable functor, i.e., an answer for Problem B. It generalizes and unifiestwo theorems that at first glance have nothing in common: [18, Theorem 2.1] isrecovered for the trivial coring C = R, while [5, Theorem 4.1] is obtained as aparticular case for U = S = R if in addition to that we impose and predefine Cto be the object of representability of R⊗R −. Example 3.5 and Corollary 3.10explain that various theorems [10, Theorem 2.4], [5, Theorem 4.1], etc. givingnecessary and sufficient conditions for a forgetful functor to be Frobenius areparticular cases of representability. As a bonus of our approach, Theorem 3.6 givesnecessary and sufficient conditions for U ⊗R − to be an equivalence of categories.The Morita equivalence between two categories of modules is recovered as a specialcase corresponding to the trivial coring C = R. Finally, Corollaries 3.11 and 3.15give necessary and sufficient conditions for two types of induction functors to beseparable functors in the case that there exists what we have called a comodule dualbasis of first (or second) kind. Moreover, (2) of Corollary 3.11 is in fact an answerfor Problem D, concerning the separability of a representable functor.

1. PRELIMINARIES

We denote by �et the category of sets and �b the category of abelian groups.All functors in this article will be covariant functors. �op will be the opposite ofa category �. Except the category �� below all categories are locally small, thatis, Hom��C�D� is a set for any objects C, D ∈ �. We denote by Nat�F�G� theclass of all natural transformations between two functors F , G � � → � and by

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Functors����� = �� the large category of all functors F � � → �. The morphismsbetween two functors F , G ∈ �� are all natural transformations � � F → G.

Let R, S be two rings. We denote by R�, �S , R�S the categories ofleft R-modules, right S-modules, �R� S�-bimodules. RHom�M�N�, HomS�M�N�,RHomS�M�N� will be the morphisms in the respective categories. For two �R� S�-bimodules P, Q ∈ R�S , we have that P ⊗S − � Q⊗S − (natural isomorphism) if andonly if P � Q (isomorphism of �R� S�-bimodules). For an R-bimodule M , we denoteby MR = �m ∈ M � rm = mr�∀r ∈ R� the set of R-centralized elements.

Representable Functors

A covariant functor F � � → �et is called representable if there exists C ∈ �,called the representing (universal or generic) object of F , such that F � Hom��C�−�in �et �. Rep����et� will be the full subcategory of Functors����et� of allrepresentable functors. The Yoneda lemma states that for any functor F � � → �etand C ∈ � the map

� Nat�Hom��C�−�� F� → F�C�� ��� �= �C�IdC� (1)

is a bijection between sets with the inverse given by

−1�x�D�f� �= F�f��x� (2)

for all x ∈ F�C�, D ∈ �, and f ∈ Hom��C�D�. As a consequence, the functor calledYoneda embedding,

Y � �op → Functors����et�� Y�C� �= Hom��C�−�� Y�f� �= Hom��f�−�(3)

for all C, D ∈ � and f ∈ Hom��C�D� is faithful and full. Thus, using thecharacterization of equivalences of categories [17], we obtain that the Yonedaembedding gives an equivalence of categories

�op � Rep����et�� C �→ Hom��C�−�

The above definition of representable functors is the classical one: it can begeneralized replacing the category �et to a variety of algebras � in the sense ofuniversal algebra [1] (for example, � can be the category of groups, abelian groups,rings, k-algebras, R�, etc.).

Definition 1.1. Let � be a variety of algebras and � � � → �et the forgetfulfunctor. A functor F � � → � is called representable if � � F � � → �et isrepresentable in the classical sense.

Let � be a category and � a variety of algebras. We shall give some examplesrelated to Problems A, B, C from the Introduction. The Yoneda lemma gives a firstanswer for Problem A in the trivial case � �= �et. In the following, we shall give asan example one of the fundamental theorems of [1]. Let R be a ring; an R-ring is a

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semigroup in the monoidal category �R�R�−⊗R −� of R-bimodules. Let R−�ingsbe the category of R-rings with R-bimodule maps that respect the multiplications onR-rings as morphisms. Then the functor

� � �R�R�op → Rep�R−�ings��b�� ��M� �= RHomR�M�−�

is an equivalence of categories ([1, Theorem 13.15]).Concerning Problem B, besides the universal constructions in mathematics

that give answers for it in the trivial case � = �et, we shall indicate two moreexamples. The first one was proven by Morita in [18, Theorem 2.1]: for a �S� R�-bimodule V the induction functor V ⊗R − � R� → S� is representable if and onlyif V is finitely generated projective as a right R-module. As explained in theIntroduction, the property of a functor to be Frobenius can be restated moreelegantly as a representability problem, predefining the object of representability.For instance, [10, Theorem 4.2] can be restated as follows: Let H be a Hopf algebraover a field and H

H�� be the category of Yetter–Drinfel’d over H . Then the forgetfulfunctor F � H

H�� → H� is representable having H ⊗H as a representing object ifand only if H is a finite dimensional and unimodular Hopf algebra.

Problem C has a positive answer for categories of modules: the tensor productof bimodules is responsible for this. Indeed, let R, S, and T be rings, U ∈ S�R,V ∈ R�T , and consider the representable functors SHom�U�−� � S� → R� andRHom�V�−� � R� → T�. Then

RHom�V�−� � SHom�U�−� � SHom�U ⊗R V�−�

For arbitrary categories, even though the statement of the Problem C is elementary,the answer is difficult to give.

Frobenius and Separable Functors

Let F � � → �, G � � → � be two functors. F is a left adjoint of G and wedenote this by F G if there exist two natural transformations � � 1� → GF and� � FG → 1�, called the unit and counit of the adjunction, such that

G��D� � �G�D� = IG�D� and �F�C� � F��C� = IF�C� (4)

for all C ∈ � and D ∈ �. The following lemma was stated in [15, p. 609, Exercise 8]for abelian categories; however, it holds for additive categories.

Lemma 1.2. Let �, � be additive categories and F � � → �, G � � → � such thatF is a left adjoint of G. Then F and G are additive functors.

Proof. F is a left adjoint, thus it preserves all coproducts. In particular, F preservesfinite coproducts between two additive categories. It follows from [3, Proposition1.3.4] that F is an additive functor. Using [17, Theorem 3, p. 85] we obtain thatG is also an additive functor (alternatively, we can view G � �op → �op as a leftadjoint of F � �op → �op and the opposite of an additive category is also an additivecategory). �

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A functor F � � → � is called a Frobenius functor [10] if there exists afunctor G that is a left and right adjoint of F . F is called separable if the naturaltransformation associated to F

� Hom��•� •� → Hom��F�•�� F�•��� C�C′�f� = F�f�

splits. Let F G be an adjoint pair. Then F is separable if and only if � � 1� → GFsplits: i.e. there exists a natural transformation � GF → 1� such that C � �C = IdC ,for all C ∈ �. Moreover, G is separable if and only if � � FG → 1� cosplits, i.e.,there exists a natural transformation � � 1� → FG such that �D � �D = IdS , for allD ∈ �. If i � R → S is a morphism of rings, then the restriction of scalars functori∗ � �S → �R is Frobenius (resp., separable) if and only if S/R is a Frobenius(resp., separable) extensions of rings in the classical sense [11]. For details and moreexamples of Frobenius or separable functors we refer to [6, 11].

Here we give some examples for Problem D. The first two are the extremecases of the problem: in the first example, any representable functor is separable,while in the second example none is. The examples below show that Problem Dessentially depends on the nature of categories � and � .

Examples 1.3.

1. Any representable functor Hom�et�A�−� � �et → �et is separable. Indeed,Hom�et�A�−� has a left adjoint A×−: the counit of the adjunction A×− Hom�et�A�−� is the evaluation map

�Y � A×Hom�et�A� Y� → Y� �Y �a� f� �= f�a�

for all Y ∈ �et, a ∈ A, and f ∈ Hom�et�A� Y�. Then � cosplits: fix a0 ∈ A and fory ∈ Y . Let cy � A → Y be the constant function cy�a� �= y, for all a ∈ A. Then

�Y � Y → A×Hom�et�A� Y�� �Y �y� �= �a0� cy�

for all Y ∈ �et, y ∈ Y is a natural transformation that cosplits �.2. Let � = rf be the category of finite groups. Then no representable functor

Homrf �G�−� � rf → �et is separable.

Indeed, assume that the functor Homrf �G�−� is separable, for a finite group G.Then it is a faithful functor, and hence G is a generator [17] in the category of finitegroups, which is a contradiction (rf has not a generator [20]).

3. Let R, S be two rings, U an �R� S�-bimodule, and ∗U �= RHom�U�R� ∈ S�R. Thenthe representable functor RHom�U�−� � R� → S� is separable if and only if thereexists

∑i ui ⊗S u

∗i ∈ �U ⊗S

∗U�R such that∑

i u∗i �ui� = 1R [9, Corollary 5.8].

We recall that an object C ∈ � has arbitrary copowers if for any set X thereexists the coproduct C�X� �= ⊕x∈XCx in �, where Cx = C, for all x ∈ X. The nextresult is folklore being just an interpretation of the bijection of the adjunction readas definition of the coproduct.

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Lemma 1.4. Let � be a category and C ∈ �. The representable functorHom��C�−� � � → �et has a left adjoint if and only if C has arbitrary copowers. Inthis case, the left adjoint of Hom��C�−� is the functor

C�−� � �et → �� X �→ C�X�

for all X ∈ �et.

Now we can give an answer to Problem D for functors that are at the sametime Frobenius and separable, evidencing the role played by representable functors.

Theorem 1.5. Let � be an abelian category and F � � → �b be a functor. Thefollowing are equivalent:

(1) F is a Frobenius and separable functor;(2) F � Hom��C�−�, for some C ∈ � having the following properties:

(a) C is a small, projective, generator and has all copowers in �;(b) End��C� is Frobenius and separable as a �-algebra.

Proof. For any object C ∈ �, we have the commutative diagram of functors

where T is the restriction of scalars functor associated to the ring extension � →End��C�.

�1� ⇒ �2� Assume that F � � → �b is Frobenius and separable. Inparticular, it has a left adjoint H � �b → �. If follows from Gabriel’s theorem [14,Chapter V] that there exists C ∈ � (namely, C �= H���) such that F � Hom��C�−�.As F � Hom��C�−� has a left adjoint it follows from Lemma 1.4 that C has allcopowers. On the other hand, F � Hom��C�−� is Frobenius: in particular, it isa left and right adjoint. Thus it is an exact functor, i.e., C is projective object in�. Being a left adjoint the functor Hom��C�−� preserves coproducts, i.e., C is asmall object in �. Now, F � Hom��C�−� is separable, thus it is faithful, that is,C is a generator in �. It follows from Mitchell’s theorem [12] that the functorHom��C�−� � � → �End��C�

is an equivalence of categories. Thus the restriction ofscalars functor T is Frobenius and separable as a composition of Frobenius andseparable functors, i.e., the ring extension � → End��C� is Frobenius and separablein the classical sense.

�2� ⇒ �1� The restriction of scalars functor T is a Frobenius and separablefunctor as so is the ring extension � → End��C� [11]. As C is a small, projectivegenerator and has all copowers in �, we can use once again the Mitchell theorem[12]: thus the functor Hom��C�−� � � → �End��C�

is an equivalence of categories,

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and hence it is a Frobenius and separable functor. It follows that Hom��C�−� �� → �b is Frobenius and separable.

Corings and Comodules

Let R be a ring and C = �C��� �� an R-coring; i.e., C is a comonoid in themonoidal category of R-bimodules �R�R�−⊗R −� R�. We denote by �C

R ,CR�, and

CR�

CR the cocomplete categories of right, left, respectively, C-bicomodules. We denote

by HomCR�M�N�, C

RHom�M�N�, and CRHomC

R�M�N� the set of all morphisms in thecategories of right, left, and respectively, C-bicomodules, for two C-comodules Mand N . A right C-coaction will be denoted by

� � M → M ⊗R C� ��m� = m�0� ⊗R m<1>

for all M ∈ �CR and m ∈ M and a left C-coaction will be denoted by

� � M → C ⊗R M� ��m� = m�−1� ⊗R m�0�

for all M ∈ CR� and m ∈ M (summation understood). The categories �C

R ,CR� and

CR�

CR are additive and cocomplete (they have all coproducts and coequalizers [6,

Proposition 18.13])Different types categories like modules over a ring, descent datum, associated

to a ring extension, comodules over a coalgebra, relative Hopf modules, Doi–Koppinen modules, or Yetter–Drinfel’d modules are special cases of comodules overvarious corings [6, 11]. For instance, if H is a Hopf algebra with an antipode S, thenH ⊗H has an H-coring structure via

l · �g ⊗ h� �= lg ⊗ h� �g ⊗ h� · l �= gl�2� ⊗ S�l�1��hl�3�

��g ⊗ h� �= g ⊗ h�1� ⊗H 1H ⊗ h�2�� ��g ⊗ h� �= �H�h�g

for all l, g, h ∈ H , and there exists an equivalence of categories

�H⊗HH � ��H

H�

where ��HH is the category of right-right Yetter–Drinfel’d modules over H .

Let R, S be two rings, C an R-coring, and CR�S be the category of all pairs

�V� �V �, where V is an �R� S�-bimodule, �V � V → C ⊗R V is a morphism of �R� S�-bimodules and a left C-coaction on V . For two objects V , W ∈ C

R�S , we denote byCRHomS�V�W� the set of morphisms in the category C

R�S , i.e., the set of all �R� S�-bimodule maps f � V → W that are also left C-comodule maps. The category S�

CR

is defined similarly.Let V ∈ C

R�S . Then we have two functors [7, Proposition 6]

V ⊗S − � S� → CR�� C

RHom�V�−� � CR� → S��

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where V ⊗S N ∈ CR� via r · �v⊗S n� �= rv⊗S n, ��v⊗S n� �= v�−1� ⊗R v�0� ⊗S n, for

all N ∈ S�, r ∈ R, v ∈ V , and n ∈ N and CRHom�V�M� ∈ S� via �s · f��v� �= f�vs�,

for all M ∈ CR�, s ∈ S, f ∈ C

RHom�V�M�, and v ∈ V . Indeed, using the fact that theleft C-coaction �V � V → C ⊗R V is also a right S-module map we can easily provethat s · f is a C-colinear map, for all s ∈ S and f ∈ C

RHom�V�M�.The following is the left version of [21, Theorem 3.2], proved over a firm ring

S, as a generalization of the classical the Eilenberg–Watts theorem for categories ofmodules. For the convenience of the reader we present a different and direct proofof �1� ⇒ �2�.

Theorem 1.6. Let R, S be two rings, C an R-coring, and F and G two functors

F � S� → CR�� G � C

R� → S�

The following are equivalent:

(1) F is a left adjoint of G;(2) There exists V ∈ C

R�S , unique up to an isomorphism in CR�S , such that

F � V ⊗S −G � CRHom�V�−�

(natural isomorphisms of functors).

Proof. �2� ⇒ �1� This is the left version of [7, Proposition 6]. The unit and thecounit of the adjunction V ⊗S − C

RHom�V�−� are given by

�N � N → CRHom�V� V ⊗S N�� �N �n��v� �= v⊗S n

for all N ∈ S�, n ∈ N , and v ∈ V and

�M � V ⊗SCRHom�V�M� → M� �M�v⊗S f� �= f�v�

for all M ∈ CR�, f ∈ C

RHom�V�M�, and v ∈ V .

�1� ⇒ �2� F is a left adjoint of G between two additive categories. It followsfrom Lemma 1.2 that they are additive functors and using [17, Theorem 3, p. 85]the adjunction bijections

�X�Y � CRHom�F�X�� Y� → SHom�X�G�Y��

are isomorphisms of abelian groups for all X ∈ S� and Y ∈ CR�.

Let V �= F�S� ∈ CR�. We shall prove that V ∈ C

R�S and G � CRHom�V�−�.

Indeed, let s ∈ S and fs � S → S, fs�x� = xs, for all x ∈ S. Then fs is amorphism in S�, and thus F�fs� � V → V is a morphism in C

R�, i.e.,

F�fs��v��−1� ⊗R F�fs��v��0� = v�−1� ⊗R F�fs��v�0�� (6)

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REPRESENTABLE FUNCTORS FOR CORINGS 1775

for all v ∈ V . We shall first prove that V ∈ CR�S , where the right S-action is given by

v · s �= F�fs��v� (7)

for all v ∈ V , s ∈ S. It is easy to show that �V� ·� is a right S-module: for example,using the fact that F is an additive functor, we have

v · �s + s′� = F�fs+s′��v� = F�fs + fs′��v�

=(F�fs�+ F�fs′�

)�v� = F�fs��v�+ F�fs′��v� = v · s + v · s′

for all v ∈ V , s, s′ ∈ S. Moreover, as F�fs� is left R-linear, for any r ∈ R, v ∈ V , ands ∈ S, we have

r�v · s� = rF�fs��v� = F�fs��rv� = �rv� · s

i.e., V is an �R� S�-bimodule. Let us show that the left C-coaction � � V → C ⊗R

V is a right S-module map, i.e., ��v · s� = ��v� · s, for all v ∈ V and s ∈ S. This isequivalent to

�(F�fs��v�

) = v�−1� ⊗R F�fs��v�0���

which is exactly (6). Hence, V ∈ CR�S , and there exists an isomorphism of abelian

groups, given by the adjunction

�Y �= �S�Y � CRHom�V� Y� → SHom�S�G�Y��

that is natural in any Y ∈ CR�. Using the fact that �S�− = � is a natural

transformation, we shall prove that �Y is an isomorphism of left S-modules, whereSHom�S�G�Y�� is a left S-module via

�s · ���x� �= ��xs�

for all Y ∈ CR�, s ∈ S, � ∈ SHom�S�G�Y��, and x ∈ S.

Indeed, for any s ∈ S, the following diagram

is commutative, i.e.,

�Y

(� � F�fs�

) = �Y ��� � fs (8)

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1776 MILITARU

for all � ∈ CRHom�V� Y�, s ∈ S. Now,

(� � F�fs�

)�v� = �

(F�fs��v�

) = ��v · s� = �s · ���v��

for all � ∈ CRHom�V� Y�, v ∈ V , i.e., � � F�fs� = s · � and

(�Y ��� � fs

)�x� = �Y ����xs� =

(s · �Y ���

)�x�

for all � ∈ CRHom�V� Y�, x, s ∈ S, i.e., �Y ��� � fs = s · �Y ���. Hence (8) can be written

as �Y �s · �� = s · �Y ���, for all s ∈ S and � ∈ CRHom�V� Y�, and we have proven that

�Y � CRHom�V� Y� → SHom�S�G�Y��

is an isomorphism of left S-modules which is natural in any Y ∈ CR�. Of course,

there exists a natural isomorphism of functors

SHom�S�G�−�� � G

from CR� to S�. Thus we have proved that there exists a natural isomorphism of

functors

G � CRHom�V�−�

On the other hand, V ⊗R − is a left adjoint of CRHom�V�−�. Using Khan’s adjoint

uniqueness theorem [17], we obtain that there exists also a natural isomorphism offunctors F � V ⊗R −. Finally, Proposition 2.1 will show that the functor

Y1 �CR�S → Functors�

(S�� CR�

)� Y1�V� �= V ⊗S −

for all V ∈ CR�S is faithful and full, and thus it preserves and reflects the

isomorphisms. Hence, V is unique up to an isomorphism in CR�S , and the proof is

complete. �

2. COMPUTING NATURAL TRANSFORMATIONS AND YONEDA TYPEEMBEDDINGS

In this section we shall prove all technical results that we shall futher useto prove the main results of the article. Let R, S be two rings, C an R-coring,V , W ∈ C

R�S , and f � V → W a morphism in CR�S . We associate to f two natural

transformations:

f ⊗S − � V ⊗S − → W ⊗S −� v⊗S n �→ f�v�⊗S n

for all N ∈ S�, n ∈ N , v ∈ V , and

CRHom�f�−� � C

RHom�W�−� → CRHom�V�−�� � �→ � � f

for all M ∈ CR�, � ∈ C

RHom�W�M�.

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Proposition 2.1. Let R, S be two rings, C an R-coring. Then:

(1) The functor

Y1 �CR�S → Functors

(S�� CR�

)� Y1�V� �= V ⊗S −

for all V ∈ CR�S is faithful and full;

(2) The functor

Y2 � �CR�S�

op → Functors�(CR�� S�

)� Y2�V� �= C

RHom�V�−�

for all V ∈ CR�S is faithful and full.

Proof. 1. Let V , W ∈ CR�S . We have to prove that

�Y1�V�W � CRHomS�V�W� → Nat

(V ⊗S −�W ⊗S −

)� �Y1�V�W �f� = f ⊗S −

for all f ∈ CRHomS�V�W� is a bijection between sets.

Let � � V ⊗S − → W ⊗S − be a natural transformation. In particular, �S �V ⊗S S → W ⊗S S is a morphism in C

R�. We define f � V → W by the formulaf �= can′ � �S � can, where can � V → V ⊗S S and can′ � W ⊗S S → W are canonicalisomorphisms. Of course, f is a morphism in C

R�. Using the fact that � is anatural transformation, we shall prove that f is also a right S-module map, andhence a morphism in C

R�S and � is uniquely determined by f with the formula� = �Y1�V�W �f�.

Let N ∈ S� and n ∈ N . Then un � S → N , un�s� �= sn is a morphism in S�.Thus the diagram

is commutative. We evaluate at v⊗S 1S , and we obtain that

�N�v⊗S n� = f�v�⊗S n

for all N ∈ S�, v ∈ V , and n ∈ N . In particular, for N �= S, we obtain that f is alsoa right S-module map and the above formula tells us that � = �Y1�V�W �f�.

2. Let V , W ∈ CR�S . We have to prove that

�Y2�V�W � CRHomS�V�W�→Nat(CRHom�W�−�� CRHom�V�−�

)� �Y2�V�W �f�= C

RHom�f�−�

for all f ∈ CRHomS�V�W� is a bijection between sets with the inverse given by

�Y−12 �V�W ��� = �W�IdW� � V → W

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1778 MILITARU

for any natural transformation � � CRHom�W�−� → C

RHom�V�−�. This follows fromthe Yoneda lemma if we replace the category �et with the category of left S-modules. The only two things we have to prove are that the maps (1), (2) fromthe Yoneda lemma work properly. More precisely, we note that if M ∈ C

R� and f ∈CRHomS�V�W�, then we can easily show that Y2�f�M � C

RHom�W�M� → CRHom�V�M�,

Y2�f�M��� = � � f , for all � ∈ CRHom�W�M� is a morphism of left S-modules. Finally,

if � � CRHom�W�−� → C

RHom�V�−� is a natural transformation, we have to provethat �W�IdW� � V → W is also a right S-module map, hence a morphism in C

R�S . Weshow that � is a natural transformation. Let s ∈ S and �s � W → W , �s�w� �= ws, forall w ∈ W . Then �s is a morphism in C

R�, thus we have a commutative diagram

Now, if we evaluate the diagram at IdW , we obtain that

�W��s� = �s � �W�IdW�

For any v ∈ V , we have

�W��s��v� = �W�s · IdW��v� =(s · �W�IdW�

)�v� = �W�IdW��vs�

and(�s � �W�IdW�

)�v� = �W�IdW��v�s�

hence �W�IdW� is also a right S-module map, and the proof is finished. �

Let U ∈ S�R and V ∈ CR�S . Then U ⊗R C ∈ S�

CR via the right C-coaction

u⊗R c �→ u⊗R c�1� ⊗R c�2�

for all u ∈ U , c ∈ C, and V ⊗S U ⊗R C ∈ CR�

CR , where the left and the right

C-coactions are defined by

v⊗S u⊗R c �→ v�−1� ⊗R v�0� ⊗S u⊗R c� v⊗S u⊗R c �→ v⊗S u⊗R c�1� ⊗R c�2�

for all v ∈ V , u ∈ U , and c ∈ C. Moreover, CRHom�V� C� ∈ S�R, where the right

R-action is given by

�f · r��v� �= f�v�r

for all f ∈ CRHom�V� C�, r ∈ R, v ∈ V . We can construct two functors

U ⊗R − � CR� → S�

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REPRESENTABLE FUNCTORS FOR CORINGS 1779

and

?⊗R − � S�R → Functors(CR�� S�

)� U �→ U ⊗R −� f �→ f ⊗R −

Contrary to (1) of Proposition 2.1 the functor ?⊗R − is not faithful and full. Inorder to make it faithful and full we have to change the category S�R as follows.Let S�

•R be the category whose objects are �S� R�-bimodules and the morphisms are

defined by

HomS�

•R�U� U ′� �= SHomC

R�U ⊗R C�U ′ ⊗R C�

for all U , U ′ ∈ S�•R. If � � U ⊗R C → U ′ ⊗R C is a map in S�

•R, we shall denote

��u⊗R c� = ∑u� ⊗R c� ∈ U ′ ⊗R C

for all u ∈ U , c ∈ C. We define a natural transformation between two inductionfunctors

Y3��� � U ⊗R − → U ′ ⊗R −�(Y3���

)M�u⊗R m� �= ∑

u� ⊗R �C

(�m�−1��

�)m�0� (9)

for all M ∈ CR�, u ∈ U , m ∈ M . It is easy to prove that Y3��� is a natural

transformation.

Theorem 2.2. Let R, S be two rings, C an R-coring. Then the functor

Y3 � S�•R → Functors

(CR�� S�

)� U �→ Y3�U� �= U ⊗R −� � �→ Y3���

for all U , U ′ ∈ S�•R and � ∈ Hom

S�•R�U� U ′� is faithful and full.

Proof. Let U , U ′ ∈ S�•R. We have to prove that the map

SHomCR�U ⊗R C�U ′ ⊗R C� → Nat

(U ⊗R −� U ′ ⊗R −)

� � �→ Y3���

is a bijection between sets. Let � � U ⊗R − → U ′ ⊗R − be a natural transformation.Hence, �M � U ⊗R M → U ′ ⊗R M is a morphism in S� for all M ∈ C

R�. In particular,�C � U ⊗R C → U ′ ⊗R C is a left S-module map. Using that � is a naturaltransformation, we shall prove that �C is a map in S�

CR and � = Y3��C�.

Let r ∈ R and fr � C → C, fr�c� �= cr, for all c ∈ C. Then fr is a morphism inCR�, and hence the diagram

is commutative, i.e., �C is also a right R-module map.

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1780 MILITARU

Let N ∈ R� and n ∈ N fixed. Then fn � C → C ⊗R N , fn�c� �= c ⊗R n, for allc ∈ C is a map in C

R�, where C ⊗R N ∈ CR� via the left C-coaction c ⊗R n �→ ��c�⊗R

n. Thus the diagram

is commutative, which means that

�C⊗RN= �C ⊗R IdN

for all N ∈ R�. Thus for any cofree left C-comodule M = C ⊗R N , �M is uniquelydetermined by �C . Now � � C → C ⊗R C is a morphism in C

R�, where C ⊗R C is aleft C-comodule via the cofree structure: c ⊗R d �→ ��c�⊗R d. Thus the diagram

is commutative. As �C⊗RC= �C ⊗R IdC we obtain that �C is a right C-comodule map,

i.e., we have proved that �C is a morphism in S�CR , that is, using our notation

∑u� ⊗R �c���1� ⊗R �c���2� = �C�u⊗R c�1��⊗R c�2� (10)

for all u ∈ U and c ∈ C. If we apply �C to the second position, we get

�C�u⊗R c� = ∑u� ⊗R ���c���1��c�2� (11)

for all u ∈ U and c ∈ C. Conversely, (10) follows immediately from (11). Let nowM ∈ C

R� with the left C-coaction � � M → C ⊗R M which is a morphism in CR�.

Hence the diagram

is commutative. As �C⊗RM= �C ⊗R IdM , we obtain

∑u� ⊗R �m���−1� ⊗R �m���0� = �C�u⊗R m�−1��⊗R m�0�

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REPRESENTABLE FUNCTORS FOR CORINGS 1781

We apply �C to the second position, and we get

�M�u⊗R m� = ∑u� ⊗R �C

(�m�−1��

�)m�0�

Thus we have proved that � = Y3��C�, that is, the map � �→ Y3��� is surjective. It isalso obviously injective, and hence the proof is finished. �

Corollary 2.3. Let R be a ring, C an R-coring, and U ∈ R�R. The following areequivalent:

(1) The induction functor U ⊗R − � CR� → R� is isomorphic to the forgetful functor

F � CR� → R�;

(2) There exists an isomorphism U ⊗R C � C in the category R�CR;

(3) There exist two R-bimodule maps p � C → U and h � U ⊗R C → R such that

h(p�c�1��⊗R c�2�

)= ��c�� h�u⊗R c�1��p�c�2�� = u��c� (12)

for all c ∈ C, u ∈ U .

Moreover, in this case, if C contains an element c such that ��c� is right invertible inR, then U is finitely generated and projective as a left R-module.

Proof. �1� ⇔ �2� We shall apply Theorem 2.2 for S = R. The functor Y3 is faithfuland full; hence, it preserves and reflects the isomorphisms. The forgetful functor Fis naturally isomorphic to the induction functor R⊗R −. Thus we obtain that U ⊗R

− � F if and only if U ⊗R − � R⊗R − in the category Functors(CR�� R�

). As Y3 is

a faithful and full functor, this is equivalent to the fact that U ⊗R C � R⊗R C = Cin the category R�

CR .

�2� ⇔ �3� The proof is given by the parametrization of (iso)morphisms inR�

CR between the objects C and U ⊗R C given by Hom-tensor type relations [6].

More precisely, the map

� � RHomR�U ⊗R C�R� → RHomCR�U ⊗R C�C�� ��h��u⊗R c� �= h�u⊗R c�1��c�2�

for all h ∈ RHomR�U ⊗R C�R�, u ∈ U , and c ∈ C is bijective with the inverse givenby

�−1�f��u⊗R c� �= �C

(f�u⊗R c�

)

for all f ∈ RHomCR�U ⊗R C�C�, u ∈ U , and c ∈ C. On the other hand, the map

� � RHomR�C�U� → RHomCR�C�U ⊗R C�� ��p��c� �= p�c�1��⊗R c�2�

for all p ∈ RHomR�C�U�, c ∈ C is bijective with the inverse

�−1�g� �= �IdU ⊗R �C� � g

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1782 MILITARU

for all g ∈ RHomCR�C�U ⊗R C�. We can easily check that two morphisms ��h� and

��p� in R�CR are inverse to each other if and only if (12) holds.

For the last statement, if ��c� is right invertible in R, then�p�c�2����c�

−1� h�?⊗R c�1��� is a pair of dual basis for U ∈ R�. �

In the next two lemmas we shall compute all natural transformations betweenan induction functor and the identity functor on the category C

R� of left C-comodules.

Lemma 2.4. Let R be a ring, C an R-coring, Z ∈ CR�R, and the induction functor

Z ⊗R − � CR� → C

R�. Then there exists a bijection between sets

Nat(1C

R�� Z ⊗R −)� C

RHomCR�C� Z ⊗R C� � C

RHomR�C� Z�

Explicitly, for any natural transformation � � 1CR�

→ Z ⊗R − there exists a unique mapp ∈ C

RHomR�C� Z� such that

�M � M → Z ⊗R M� �M�m� = p�m�−1��⊗R m�0� (13)

for all M ∈ CR� and m ∈ M .

Proof. Let � � 1CR�

→ Z ⊗R − be a natural transformation. In particular, �C � C →Z ⊗R C is a morphism in C

R�. We shall prove that �C is in fact a morphism in CR�

CR

and � is uniquely determined by �C .Let r ∈ R and fr � C → C, fr�c� �= cr, for all c ∈ C. Then fr is a morphism in

CR�, and hence the diagram

is commutative, i.e., �C is also a right R-module map.Let N ∈ R� and n ∈ N fixed. Then fn � C → C ⊗R N , fn�c� �= c ⊗R n, for all

c ∈ C is a map in CR�. Thus the diagram

is commutative, which means that

�C⊗RN= �C ⊗R IdN

for all N ∈ R�.

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REPRESENTABLE FUNCTORS FOR CORINGS 1783

On the other hand, � � C → C ⊗R C is a morphism in CR�. Thus the diagram

is commutative. As �C⊗RC= �C ⊗R IdC , we obtain that �C is a right C-comodule

map, i.e., we have proven that �C is a morphism in CR�

CR .

Let �M� �� ∈ CR�; then � � M → C ⊗R M is a morphism in C

R�, so the diagram

is commutative. We shall temporarily denote �M�m� = ∑mZ ⊗R mM . As �C⊗RM

=�C ⊗R IdM , we obtain

∑mZ ⊗R �mM��−1� ⊗R �mM��0� = �C�m�−1��⊗R m�0� (14)

for all m ∈ M . If we apply �C to the second position in (14), we obtain

�M�m� = �IdZ ⊗R �C���C�m�−1��⊗R m�0�� (15)

for all m ∈ M , and hence �M is uniquely determined by �C via formula (15); hencethe first bijection is proven.

The second bijection is given by Hom-tensor type relations [6]. More precisely,the map

� � CRHomR�C� Z� → C

RHomCR�C� Z ⊗R C�� ��p��c� �= p�c�1��⊗R c�2� (16)

for all p ∈ CRHomR�C� Z�, c ∈ C is bijective with the inverse

�−1�g� �= �IdZ ⊗R �C� � g

for all g ∈ CRHomC

R�C� Z ⊗R C�. Thus any �C ∈ CRHomC

R�C� Z ⊗R C� has the form�C�c� = p�c�1��⊗R c�2�, for a unique p ∈ C

RHomR�C� Z�. With this presentation, theformula (15) takes the form (13), and the proof is complete. �

Similar to Lemma 2.4, we shall compute the class of all naturaltransformations Nat

(Z ⊗R −� 1C

R�

). For any object Z ∈ C

R�R, we denote by

•RHomR�Z ⊗R C�R�

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1784 MILITARU

the set of all R-bimodule maps h � Z ⊗R C → R satisfying the compatibilitycondition

z�−1�h�z�0� ⊗R c� = h�z⊗R c�1��c�2� (17)

for all z ∈ Z and c ∈ C.

Lemma 2.5. Let R be a ring, C an R-coring, Z ∈ CR�R, and the induction functor

Z ⊗R − � CR� → C

R�. Then there exists a bijection between sets

Nat(Z ⊗R −� 1C

R�

)� CRHomC

R�Z ⊗R C�C� � •RHomR�Z ⊗R C�R�

Explicitly, for any natural transformation � � Z ⊗R − → 1CR�

, there exists a uniquemap h ∈ •

RHomR�Z ⊗R C�R� such that

�M � Z ⊗R M → M� �M�z⊗R m� = h�z⊗R m�−1��m�0� (18)

for all M ∈ CR�, m ∈ M , and z ∈ Z.

Proof. The last bijection follows from Hom-tensor type relations [6]. Moreprecisely, the map

� � •RHomR�Z ⊗R C�R� → C

RHomCR�Z ⊗R C�C�� ��h��z⊗R c� �= h�z⊗R c�1��c�2�

for all h ∈ •RHomR�Z ⊗R C�R�, z ∈ Z, and c ∈ C is bijective with the inverse given

by

�−1�f��z⊗R c� �= �C

(f�z⊗R c�

)

for all f ∈ CRHomC

R�Z ⊗R C�C�, z ∈ Z, and c ∈ C.Let now � � Z ⊗R − → 1C

R�be a natural transformation. In particular, �C �

Z ⊗R C → C is a morphism in CR�. Using that � is a natural transformation, we shall

prove that �C is a morphism in CR�

CR and � is uniquely determined by �C . We shall

use the same steps from the proof of Lemma 2.4.Let r ∈ R and fr � C → C, fr�c� �= cr, for all c ∈ C. Then fr is a morphism in

CR�, and hence the diagram

is commutative, i.e., �C is also a right R-module map.

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Let N ∈ R� and n ∈ N . Then fn � C → C ⊗R N , fn�c� �= c ⊗R n, for all c ∈ Cis a map in C

R�. Thus the diagram

is commutative, which means that

�C⊗RN= �C ⊗R IdN

for all N ∈ R�.Let �M� �� ∈ C

R�; then � � M → C ⊗R M is a morphism in CR� so the diagram

is commutative. Using that �C⊗RM= �C ⊗R IdM , we obtain if we evaluate the last

diagram at z⊗R m

�(�M�z⊗R m�

)= �C�z⊗R m�−1��⊗R m�0� (19)

In particular, for �M� �� = �C���, we obtain that �C is a morphism in CR�

CR

and, if we apply �C to the first position in (19), we get

�M�z⊗R m� = �C

(�C�z⊗R m�−1��)m�0� (20)

for all M ∈ CR�, z ∈ C, m ∈ M , i.e., the first bijection from the statement.

Now, from the first part of the proof, for any �C ∈ CRHomC

R�Z ⊗R C�C� thereexists a unique map h ∈ •

RHomR�Z ⊗R C�R� such that �C�z⊗R c� = h�z⊗R c�1��c�2�,for all z ∈ Z and c ∈ C. Using this formula for �C , Eq. (20) takes the form (18), andthe proof is finished. �

Remarks 2.6. Using the same steps form the proof of Lemmas 2.4 and 2.5 wecan also compute the center of the category C

R�, i.e., the class of all naturaltransformations on the identity functor of C

R�. More precisely, we have

�(CR�

)�= Nat

(1C

R�� 1C

R�

)� CRHomC

R�C�C�

Having in mind the module case, this ring of all endomorphisms of C in the categoryof C-bicomodules can be called the center of the coring C.

Corollary 2.7. Let R be a ring, C an R-coring, and Z ∈ CR�R. The following are

equivalent:

(1) The induction functor Z ⊗R − � CR� → C

R� is isomorphic to the identity functor 1CR�

of the category CR�;

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1786 MILITARU

(2) There exists an isomorphism Z ⊗R C � C in the category CR�

CR of C-bicomodules;

(3) There exists a pair �p� h�, where p ∈ CRHomR�C� Z�, h ∈ •

RHomR�Z ⊗R C�R� suchthat

h(p�c�1��⊗R c�2�

)= ��c�� h�z⊗R c�1��p�c�2�� = u��c� (21)

for all c ∈ C, z ∈ Z.

Moreover, in this case, if C contains an element c such that ��c� is right invertible inR, then Z is finitely generated and projective as a left R-module.

Proof. �1� ⇒ �2� Let � � 1CR�

→ Z ⊗R − and � � Z ⊗R − → 1CR�

be a pair ofnatural transformations inverse each to other. From the proof of Lemmas 2.4and 2.5, �C and �C are isomorphisms inverse to each other between C and Z ⊗R Cin the category C

R�CR .

�2� ⇒ �3� The pair of maps �p� h� satisfying (21) parameterizes theisomorphisms between C and Z ⊗R C in the category C

R�CR using Hom-tensor type

relations from the proofs of Lemmas 2.4 and 2.5.

�3� ⇒ �1� Let �p� h� be such a pair of maps. Then

�M � M → Z ⊗R M� �M�m� = p�m�−1��⊗R m�0�

for all M ∈ CR� and m ∈ M is a natural isomorphism between the functors 1C

R�and

Z ⊗R − with the inverse

�M � Z ⊗R M → M� �M�z⊗R m� = h�z⊗R m�−1��m�0�

for all M ∈ CR�, m ∈ M , and z ∈ Z. �

3. REPRESENTABLE FUNCTORS FOR CORINGS: APPLICATIONS

In this section we shall use all technical results proven before in order to obtainthe main theorems of the article. First, we shall give an answer to Problem A.

Theorem 3.1. Let R, S be rings, C an R-coring, and Rep�CR�� S�� be the categoryof all representable functors C

R� → S�. Then the functor

Y � �CR�S�op → Rep�CR�� S��� Y�V� �= C

RHom�V�−�

is an equivalence of categories.

Proof. It follows from �2� of Proposition 2.1 that Y is a faithful and full functor.Let G ∈ Rep�CR�� S�� be a representable functor. C

R� is a cocomplete category,as it has all coproducts and coequalizers [6, Proposition 18.13]; thus we canapply Freyd’s theorem [1, Theorem 8.14] to obtain that G is a right adjoint.Using Theorem 1.6, we get that G � C

RHom�V�−� = Y�V�, for some V ∈ CR�S , i.e.,

Y is surjective on objects. It follows that Y is an equivalence of categories [17,Theorem 1, p. 93]. �

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Theorem 3.1 can be applied to all categories that are special cases ofcategories of comodules over a coring: modules, graded modules by G-sets, relativeHopf modules, Doi–Koppinen modules, Yetter–Drinfel’d modules, modules overentwined structures, etc. We shall only describe the case of Yetter–Drinfel’dmodules.

Corollary 3.2. Let H be a Hopf algebra and HH�� be the category of left-left Yetter–

Drinfel’d modules. Then the functor

Y � �HH��H�op → Rep�HH��� H��� Y�V� �= H

HHom�V�−�

is an equivalence of categories.

Let R, S be rings, C an R-coring. The following problems seem to be hopeless.

Question 1: Describe the category Rep�S�� CR�� of all representable functors

S� → CR�.

Question 2: Describe all right adjoint functors G � S� → CR�.

Remark 3.3. We note that there exists a partial answer to the second question thatcan be obtained from [6, Theorem 23.2]: if C ∈ �R is flat, G preserves all colimits,and its left adjoint F preserves kernels, then

G � V ⊗S −� F � M�C−

for some V ∈ CR�S , M ∈ S�

CR . Unfortunately the assumptions of this theorem are

very restrictive as right adjoint functors usually do not preserve colimits (theypreserve limits) and left adjoint functors usually do not preserve kernels (theypreserve cokernels).

We shall indicate an answer for Problem C.

Corollary 3.4. Let R, S, T be rings, C an R-coring, and F � CR� → S�, G � S� → T�

representable functors. Then G � F � CR� → T� is a representable functor.

Proof. Theorem 3.1 gives that there exists V ∈ CR�S such that F � C

RHom�V�−�.We apply once again Theorem 3.1 for the trivial coring C = R, and we obtainthat there exists W ∈ S�T such that G � SHom�W�−�. Now the proof follows fromTheorem 3.1 taking into account that there exists a natural isomorphism of functorsgiven by the Hom-tensor adjunction

SHom�W�−� � CRHom�V�−� � C

RHom�V ⊗S W�−��

where V ⊗S W ∈∈ CR�T via the left C-coaction given by v⊗S w �→ v�−1� ⊗R v�0� ⊗S w,

for all v ∈ V and w ∈ W . �

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1788 MILITARU

The Induction Functor

Let R, S be rings, C an R-coring, and U ∈ S�R. In the last part of the articlewe shall give necessary and sufficient conditions for the induction functor

U ⊗R − � CR� → S�

to be: a representable functor, an equivalence of categories, or a separable or aFrobenius functor.

Examples 3.5. Let us give the motivation for the first problem. Consider U �=S = R. Then R⊗R − � F , where F � C

R� → R� is the forgetful functor. We have theadjoint pairs of functors

F = R⊗R − C ⊗R − CRHom�C�−�

Now [5, Theorem 4.1] gives three necessary and sufficient conditions for theforgetful functor F � R⊗R − to be a Frobenius functor. This can be restated asF � R⊗R − is a representable functor having C as an object of representability.In the following, we shall address the general case of an arbitrary inductionfunctor; moreover, we shall not impose restrictive conditions regarding the object ofrepresentability.

First, we shall give necessary and sufficient conditions for U ⊗R − to be anequivalence of categories. The Morita theorem for equivalence between two modulecategories can be recovered as a special case for the trivial coring C = R. Morita-type theorems for categories of comodules over corings were also proved in [4, 7].The next theorem is not a special case of them. The proof we give is very short andelementary, being based on Theorem 1.6 and Corollary 2.7.

Theorem 3.6. Let C be an R-coring and U ∈ S�R. The following are equivalent:

(1) U ⊗R − � CR� → S� is an equivalence of categories;

(2) There exists V ∈ CR�S such that:

(i) U ⊗R V � S, isomorphism in S�S;(ii) V ⊗S U ⊗R C � C, isomorphism in C

R�CR;

(3) There exists a triple �V� p� h�, where V ∈ CR�S , p ∈ C

RHomR�C� V ⊗S U�, h ∈RHomR�V ⊗S U ⊗R C�R� such that:

(i) U ⊗R V � S, isomorphism in S�S;(ii) v�−1�h�v�0� ⊗S u⊗R c� = h�v⊗S u⊗R c�1��c�2�;(iii) h�p�c�1��⊗R c�2�� = ��c�;(iv) h�v⊗S u⊗R c�1��p�c�2�� = v⊗S u��c�

for all v ∈ V , u ∈ U , c ∈ C.

Proof. �1� ⇔ �2� First we note that, if the functor U ⊗R − � CR� → S� is an

equivalence of categories, then its inverse F is a left (and a right) adjoint. Using

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REPRESENTABLE FUNCTORS FOR CORINGS 1789

Theorem 1.6, we obtain that there exists V ∈ CR�S , unique up to an isomorphism in

CR�S , such that F � V ⊗S −. Thus �1� can be restated as: �U ⊗R −� V ⊗S −� is anequivalence of categories inverse each other.

Let F �= V ⊗S − and G �= U ⊗R −. Then F �G = V ⊗S U ⊗R − = Z ⊗R −,where Z �= V ⊗S U ∈ C

R�R. Using Corollary 2.7, we obtain that F �G � 1CR�

ifand only if �ii� of �2� holds. On the other hand, G � F = U ⊗R V ⊗S − = T ⊗S −,where T �= U ⊗R V ∈ S�S . Thus, G � F � 1

S�= S ⊗S − if and only if U ⊗R V � S,

isomorphism in S�S .

�2� ⇔ �3� The pair of maps �p� h� and the conditions (ii)–(iv) of �3� give theparametrization of isomorphisms in C

R�CR between V ⊗S U ⊗R C and C according to

Corollary 2.7 applied for Z �= V ⊗S U ∈ CR�R. The condition (ii) in �3� expresses the

fact that h ∈ •RHomR�V ⊗S U ⊗R C�R�.

In order to study the representability of the induction functor U ⊗R − we needto introduce the following concept.

Definition 3.7. Let R, S be two rings, C an R-coring, U ∈ S�R, and V ∈ CR�S .

A pair �e� h�, where e = ∑e1 ⊗ e2 ∈ (

U ⊗R V)S, h ∈ RHomR�V ⊗S U ⊗R C�R�,

such that

v�−1�h�v�0� ⊗S u⊗R c� = h�v⊗S u⊗R c�1��c�2� (22)∑

e1h�e2 ⊗S u⊗R c� = u��c� (23)∑

h�v⊗S e1 ⊗R e2�−1��e

2�0� = v (24)

for all v ∈ V , u ∈ U , c ∈ C is called a comodule dual basis of first kind for �U� V�.

Remarks 3.8.

1. We shall look at the module case in order to explain the terminology. Let C �= R,U ∈ S�R, V �= U ∗ = HomR�U�R� ∈ R�S , and h the evaluation map

h �= evU � U ∗ ⊗S U → R� u∗ ⊗S u �→ �u∗� u�

There exists e = ∑i ui ⊗R u∗

i ∈ �U ⊗R U ∗�S such that �e� evU � is a comodule dualbasis of first kind for �U�U ∗� if and only if �ui� u

∗i � is a dual basis for U ∈ �R.

This is equivalent to U is finitely generated projective as a right R-module.2. Let �e� h� be a comodule dual basis of first kind for �U� V�. Then V is finitely

generated projective as a left R-module: indeed, it follows from (24) that �h�?⊗S

e1 ⊗R e2�−1��� e2�0�� is a dual basis for V as a left R-module.

Moreover, if C contains an element c such that ��c� is right invertible in R,then U is finitely generated and projective as a right R-module: it follows from (23)that �e1� h�e2⊗S?⊗R c��c�−1�� is a pair of dual basis for U as a right R-module.

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1790 MILITARU

3. Using Example 3.5, Theorem 3.9 below, and [5, Theorem 4.1], we obtain thefollowing: let U �= S �= R. Then there exists a comodule dual basis of the firsttype for �R�C� if and only if C is finitely generated projective as a left R-moduleand the extension R → C∗ = HomR�C�R� is a Frobenius extension of rings in theclassical sense.

Let X ∈ S�S . We recall two well-know results (in fact they are also specialcases of Lemma 2.4 and Lemma 2.5 for the trivial coring C = R). For anynatural transformation � � 1

S�→ X ⊗S − there exists a unique element e ∈ XS �=

�x ∈ X � sx = xs�∀s ∈ S� such that

�N � N → X ⊗S N� �N �n� = e⊗S n

for all N ∈ S�, and n ∈ N , and for any natural transformation � � X ⊗S − → 1S�,

there exists a unique map E ∈ SHomS�X� S� such that

�N � X ⊗S N → N� �N�x⊗S n� = E�x�n

for all N ∈ S�, x ∈ X and n ∈ N .Now we are ready to give an answer to Problem B for an induction functor

connecting the categories of comodules and modules. It generalizes and unifies twotheorems that at first glance have nothing in common: [18, Theorem 2.1] is recoveredfor the trivial coring C �= R, and [5, Theorem 4.1] is obtained as special case forU �= S �= R if we predefine C to be the object of representability of the inductionfunctor in the next theorem.

Theorem 3.9. Let C be an R-coring and U ∈ S�R. The following are equivalent:

(1) The induction functor U ⊗R − � CR� → S� is representable;

(2) There exists V ∈ CR�S such that V ⊗S − is a left adjoint of U ⊗R −;

(3) There exists �V� e� h�, where V ∈ CR�S and �e� h� is a comodule dual basis of first

kind for �U� V�.

In this case U ⊗R − � CRHom�V�−� and V is finitely generated and projective as a left

R-module.

Proof. �1� ⇔ �2� It follows from Theorem 3.1 that a representable functor CR� →

S� is isomorphic to CRHom�V�−�, for some V ∈ C

R�S . Now, V ⊗S − is a left adjointof C

RHom�V�−�; hence the conclusion follows from Khan’s theorem of uniquenessof adjoints [17].

�2� ⇔ �3� Let V ∈ CR�S . We shall prove that V ⊗S − is a left adjoint of U ⊗R

− if and only if there exists �e� h� a comodule dual basis of first kind for �U� V�.Indeed, for any natural transformation � � 1

S�→ U ⊗R V ⊗S − there exists a

unique element e = ∑e1 ⊗R e2 ∈ �U ⊗R V�S such that

�N � N → U ⊗R V ⊗S N� �N �n� =∑

e1 ⊗R e2 ⊗S n (25)

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REPRESENTABLE FUNCTORS FOR CORINGS 1791

for all N ∈ S� and n ∈ N . On the other hand, if we apply Lemma 2.5 for Z �= V ⊗S

U we obtain that, for any natural transformation � � V ⊗S U ⊗R − → 1CR�

, thereexists a unique h ∈ •

RHomR�V ⊗S U ⊗R C�R� such that

�M � V ⊗S U ⊗R M → M� �M�v⊗S u⊗R m� = h�v⊗S u⊗R m�−1��m�0� (26)

for all M ∈ CR�, m ∈ M , v ∈ V , and u ∈ U . We note that (22) means that

h ∈ •RHomR�V ⊗S U ⊗R C�R�.We shall prove that the above pair of natural transformations ��� �� meets the

condition of adjunction (4) if and only if (23) and (24) hold. We denote G = U ⊗R −and F = V ⊗S −. By a direct calculation, we have G��M� � �G�M� = IdG�M�, for allM ∈ C

R� if and only if

u⊗R m = ∑e1h�e2 ⊗S u⊗R m�−1��⊗R m�0� (27)

for all M ∈ CR�, m ∈ M , and u ∈ U . Now, (27) follows from (23). Conversely, if we

consider M �= C and apply Id⊗R � to (27), we obtain (23).Finally, �F�N� � F��N � = IdF�N�, for all N ∈ S� if and only if

v⊗S n = ∑h�v⊗S e

1 ⊗R e2�−1��e2�0� ⊗S n

for all for all N ∈ S�, v ∈ V , n ∈ N , and this condition is obviously equivalent to(24). �

Corollary 3.10. Let R be a ring, C an R-coring. The following are equivalent:

(1) The forgetful functor F � CR� → R� is representable;

(2) There exists �V� e� h�, where V ∈ CR�R, e ∈ VR, and h ∈ RHomR�V ⊗R C�R�, such

that

v�−1�h�v�0� ⊗R c� = h�v⊗R c�1��c�2� (28)

h�e⊗R c� = ��c� (29)

h�v⊗R e�−1��e�0� = v (30)

for all v ∈ V , c ∈ C.

Proof. We apply Theorem 3.9 for U = S = R. In this case, the induction functorR⊗R − is isomorphic to the forgetful functor. The conditions (28), (29), (30) meanthat �e� h� is a comodule dual basis of first kind for �R� V�. �

Corollary 3.10 generalizes [8, Corollary 5.6] that is obtained as a special caseimposing that C be the representing object of F .

Corollary 3.11. Let R, S be two rings, C an R-coring, U ∈ S�R, and V ∈ CR�S .

Assume that there exists �e� h� a comodule dual basis of first kind for �U� V�. Then:

(1) The induction functor V ⊗S − � S� → CR� is separable if and only if there exists

E ∈ SHomS�U ⊗R V� S� such that E�e� = 1.

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1792 MILITARU

(2) The induction functor U ⊗R − � CR� → S� is separable if and only if there exists

p ∈ CRHomR�C� V ⊗S U� such that:

h�p�c�1��⊗R c�2�� = ��c�

for all c ∈ C.

Proof. With our assumptions V ⊗S − is a left adjoint of U ⊗R − (Theorem 3.9)with the unit and counit given by (25) and (26).

(1) Being a left adjoint, V ⊗S − is a separable functor if and only if theunit � of the adjunction V ⊗S − U ⊗R − splits, that is, there exists � U ⊗R

⊗S− → 1S�

a natural transformation such that N � �N = IdN for all N ∈ S�. Sucha natural transformation is uniquely defined by a map E ∈ SHomS�U ⊗R V� S� viathe formula N �u⊗R v⊗S n� = E�u⊗R v�n, for all N ∈ S�, u ∈ U , v ∈ V , and n ∈ N .It is easy to see that N splits �N if and only if E�e� = 1S .

(2) U ⊗R − is a right adjoint: hence, it is separable if and only if thecounit � of the adjunction V ⊗S − U ⊗R − cosplits; that is, there exists a naturaltransformation � � 1C

R�→ V ⊗S U ⊗R − such that �M � �M = IdM for all M ∈ C

R�. Itfollows from Lemma 2.4 that such a natural transformation � is uniquely definedby a map p ∈ C

RHomR�C� V ⊗S U� such that

�M � M → V ⊗S U ⊗R M� �M�m� = p�m�−1��⊗R m�0�

for all M ∈ CR� and m ∈ M . Now, we can prove directly that � cosplits � if and only

if

h(p�m�−1��1��⊗R m�−1��2�

)m�0� = m

for all M ∈ CR� and m ∈ M . This condition is obviously equivalent (take M = C and

apply the counit of C on the second position, the converse is trivial) to h�p�c�1��⊗R

c�2�� = ��c� for all c ∈ C.

Definition 3.12. Let R, S be two rings, C an R-coring, U ∈ S�R, and V ∈ CR�S .

A pair of maps �p� E�, where p ∈ CRHomR�C� V ⊗S U�, E ∈ SHomS�U ⊗R V� S� such

that the diagrams

are commutative is called a comodule dual basis of the second kind for �U� V�.

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Examples 3.13.

1. Let C �= R be the trivial coring, and V ∈ R�S , U �= V ∗ ∈ S�R its right dual.Consider the evaluation map

E � V ∗ ⊗R V → S� E�v∗ ⊗R v� = �v∗� v�

Then there exists p ∈ RHomR�R� V ⊗S V∗� such that �p� E� is a comodule dual

basis of the second kind for �V ∗� V� if and only if V is finitely generated andprojective as a right S-module.

2. Let U = S = R and V ∈ CR�R. Then �p� E� is a comodule dual basis of the second

kind for �R� V� if and only if p � C → V is a morphism in CR�R, E � V → R is an

R− R-bimodule map such that

E � p = �C� p�v�−1��E�v�0�� = v

for all v ∈ V . In particular, for V = C, we obtain that �IdC� �C� is a comoduledual basis of the second kind for �R�C�.

The reverse side of the adjunction of the same induction functors is alsointeresting as follows.

Theorem 3.14. Let R, S be two rings, C an R-coring, U ∈ S�R, and V ∈ CR�S . The

following are equivalent:

(1) The induction functor U ⊗R − � CR� → S� is a left adjoint of V ⊗S − � S� → C

R�;(2) There exists �p� E� a comodule dual basis of the second kind for �U� V�.

Proof. It follows from Lemma 2.4 for Z = V ⊗S U that a natural transformation� � 1C

R�→ V ⊗S U ⊗R − is uniquely defined by a map p ∈ C

RHomR�C� V ⊗S U� suchthat

�M � M → V ⊗S U ⊗R M� �M�m� = p�m�−1��⊗R m�0� (31)

for all M ∈ CR� and m ∈ M .

A natural transformation � � U ⊗R ⊗S− → 1S�

is uniquely defined by a mapE ∈ SHomS�U ⊗R V� S� such that

�N �u⊗R v⊗S n� = E�u⊗R v�n (32)

for all N ∈ S�, u ∈ U , v ∈ V and n ∈ N .Now we shall prove that ��� �� given by (31) and (32) fulfill the condition

of adjunction (4) if and only if the pair of maps �p� E� that defines the naturaltransformations � and � is a comodule dual basis of the second kind for �U� V�.We denote F = U ⊗R − and G = V ⊗S −, and we shall adopt the notation p�c� =∑

p�c�V ⊗ p�c�U ∈ V ⊗S U , for all c ∈ C.

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1794 MILITARU

By a direct calculation, we have G��N� � �G�N� = IdG�N�, for all N ∈ S� if andonly if

v⊗S n = ∑p�v�−1��

VE(p�v�−1��

U ⊗R v�0�)⊗Sn (33)

for all N ∈ S�, n ∈ N , and v ∈ V . Now, (33) is equivalent (take N = S, n = 1S) tothe fact that the left diagram of Definition 3.12 is commutative.

On the other hand, �F�M� � F��M� = IdF�M�, for all M ∈ CR� if and only if

u⊗R m = ∑E(u⊗R p�m�−1��

V)p�v�−1��

U ⊗R m�0� (34)

for all M ∈ CR�, u ∈ U , and m ∈ M . Now, (34) is equivalent to the fact that the right

diagram of Definition 3.12 is commutative. Indeed, if we take M = C and m = c ∈ Cand apply �C to (34), we obtain the commutativity of the diagram. The converse isstraightforward. �

The fact that the forgetful functor F � CR� → R� has a right adjoint [5,

Lemma 3.1] is a special case of Theorem 3.14 as �IdC� �C� is a comodule dual basisof the second kind for �R�C�. Moreover, the following Corollary is a generalizationof [5, Theorem 3.3] and [5, Theorem 3.5] which are obtained if we consider U �=S �= R and V �= C taking into account that �IdC� �C� is a comodule dual basis ofthe second kind for �R�C�.

Corollary 3.15. Let R, S be two rings, C an R-coring, U ∈ S�R, and V ∈ CR�S .

Assume that there exists �p� E� a comodule dual basis of the second kind for �U� V�.Then:

(1) The induction functor V ⊗S − � S� → CR� is separable if and only if there exists an

element e ∈ �U ⊗R V�S such that E�e� = 1;(2) The induction functor U ⊗R − � C

R� → S� is separable if and only if there existsh ∈ RHomR�V ⊗S U ⊗R C�R� s.t.:

v�−1�h�v�0� ⊗S u⊗R c� = h�v⊗S u⊗R c�1��c�2�

h(p�c�1��⊗R c�2�

) = ��c�

for all v ∈ V , u ∈ U , c ∈ C.

Proof. With our assumptions, U ⊗R − is a left adjoint of V ⊗S − (Theorem 3.14)with the unit and counit given by (31) and (32). Using Lemma 2.5, the proof followssimilarly to the one of Corollary 3.11. �

Remarks 3.16. In general, the separability of an induction functor V ⊗S − � S� →CR� is still an open problem even for the category of modules, i.e., for the trivialcoring C �= R. [9, Corollary 5.11] solved the problem only for finitely generatedand projective modules. All four statements of Corollary 3.11 and Corollary 3.15generalize their result.

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REPRESENTABLE FUNCTORS FOR CORINGS 1795

Theorem 3.9 and Theorem 3.14 give us the following corollary.

Corollary 3.17. Let R, S be two rings, C an R-coring, and U ∈ S�R. The followingare equivalent:

(1) The induction functor U ⊗R − � CR� → S� is a Frobenius functor;

(2) There exists(V� �e� h�� �p� E�

), where V ∈ C

R�S and �e� h� (resp., �p� E�) is acomodule dual basis of the first kind (resp., the second kind) for �U� V�.

ACKNOWLEDGMENTS

This work was supported by CNCS-UEFISCDI grant no. 88/05.10.2011“Hopf algebras and related topics.”

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