Galois Theory for Corings and Comoduleshomepages.vub.ac.be/~jvercruy/phd.pdf · 5.4. Structure...

FACULTEIT WETENSCHAPPENDEPARTEMENT WISKUNDE

Galois Theory for Corings and Comodules

Proefschrift voorgelegd aan de Faculteit Wetenschappen,voor het verkrijgen van de graad van Doctor in de Wetenschappen door

Joost Vercruysse

Promotor: Prof. S. Caenepeel

Academiejaar 2006-2007

Whenever there is existence,There shall be co-existence...

Preface

This work is written as a PhD thesis under the supervision of Professor Stefaan Caenepeel. Thethesis was defended on March 9, 2007 at the Vrije Universiteit Brussel (Free University of Brussels,VUB). The members of the jury were Tomasz Brzezinski, Stefaan Caenepeel, Eric Jespers, RudgerKieboom (secretary), Claudia Menini, Michel Van den Bergh (president) and Yinhuo Zhang. Iwould like all of them for the effort they have put into this job.

The present version of this thesis is slightly different from the one orriginally presented onMarch 9. Some typograffical and mathematical inconsistencies have been cured. In Chapter 4,some of the structure theorems have been reformulated in a way that makes the flatness conditionon the coring less prominent. Let me thank the members of the jury of their useful suggestionsand as well Jawad Abuhlail for his remark on local projectivity for modules generated by countablesets (see Remark 2.55).

Since without any doubt many mistakes and misprints did still survive endless proofreading,all corrections and remarks are welcomed at [email protected].

Brussel/Gent, April 2007.

iii

mailto:[email protected]

Acknowledgements

Ik ben blij dat jij in mijn team zit.– Guido Pallemans, Eilandbewoner

A thousand words of gratitude for...

Jan Van Geel, my supervisor at Universiteit Gent during my licentiaatsthesis. He was thefirst who learnt me about (classical) Galois theory and in this way guided meinto a fascinating algebraic world. He also informed me about an open positionat the VUB for which I applied. So, without him, this thesis would have neverexisted for different reasons.

Stef Caenepeel, my current supervisor. Beside the mathematics he learnt me, it was a greatpleasure to join him during conferences and foreign visits, especially when thebottles at the conference dinner turned out to be too small.

Gabriella (Gabi) Bohm and Jose (Pepe) Gomez-Torrecillas, with whom I had and have thechance to collaborate. I want to thank them for inviting me twice to their homeinstitute in Budapest and Granada, where I had very pleasing stays. I learnt alot from these collaborations and their contribution to this work is of great value.

Laiachi El Kaoutit, Miodrag Iovanov and Shuanhong Wang, my other collaborators.Beside their contribution to this work I am grateful to them for the pleasantmoments when they visited Brussels, and to Mio for the hospitality in Bucharest.

Erwin De Groot and Kris Janssen, my friends and roommates during the working daysat VUB and even longer during conferences. I enjoyed a lot to talk with themabout mathematics, play frozen bubble or laugh with things that should not bementioned here.

Philippe Cara, one of my other nice colleagues, who is always willing to help with anycomputer- or LATEX-problem.

My dear parents, for their continuous support.

Finally..., special thanks to my best friends Annelies, Ilse, Stijn, Jarno, Thomas and of coursemost especially to Carolien, for all the fine moments when mathematics was faraway.

v

Table Of ContentsAcknowledgement vNotational conventions xiiiIntroduction 1Part I : Algebraic and Categorical Constructions

1. Algebras in Monoidal Categories and Bicategories 92. Local Projectivity versus local algebraic Structures 353. Corings and Comodules 73

Part II : Galois Theory5. Galois Comodules 1136. Morita Theory for Corings 1457. Cleft Bicomodules 167

Part III : Frobenius and Separable Functors8. Separable Functors and relative Cohomology 1939. Co-Frobenius Corings and related Functors 211

10. Applications to Galois Theory 235Appendix

A. Nederlandse samenvatting 249

vii

Contents

Preface iii

Acknowledgements v

Table Of Contents vii

Notational Conventions xiii

Introduction 1

Chapter 1. Algebras in Monoidal Categories and Bicategories 91.1. Bicategories and monoidal categories 91.1.1. Bicategories 91.1.2. Bicategories versus 2-categories 121.1.3. Monoidal categories 151.2. Monads and algebras 171.2.1. Monads, algebras and modules 171.2.2. Comonads, coalgebras and comodules 191.3. Bicategories of bimodules 221.3.1. Adjoint pairs and Morita contexts 221.3.2. The bicategories of Eilenberg-Moore objects 231.3.3. Locally finite duality 261.3.4. The bicategory of bi(co)modules 281.4. Enriched bicategories 31References 34

Chapter 2. Local Projectivity versus local algebraic Structures 352.1. Colimits and split direct systems 352.2. Non-unital rings 382.2.1. The Dorroh extension 382.2.2. Firm rings 392.2.3. Rings with local units 462.2.4. Rings with idempotent local units 482.2.5. The M -adic and finite topology 512.3. Projectivity 522.3.1. Firmly projective modules 522.3.2. Weakly locally projective modules 572.3.3. Strongly locally projective modules 592.3.4. Local projectivity versus local units 602.3.5. Representations of rings with local units 642.4. Local structure maps 652.4.1. Corings with local comultiplications 652.4.2. Corings with local counits 692.4.3. Rings with local multiplication 70References 72

Chapter 3. Corings and Comodules 733.1. Basic properties of corings and comodules 733.1.1. Definitions 73

ix

x CONTENTS

3.1.2. Adjunctions 753.1.3. The Dorroh coring 783.2. Corings and entwining Structures 803.2.1. Entwining structures and entwined modules 803.2.2. Factorizable enwining structures 863.3. Rational modules 883.3.1. R-rational modules 883.3.2. The rational functor 933.4. Comatrix corings 953.4.1. Comatrix corings over firm rings 953.4.2. Locally finite duality 1003.4.3. Comatrix corings and cotriples coming from adjunctions 1013.5. Corings from colimits 1023.5.1. Corings from colimits 1023.5.2. Colimit comatrix corings 1033.5.3. Factorizing split direct systems 107References 110

Chapter 4. Galois Comodules 1134.1. Introduction : Motivating problems 1134.1.1. The descent problem 1134.1.2. Hopf-Galois theory 1144.2. Galois comodules 1154.2.1. Comonadic-Galois comodules 1154.2.2. The canonical cotriple morphism 1174.2.3. Firm Galois comodules 1204.2.4. Comonadic-Galois versus firm Galois comodules 1224.2.5. Structure theorems 1254.2.6. Applications 1354.3. Galois theory in bicategories: a unifying approach 1394.3.1. Push-out and pull-back functors 1394.3.2. Weak and strong structure theorems 1414.3.3. examples 142References 144

Chapter 5. Morita Theory for Corings 1455.1. Finite Galois theory and Morita theory 1455.1.1. Finite Galois theory 1455.1.2. Morita theory in the bicategory of bimodules 1465.2. The dual of the canonical map 1475.3. Morita contexts associated to a comodule 1495.3.1. The *-Morita context associated to a comodule 1495.3.2. The Morita context associated to a comodule 1505.3.3. Morita contexts associated to a bimodule 1545.4. Structure theorems 1565.5. Application: Morita contexts associated to a grouplike element 1615.5.1. Grouplike characters 1615.5.2. Grouplike elements 164References 166

Chapter 6. Cleft Bicomodules 1676.1. Some remarks on notation and coring extensions 1676.2. Morita theory for coring extensions 1686.3. Weak and strong structure theorems 1766.4. Cleft bicomodules 1786.5. Examples 179

CONTENTS xi

6.5.1. Cleft entwining structures 1796.5.2. Cleft extensions of algebras by a coalgebra 1826.5.3. Cleft extensions of algebras by a Hopf algebra and the fundamental theorem 1826.5.4. Cleft weak entwining structures 1836.5.5. Cleft extensions by partial group actions 1846.5.6. Cleft entwining structures over arbitrary base 1856.5.7. Cleft extensions of algebras by a Hopf algebroid 1866.5.8. Cleft factorization structures 187References 189

Chapter 7. Separable Functors and relative Cohomology 1937.1. Separability and relative injectivity 1937.2. Bicomodules and separability 1947.3. Coderivations and cointegrations 1987.4. Cohomology for bicomodules 2027.5. Applications 2067.5.1. Coseparable corings 2067.5.2. Coseparable coalgebra co-extensions 207References 210

Chapter 8. Co-Frobenius Corings and related Functors 2118.1. Elementary results 2118.1.1. Direct sums and direct products 2118.1.2. Frobenius corings 2128.1.3. Morita contexts 2128.2. Locally adjoint functors 2138.2.1. Action of a set of natural transformations on a category 2138.2.2. Locally adjoint functors 2148.3. The induction functor 2158.3.1. Adjunctions 2158.3.2. Description of sets of natural transformations 2158.3.3. The Yoneda-approach 2188.3.4. The coproduct functor 2208.4. Characterizations of co-Frobenius and quasi-co-Frobenius corings 2238.4.1. Locally Frobenius corings 2238.4.2. Characterization of Frobenius corings 2268.4.3. Quasi-co-Frobenius corings and related functors 228References 233

Chapter 9. Applications to Galois Theory 2359.1. Separable corings and Galois comodules 2359.2. Frobenius properties and Morita theory for comodules 2379.2.1. Frobenius corings 2379.2.2. Co-Frobenius corings 2419.3. The coring as a Galois comodule 2429.3.1. The coring as a finite Galois comodule 2429.3.2. The coring as an infinite Galois comodule: rationality properties 2449.3.3. The coring as an infinite Galois comodule: quasi co-Frobenius corings 246References 247

Appendix A. Nederlandse Samenvatting 2491.1. Ringen en coringen 2491.2. Comatrix coringen en Galois comodulen 2511.3. Separabiliteitseigenschappen en Frobenius coringen 251

Bibliography 253

xii CONTENTS

Index 256

Notational Conventions

Willen we dat afspreken ?– Lydia Protut

Rings and corings. We will usually denote by k a commutative ring, it will only be a fieldif it is explicitly mentioned. Capitals A,B,A′, B′, .. or R,R′, S, S′ will be used for associativerings (k-algebras). Mostly we will preserve capitals A,B for rings with unit (i.e. an element1 such that 1a = a1 for all a in the algebra) and R,S for firm rings. However we mentionexplicitly what is the case as much as possible. Capitals C,D are also used to denote coalgebras(with commutative basering). This is in contrast to corings, which are denoted by gothic capitalsC,D, . . . Modules and comodules, whether over algebras, coalgebras or corings, are always denotedby capitals M,N,P, . . .. By Σ we will denote a comodule whose Galois theory is studied.

Categories and functors. We denote categories with capitals in calligraphic font, A,B, C,M.In particular, the category of firm left A-modules, firm right B-modules and firm A-B bimodulesover rings A and B are respecively denoted by AM, MB and AMB. The category of arbitrary leftA-modules, right B-modules and A-B bimodules over rings A and B are respecively denoteted

by AM,MB,AMB. In the same way, the categories of left C-comodules, right D-comodules andC-D bicomodules over corings C and D are denoted by CM, MD and CMD.

Let C and D be two categories. A functor F : C → D will be the name for a covariantfunctor, it will only be a contravariant functor if it is explicitly mentioned. Consider two functorsF,G : C → D. The class of all natural transformations between F and G is denoted by Nat(F,G).

Finally, 1 denotes the discrete one-object category, i.e. 1 is the category with one object thatwe will denote by ∗ and with only one morphism, the identity on ∗.

Morphisms. Let X and Y be two objects of a category C then HomC(X,Y ) is the notationfor all morphisms in C from X to Y . In particular, for rings A and B, we denote by AHom(X,Y ),HomB(X,Y ) and AHomB(X,Y ) the sets of left A-linear, right B-linear and A-B bilinear maps be-tween modulesX and Y . Similarly, for corings C and D, we denote by CHom(X,Y ), HomD(X,Y ),CHomD(X,Y ) are sets of left C, right D and C-D colinear morphisms between X and Y .

For an object X in a category C, X will also be the name for the identity morphism on X. Insome cases, where this would cause too much confusion or to not overload the notation, we willdenote the identity on X by 1X or just by 1.

xiii

Introduction

Get on with it!– Monthy Python, ‘Search for the Holy Grail’

Recall the sentence that is stated on the first page of this book : “Whenever there is existence,there shall be co-existence” (this quote is due to Raimundas Vidunas, a former visitor at theVrije Universiteit Brussel). In mathematics, especially for category theorists, this expression is anevidence, since every mathematical idea, can be dualized, i.e. every construction in a categoryA can be performed in the opposite category Aop, where arrows are reversed. This principle isstanding at the origin of the notion of a coalgebra (co-algebra). A k-algebra over a commutativebase ring k is a monoid in the monoidal category Mk. Hence a coalgebra is a monoid in thecategory Mop

k . In the same way, A-rings (being monoids in the monoidal category of bimodulesover a not necessarily commutative base ring A) can be dualized to corings and modules dualizeto comodules. Although corings are just the dual notion of the well-understood rings, the wholetheory of corings cannot be directly obtained from the theory of rings, by use of a general dualityprinciple. The reason is simple: by dualizing rings to corings, we dualize in the first place thecategory of modules and this category is not self-dual. A first difference between rings and coringsthat can be easily understood from the definition is the following. For every ring R, the ring ofintegers Z can play the role of base ring for R, since Z → R is a ring morphism. For corings theredoes not exist such a universal base ring, nor does there exist a universal coring Ω which allows acoring morphism C → Ω, for all corings C.

The main goal of this work is to study the relation between the ‘world’ and the ‘co-world’.Explicitly, we want to examine functors between categories of modules and categories of comodules(over corings). Module categories have been studied for decades, therefore they are consideredinteresting for numerous reasons. Coring theory has not such a long standing tradition, but stillcorings and comodules are known to be relevant in several subfields of abstract algebra and evennon-commutative geometry. Corings were defined by Sweedler [109] in 1975, who introducedthese structures in order to formulate a theorem, which in a dual form gives rise to the Jacobson-Bourbaki Theorem. However, because of a lack of examples during the next 25 years, only aboutfive articles appeared that really dealt with corings (e.g. [76], [83], [91]). Corings surfaced for amoment in the beginning of the eighties, under the name bocs, in the work of Kleiner [82] andRojter [101]. Corings became almost forgotten, but this situation came to an abrupt end justbefore the end of the second millenium, when Takeuchi observed that new examples of corings canbe constructed out of entwining structures. Takeuchi did not publish this observation, but we canread in a mathematical review by Masuoka [110] the following.

The following observation due to Takeuchi helps our understanding of an entwin-ing structure (A,C)ψ : ψ makes A ⊗ C into a right A-module such that giventhe obvious left A-module structure and the comultiplication and the counitarising from those of C, it forms an A-coring, and a right (A,C)ψ-module isidentified naturally with a right A⊗ C-comodule.

In his well-known article “Structure of Corings: Induction functors, Maschke-type theorem, andFrobenius and Galois-type properties”, Brzezinski explores this idea further and in this way theobservation of Takeuchi, together with Brzezinski’s paper gave rise to a renewed interest in corings

1

2 INTRODUCTION

that is still expanding today. The main reason why the connection with entwining structures makescorings so attractive, is that it turned out that entwining structures and entwined modules providea lucid framework to study all types of Hopf modules that were constructed during the last decadesof the 20th century, such as Yetter-Drinfeld modules, Long dimodules, graded modules, Doi-Hopfmodules. In fact, by passing from entwining structures and entwined modules to corings andcomodules, although it turned out that most structural theorems could be preserved, calculationsbecame easier and more transparent and many results could be clarified as they were presented ina broader perspective.

The thesis is devided into three parts, each part consists of three chapters. In the first part,we create the algebraic and categorical tools that are needed in the two other parts. Part II, is theheart of the thesis and deals with Galois theory. In the last part, we study some special types offunctors between module and comodule categories and their implications to Galois theory.

We start by recalling in Chapter 1 the definitions of algebras, coalgebras and their representa-tions. We prefer to give a general treatment in bicategories, since this turns out to be the rightperspective for Galois theory as well (see below).

Chapter 2, is devoted to some ring theoretic aspects. In particular, we are interested in modulesthat are not finitely generated, but that still satisfy a certain projectivity condition. There is aclose connection between these notions of ‘local projectivity’ and rings with ‘local units’ (remarkthat we use the term ‘(local) unit’ for elements e in a ring R that satisfy the relation r · e = rfor a set of elements r ∈ R, and not for invertible elements e in R). In the literature, severalnotions of local projectivity have appeared. A first notion is due to Zimmermann-Huisgen [119],and is equivalent to the so-called α-condition. We will call modules that satisfy this kind of localprojectivity weakly locally projective modules. A second notion (which we will call strongly localprojectivity) is due to Abrams [1], and is related to rings with idempotent local units by Moritatheory (see [7]). In Chapter 2, we further investigate the relations between local projectivity andlocal units. We introduce an even more general notion of projectivity wich we call firm projectivityand which is related to firm rings. Our main observation is that firmly projective modules are inbijective correspondence with pairs of adjoint functors between categories of modules over firmrings. If we specify the firm base ring to a ring with local units, we obtain a module that satisfiesthe corresponding type of projecitivity. If the ring has a unit, then the module is finitely generatedand projective.

In Chapter 3, the last Chapter of Part I, we make a basic study of corings and comodules.We develop in this chapter all tools from the general theory of corings that we will need in PartsII and III. We explain in more detail the relation between corings and entwining structures, thatwe already mentioned before. We develop the theory of rational modules, which will be importantif we study Galois theory of corings by means of Morita theory in Chapter 5. The main part ofChapter 3 is devoted to the construction of comatrix corings. The first type of comatrix corings wasintroduced in [63]. Starting from a B-A bimodule that is finitely generated and projective as rightA-module El Kaoutit and Gomez-Torrecillas constructed an A-coring Σ⊗BΣ∗ that generalizes thecanonical coring that appeared in Sweedler’s original paper [109]. Although it seemed from theirconstruction that it is necessary that Σ is finitely generated and projective, several attempts weremade to drop the finiteness condition in this construction. In [64] El Kaoutit and Gomez-Torrecillasconstruct a type of infinite comatrix corings that are built from a direct sum of B-A bimodulesthat are finitely generated and projective as right A-modules. In [113], the author constructedcorings with local comultiplications, starting from a B-A bimodule that is locally projective asright A-module. Combining the techniques of both approaches, Gomez-Torrecillas and the authorconstructed corings out of firmly projective modules. This approach generalizes the constructionof [64] and has been specified in [41] to a construction of corings out of colimits in a joint workwith Caenepeel and De Groot.

Part II deals with the main subject of this work: Galois theory for Corings and Comodules.As mentioned at the beginning, Galois theory describes properties of functors between categoriesof modules and categories of comodules, i.e. it relates the objects of Chapter 2 with the objectsof Chapter 3, using the language we introduced in Chapter 1. In Chapter 4, we give a generaltreatment of the theory of Galois comodules. This theory originated from the classical work onGalois extensions of commutative fields. This theory has been extended to a Galois theory for

INTRODUCTION 3

commutative rings by Auslander and Goldman [10] and by Chase, Harrison and Rosenberg [50]. Agroup action can be generalized to a Hopf algebra (co)action. This leads to the Hopf-Galois theory,developed first for finitely generated and projective Hopf algebras (see [51] and [85]) and later forarbitrary Hopf algebras (see [58] and [104]). As we explained, corings provide a general frameworkto explain many results of Hopf algebra theory in a simple and clarifying way. In this repect, itis no surprise that Hopf-Galois theory has a formulation in terms of corings. This was shown in[28], where a Galois theory is developed for corings with a grouplike element. To a ring morphismi : B → A, we can associate an A-coring, the so-called canonical Sweedler coring. A morphismfrom this coring to another A-coring C is completely determined by a grouplike element g ∈ C. Ifthis morphism is an isomorphism, we say that (C, g) is a Galois coring. We can construct a pairof adjoint functors between the categories MB and MC and formulate sufficient and necessaryconditions for this pair to be an equivalence of categories. The development of Galois theory interms of corings provides as well an elegant formulation of descent theory. El Kaoutit and Gomez-Torrecillas [63] introduced a yet more general version of Galois theory, replacing the grouplikeelement by a right C-comodule Σ that is finitely generated and projective as right A-module. Theyconstruct an A-coring out of Σ, called the comatrix coring. We call Σ a Galois comodule if thecanonical coring morphism from the comatrix coring to C is bijective (see also [29], [42]). Thereexists an adjunction between the categories MT and MC, where T = EndC(Σ). Several attemptswere made to drop or weaken the finiteness condition on Σ and construct infinite versions of Galoistheory for comodules. El Kaoutit and Gomez-Torrecillas introduced Galois comodules that are(infinite) direct sums of finitely generated and projective right A-modules [64], Caenepeel, DeGroot and the author generalized this to a method to construct corings out of colimits and Gomez-Torrecillas and the author developed a theory of Galois comodules over firm rings and Wisbauerintroduced a functorial definition for a Galois comodule [117]. In Chapter 4, we discuss thesedifferent theories and their relation.

During the last years, other versions and generalizations of Hopf-Galois theory have beenformulated (see [31], [118], [35]) and it became clear that the proper framework to study all thesetheories in a general way, is the framework of bicategories. Therefore, we develop in the secondsection of Chapter 4, a Galois theory in general bicategories and reduce other Galois theories asspecial cases. This general formulation allows us as well to clarify the relation between Galoistheory for corings and the theory of (co)tripleability of functors developed by Beck.

In Chapter 5 we treat some special cases of Galois theory. If the coring satisfies a finitenesscondition (whether the coring is finitely generated and projective over its base ring or the coringis locally projective over its base ring such that the rational part of the dual ring is dense withrespect to the finite topology), the category of comodules can be described as a category ofmodules. Hence, Galois theory, handling about functors between the category of comodules anda category of modules, reduces to Morita theory, which studies functors between two modulecategories. The Morita theory that we develop generalizes a long standing tradition of applyingMorita theory in Hopf-Galois theory (see [53], [57]).

We extend the Morita theory of Chapter 5 further in Chapter 6, where we associate a Moritacontext to any coring extension (D : L) of (C : A) and an L-C bicomodule. If D = L = k isthe trival k-coring, then this Morita context reduces to the one from Chapter 5. The new Moritacontext allows us to develop the theory of cleft bicomodules, which unifies all previously knownnotions of cleft extensions. Cleft bicomodules provide Galois comodules for which the associatedpair of adjoint functors between the category of comodules and the category of modules overthe endomorphism ring of the Galois comodule is always an equivalence. This generalizes thefundamental theorem for Hopf algebras, which states that the category of Hopf modules over aHopf k-algebra H is equivalent to the category of modules over the base ring k.

Part III is devoted to the study of separable and Frobenius functors. Both separablity andFrobenius properties have interesing implications for Galois theory. In the first two chapters ofPart III we develop the theory of separable Frobenius, co-Frobenius and quasi co-Frobenius coringsin a functorial way. In the last chapter we discuss their implications on the Galois theory ofcomodules.

In [80] Jonah studied the second and the third cohomology groups of coalgebras defined in a,not necessarily Abelian, multiplicative category (see also [9]). Kleiner gave in [84] a cohomological

4 INTRODUCTION

characterization of separable algebras using integrations. Another approach via derivations wasgiven by Barr and Rinehart in [12]. This last one has been dualized to the case of coseparablecoalgebras by Doi [56]. Nakajima [94] showed that Doi’s result can be extended to coalgebraextensions (or coextensions) with a cocommutative base coalgebra. In [76], Guzman used Jonah’smethods to generalize Doi’s characterization for corings over an arbitrary base ring and unified thiswith a dualization of Kleiner’s approach of cointegrations. This gives rise to a nice characteriza-tion of coseparable corings in terms of cohomology, derived functors and both cointegrations andcoderivations. Unfortunately this last characterization cannot be applied to coalgebra coextensions,and Nakajima’s results are not recovered.

The common framework behind Guzman’s and Nakajima’s approach is the fact that both cosep-arable corings and coseparable coalgebra coextensions can be interpreted as cotriples (comonads)with a separable forgetful functor (in the sense of [95]). In all situations discussed before, the mul-tiplicative base category was additive with cokernels and arbitrary direct sums, and the (co)triplefunctor preserved cokernels and direct sums. In Chapter 7 we will approach the problem fromthis cotriple point of view. We work with a cotriple over a Grothendieck category (not necessar-ily multiplicative) whose underlying functor fits in the above mentioned class of functors. Thesefunctors were studied in relation with corings in [69]. We will present a generalization of Guz-man’s characterization in this situation, and as a particular application we also give, under differentassumptions, Nakajima’s result.

Frobenius and co-Frobenius coalgebras and Hopf algebras, Frobenius ring extensions and Frobe-nius bimodules have been intensively studied over the last decades. In [32], [33] the close relationbetween Frobenius extensions, Frobenius bimodules and Frobenius corings is discussed. Althoughthe name indicates differently, the co-Frobenius property of a coring is a weakening and not adualization of the Frobenius property. In particular, although the Frobenius property is left-rightsymmetric, the co-Frobenius property is not. Nevertheless, coalgebras over a base field which areat the same time left and right co-Frobenius can be understood as a dual version of Frobeniusalgebras. Indeed, for a Frobenius k-algebra A, the functors HomA(−, A) and Homk(−, k) fromMA to AM are naturally isomorphic (see [54]), for a left and right co-Frobenius k-coalgebra C,the functors HomC∗(−, C∗) and Homk(−, k) from MC to C∗M are naturally isomorphic (see[78]).

Frobenius corings have a very nice characterization in terms of Frobenius functors. This resultsays that a coring C is Frobenius if and only if the forgetful functor F : MC → MA is at thesame time a left and right adjoint of the induction functor − ⊗A C. An overview of most resultswith regard to this subject can be found in [46]. A similar categorical interpretation of (quasi)co-Frobenius coalgebras and corings remained somewhat mysterious.

In Chapter 8, we will provide this categorical description of quasi co-Frobenius corings andwe will generalize some results of [78]. For this reason, we construct several Morita contexts.Starting from the observation (see [21, Remark 3.2]) that a Morita context can be identified witha (k-linear) category with two objects, we construct a Morita context relating a coring C withits dual C∗. This context describes the Frobenius property of the coring. More precisely, if thereexists a pair of invertible elements in this Morita context, then the coring is exactly a Frobeniuscoring. A similar Morita context relates representable functors, such as those used in [78] and[54] to describe (co-)Frobenius properties. A last type of Morita contexts, that is constructed in adifferent way, describes the adjuction property of a pair of functors. More precisely, if there exists apair of invertible elements in this Morita context, then the pair of functors is exactly an adjoint pair.By relating these Morita contexts with (iso)morphisms of Morita contexts, we recover the resultthat a coring is Frobenius if and only if the forgetful functor and the induction functor make up aFrobenius pair if and only if certain representable functors are isomorphic. The advantage of ourpresentation is that it clarifies underlying relations between the different equivalent descriptions ofthe Frobenius property, and that these relations can be described even if the coring is not Frobenius.In particular, using these Morita contexts, we can formulate a categorical interpretation of (quasi)co-Frobenius corings.

In Chapter 9, the final chapter of this thesis, we study how Galois theory of a coring isaffected by separibility and Frobenius properties of corings. In Hopf-Galois theory, which can begeneralized to corings as we already discussed, there exists a stronger version of the usual structure

INTRODUCTION 5

theorem, known as Scheider’s Theorem I. It says that under certain conditions, the surjectivity ofthe canonical map is sufficient to obtain its bijectivity and even an equivalence of categories.We discuss some generalizations of these theorems in the context of firm Galois comodules inSection 9.1, where we make use of the characterization of separable corings.

The Morita context that we introduce in Chapter 5 is a generalization of a Morita contextintroduced by Doi [57]. Morita contexts similar to the one of Doi were studied by Cohen, Fischmanand Montgomery in [52] and [53]. These are different from the one of Doi, in the sense that thetwo connecting modules in the context are equal to the underlying algebra A. On the other hand,they are more restrictive, in the sense that they only work for finite dimensional Hopf algebras overa field (see [52]) or Frobenius Hopf algebras over a commutative ring (see [53]). In Section 9.2,we study the Morita context associated to a Frobenius coring with a fixed comodule Σ. It turnsout that the connecting modules in the context are then precisely Σ and its right dual Σ∗; in thecase where Σ = A, the two connecting modules are isomorphic to A. This clarifies the relationshipbetween the Morita contexts of [57] on the one hand and [52] and [53] on the other hand. Weakerresults are obtained in the situation where C is co-Frobenius.

If the coring satisfies some finiteness conditions, then we can study its Galois properties ascomodule over itself. This allows us to obtain a formulation, by means of Galois theory, of theequivalence between the category of comodules over a coring and the category of modules over thedual of the coring, if the coring is finitely generated and projective over its base ring and betweenthe category of comodules over the coring and the category of modules over the rational part ofthe dual if the coring is locally projective over its base ring. Another very interesting result is acharacterization of quasi co-Frobenius corings by means of Galois theory.

IPart I :

Categorical and AlgebraicConstructions

The Question is: “What is a Mannahmannah ?”.- The Question is: “Who cares ?”.

– Statler & Waldorf

Chapter 1Algebras in Monoidal Categories and

Bicategories

In this Chapter we introduce the basic algebraic framework that will be used throughout thisbook. In the first Section we study bicategories, monoidal categories and the way their structureis related. We give a proof for the well-known result among category theorists that any bicategoryis equivalent to a 2-category. In Section 1.2, we show how the notion of an algebra arises naturallyin this general setting and coalgebras are defined in a dual way. In Section 1.3 we compare severalpossible definitions of a bicategory of bimodules associated to a given bicategory. In the finalSection, we discuss some special features of enriched bicategories, such as the construction of thedual monad of a comonad and an interesting relation between Morita contexts and adjoint pairs.

For a general introduction to the theory of categories and functors we refer to the “classics”[25] and [88].

1.1. Bicategories and monoidal categories

1.1.1. Bicategories. Let us first recall the following terminology. A Hom-Class category isa category A with a class of objects and such that for any two objects A,B ∈ A, all morphismsbetween A and B, constitute a class HomA(A,B). In a Hom-Set category we make the restrictionthat HomA(A,B) is a set for all A,B ∈ A. Finally, a small is a Hom-Set category that containsonly a set of objects.

To avoid set-theoretical problems we will try to omit Hom-Class categories as most as possible.For this reason, if we speak about a category, then we will mean a Hom-Set category, unless it ismentioned differently.

A bicategory B consists of the following data.

(i) A class of objects A,B, . . . which are called 0-cells (or objects).(ii) For every two objects A and B, there exists a category HomB(A,B) = Hom(A,B), whose

class of objects we denote by Hom1(A,B) and which are called 1-cells. We denote f : A→ Bfor a 1-cell f ∈ Hom1(A,B). Take two 1-cells f, g ∈ Hom1(A,B). The set of morphismsfrom f to g in the category Hom(A,B) is denoted by AHomB

2 (f, g). We call these morphisms2-cells and denote them as α : f ⇒ g. We will denote the composition of morphisms in thecategory Hom(A,B) by , i.e. for all f, g, h ∈ Hom1(A,B) such that α : f ⇒ g andβ : g ⇒ h, we have β α : f ⇒ h. This composition will now be called the verticalcomposition of 2-cells.

(iii) For any three objects A,B,C ∈ B, there exist a functor

cABC : Hom(A,B)×Hom(B,C) → Hom(A,C).

9

10 CHAPTER 1. ALGEBRAS IN MONOIDAL CATEGORIES AND BICATEGORIES

For all f ∈ Hom1(A,B) and g ∈ Hom1(B,C), we denote cABC(f, g) = f•Bg ∈ Hom1(A,C).For all α ∈ AHomB

2 (f, g) and β ∈ BHomC2 (h, k), we denote cABC(α, β) = α•Bβ : f •B h⇒

g •B k. This composition will be called the horizontal composition of 2-cells.(iv) For any object A ∈ B, there exists a functor

11A : 1 → Hom(A,A),

where 1 denotes the discrete category with 1 object ∗. We will denote 11A(∗) just by 11A.(v) For any four objects A,B,C,D ∈ B, there exists a natural isomorphism

(1) αABCD : cACD (cABC ×Hom(C,D)) ⇒ cABD (Hom(A,B)× cBCD).

For any two objects A,B ∈ B, there exist two natural isomorphisms

λAB : Hom(A,B) ⇒ cAAB (11A ×Hom(A,B)),(2)

ρAB : Hom(A,B) ⇒ cABB (Hom(A,B)× 11B).

For all compatibility conditions we refer to e.g. [25, section 7.7] or [16], where the notion of abicategory was introduced.

For all objects A,B,C ∈ B, we obtain from the functorality of cABC the interchange law , i.e.

(3) (α•Bβ) (γ•Bδ) = (α γ)•B(β δ),for α ∈ AHomB

2 (a, c), β ∈ BHomC2 (b, d), γ ∈ AHomB

2 (c, e) and δ ∈ BHomC2 (d, f). From (3) one

immediately deduces that for all α ∈ AHomB2 (a, c) and β ∈ BHomC

2 (b, d),

(4) (α•Bb) (c•Bβ) = α•β = (a•Bβ) (α•Bd).A 2-category is a bicategory such that the isomorphisms αABCD, λAB and ρAB are identities

for all choices of A,B,C,D. In particular, 11A = A for all objects A of a 2-category.To any bicategory B one can associate new bicategories denoted by Bop, Bco and Bcoop. These

are constructed by taking respectively opposite composition for the 1-cells, for vertical compositionof 2-cells and for both.

Examples 1.1. (1) Let A be an ordinary category. Then A is also a bicategory, if weconsider HomA(A,B) as a discrete category for any two objects A,B ∈ A. In this wayA is even a 2-category.

(2) The second basic example is the bicategory Cat of small categories, functors and naturaltransformations. This is again a 2-category. Remark our convention to write the compo-sition of 1-cells in a ‘covariant’ way. This has the important implication if we computethe compostion of functors and the horizontal composition of natural transformations inCat. Let A,B and C be categories and F : A → B and G : B → C functors. Then wewill denote

(5) F •B G = GF : A → Cfor the composite functor. In the same way, for categories A,B and C, functors F,G :A → B and H,K : B → C and natural transformations α : F → G and β : H → K, wewill denote

(6) α•Bβ = βα : HF → KG,

where the right hand side is the Godement product of natural transformations.(3) If F and G are two functors between the Hom-Set categories A and B, then Nat(F,G)

is in general no longer a set but a class. Consequently, if we consider a 2-categorywhose 0-cells are all Hom-Set categories, 1-cells are functors and 2-cells are naturaltransformations, then the Hom-category between two 0-cells in this 2-category is a Hom-Class category. We will denote this 2-category by CAT.

(4) A particulary interesting sub-2-category of CAT is constructed as follows. The 0-cellsconsist of all Grothendieck categories and the 1-cells are so-called right continuous func-tors, i.e. functors that preserve colimits. By [49, Lemma 5.1] the Hom-categories of this2-category are Hom-Set categories.

1.1. BICATEGORIES AND MONOIDAL CATEGORIES 11

We will give more examples of bicategories in the following sections.Let A and B be two bicategories. A lax Functor F : A → B consists of the following data:

(i) for every 0-cell A ∈ A, a 0-cell FA ∈ B;(ii) for every pair of 0-cells A,B ∈ A a functor

FAB : HomA(A,B) → HomB(FA,FB);

(iii) for every triple of 0-cells A,B,C ∈ A, a natural transformation

(7) γABC : cFA,FB,FC (FAB × FBC) ⇒ FAC cABC ;

(iv) for every 0-cell A ∈ A, a natural transformation

(8) δA : 11FA ⇒ FAA 11A;

such that the following diagram commutes for all objects A,B,C,D in B,

(9) (FAB(f) •FB FBC(g)) •FC FCD(h)γ2 //

αFA,FB,FC,FD

FAD((f •B g) •C h)

FAD(αABCD)

FAB(f) •FB (FBC(g) •FC FCD(h))γ1 // FAD(f •B (g •C h))

where γ1 = γACD (γABC•FCFCD(h)) and γ2 = γABD (FAB(f)•FBγBCD). For the othercoherence axioms that γ and δ have to satisfy, we refer to [25, section 7.5].

If all γABC and δA are natural isomorphisms, then F is called a pseudo-functor .If F : A → B is a lax functor, and P a property of functors, then we say that F is locally

P , if FAB is P for all A,B ∈ A. In this way we can speak about locally an equivalence, locallyfaithful ,... .

Let us introduce the following notation. For a 1-cell f : A → B in a bicategory A, we find afunctor − • f : HomA(C,A) → Hom(C,B).

Consider two lax functors F,G : A → B. A lax natural transformation α : F ⇒ G consists ofthe following data:

(i) for every 0-cell A ∈ A, a 1-cell αA : FA→ GA;(ii) for every pair of objects A,B ∈ A, a natural transformation

τAB : (αA • −) GAB ⇒ − • αB FAB;

where α and τ satisfy coherence axioms for which we refer to [25, section 7.5].

When F and G are pseudo functors, and each τAB is a natural isomorphism, then α is calleda pseudo natural transformation.

Consider two lax natural transformations α, β : F ⇒ G between the lax functor F,G : A → B.A modification Ξ : α ; β is a family

ΞA : αA ⇒ βA

of 2-cells in B, indexed by 0-cells of A. We require this family to satisfy the following property:for every pair of 1-cells f, g : A→ A′ in A and every 2-cell σ : f ⇒ g the equality

ΞA•GAGσ = Fσ•FA′ΞA′holds in B.

With the above definitions, we can construct new bicategories.Let A and B be two bicategories, then we denote by Lax(A,B) the bicategory whose 0-cells

are lax functors, 1-cells are lax natural transformations and 2-cells are modifications. Consideringpseudo functors, pseudo natural transformations and modifications we obtain a sub-bicategoryof Lax(A,B), that is denoted as [A,B]. If B is a 2-category, then Lax(A,B) and [A,B] are2-categories as well. In general, the Hom-categories of Lax(A,B) and [A,B] can be Hom-Classcategories, even if the Hom-categories of A and B are Hom-Set categories.

Examples 1.2. (i) Let B be any bicategory and Ω a 0-cell in B. We can construct thepseudo functor

(10) RepΩ : B → CAT,

as follows.


- For a 0-cell A ∈ B, we define RepΩ(A) = Hom(Ω, A);- for a 1-cell a : X → Y in B, define

RepΩ(a) = − •X a : Hom(Ω, X) → Hom(Ω, Y )(x : Ω → X) 7→ (x •X a : Ω → Y )(ϕ : x→ x′) 7→ (ϕ•Xa : x •X a→ x′ •X a)

- for a 2-cell α ∈ XHomY2 (a, a′), where a, a′ ∈ Hom1(X,Y ), define a RepΩ(α) = α ∈

Nat(− •X a,− •X a′) as follows

αx = x•Xα : x •X a→ x •X a′,

αϕ = ϕ•Xα : f •X a→ g •X a′;

for all x ∈ Hom1(Ω, X) and ϕ ∈ ΩHomA2 (f, g).

Dually, we can define a pseudofunctor

ΩRep : Bop → CAT,

by ΩRep(A) = Hom(A,Ω), ΩRep(a) = a •Y − and ΩRep(α) = α•Y−.(ii) Given a 1-cell f : Ω′ → Ω we can construct a pseudo natural transformations

Repf = f • − : RepΩ ⇒ RepΩ′ ;

fRep = − • f : Ω′Rep ⇒ ΩRep.

(iii) Starting from a 2-cell α ∈ Ω′HomΩ2 (f, g), we obtain modifications

Repα = α•− : Repf ; Repg;

αRep = −•α : fRep ; gRep.

Let A and B be two 0-cells in a bicategory B. We call A and B internally equivalent inB, if there exist 1-cells f ∈ Hom1(A,B) and g ∈ Hom1(B,A) and two isomorphisms η ∈AHomA

2 (A, f •B g) and ε ∈ BHomB2 (g •A f,B).

Let A and B be two bicategories. A bi-equivalence from A to B consists of a pair of pseudofunctors F : A → B and G : B → A together with an internal equivalence between 1 and GF in[A,A] and an internal equivalence between F G and 1 in [B,B]. In other words, there must existpseudo natural transformations α : 1 → G F , β : G F → 1, γ : F G→ 1 and δ : 1 → F Gand modifications 1 ∼= β α, α β ∼= 1, 1 ∼= δ γ and γ δ ∼= 1.

An equivalent condition for a bi-equivalence is the following. A pseudo functor F : A → Bis a bi-equivalence if and only if F is locally an equivalence and surjective up-to-equivalence onobjects. This latter condition means that for any 0-cell B ∈ B we can find a 0-cell A ∈ A suchthat FA and B are internally equivalent in B.

1.1.2. Bicategories versus 2-categories. Computations in a bicategory B can become quitehard because of the presence of the associativity isomorphisms αABCD and the unit isomorphismλAB and ρAB. Since in a 2-category A these isomorphisms are just the identities, computations inA are much simpler. For this reason we would like to find a way to tranfer the more complicatedcomputations in B to the easier setting of A. A well-known result that belongs to the folkloreamong category-theorists tells that this is always possible (see e.g. [108], [89]). For sake ofcompleteness, we provide now a full prove of this construction.

Let F : B → A be a locally faithful pseudo functor. Suppose we want to verify an equation of2-cells in B of the form

(11) α1 αABCD = α2,

where α1 ∈ AHomD2 ((f •B g) •C h, p), α2 ∈ AHomD

2 (f •B (g •C h), p), with f ∈ Hom1(A,B),g ∈ Hom1(B,C), h ∈ Hom1(C,D) and p ∈ Hom1(A,D). Apply the pseudo functor F on thisequation, then we obtain the following diagram in the 2-category A,

(12) (FAB(f) •FB FBC(g)) •FC FCD(h)γ2 // FAD((f •B g) •C h)

FAD(α2) //

FAD(αABCD)

FAD(p)

FAB(f) •FB (FBC(g) •FC FCD(h))γ1 // FAD(f •B (g •C h))

FAD(α1)

44jjjjjjjjjjjjjjjjj


where γ1 and γ2 are isomorphisms given by compositions of the natural isomorphisms γ from(7). The inner quadrangle is commutative by the coherence axiom on γ, see (9). Since A isa 2-category, γ−1

1 FAD(αABCD) γ2 is just the identity. Denote β1 = FAD(α1) γ1 andβ2 = FAD(α2) γ2. Then, the diagram (12) commutes if and only if β1 = β2 if and only ifFAD(α1) = FAD(αABCD) FAD(α2). Since F is locally faithful, in particular FAD is a faithfulfunctor. This implies that if FAD(α1) = FAD(αABCD)FAD(α2), if and only if α1 = αABCD α2.So (11) is satisfied in B if and only if β1 = β2 commutes in A, i.e. the equivalent condition in the2-category holds true.

We can follow the same reasoning for any diagram of 2-cells that is constructed in B. Thisjustifies the following statement.

Proposition 1.3. Every calculation that holds in 2-categories can be repeated in a bicategoryfor which there exists a locally faithful pseudo functor to a 2-category.

Consider the following sub-2-category Cat(B) of CAT

- The 0-cells are of the form Hom(A,B) with A and B 0-cells in B;- the 1-cells are functors of the form RepΩ(a) = − • a : Hom(Ω, A) → Hom(Ω, B), witha ∈ Hom1(A,B) as in Example 1.2(i).

- the 2-cells RepΩ(a) ⇒ RepΩ(a′) are natural transformations in Nat(− • a,− • a′) of the form

RepΩ(α) where α ∈ XHomY2 (a, a′).

In particular, we see that if the 2-cells between two given 1-cells determine a set in B, the sameholds in Cat(B), i.e. the Hom-categories in Cat(B) are Hom-Set categories if and only if Hom-categories in B are Hom-Set categories.

Now we can construct the bicategory [B,Cat(B)] which consists of pseudo-functors, pseudonatural transformations and modifications from B to Cat(B). From the construction of Cat(B) itis clear that the representable pseudo functors, pseudo natural transformations and modificationsas defined in Example 1.2 are in [B,Cat(B)]. Now define

Rep : B → [B,Cat(B)]A 7→ Rep(A) = RepAa 7→ Rep(a) = Repaα 7→ Rep(α) = Repα

for any 0-cell A, 1-cell a and 2-cell α in B, where we used again the notation of Example 1.2.

Lemma 1.4. With notation as above, Rep : Bop → [B,Cat(B)] is a locally faithful pseudofunctor.

Proof. Let us first check that Rep is a pseudo functor. By definition of the representablesin Example 1.2, we see that Rep is well-defined as a pseudo functor Bop → [B,CatB] oncewe have found natural isomorphisms γABC and δA as in (7) and (8). Take any A,B,C ∈ B,f ∈ Hom1(A,B) and g ∈ Hom1(B,C). Consider Repf , Repg and Repf•Bg. For any X ∈ Band h ∈ Hom1(X,A) we obtain from (1) a natural isomorphism

αXABC : Repf•g(h) = (f • g) • h⇒ f • (g • h) = Repf • (Repg(h)).

In the same way, for any A ∈ B we find for all X ∈ B and all f ∈ Hom1(X,A) that λAB (2)induces a natural isomorphism

λAB : Rep11A(f) = 11A •A f ⇒ f.

We conclude that Rep is a pseudo functor.Recall that a pseudo functor F : B → A is faithful if and only if the usual functor FAB :

HomB(A,B) → HomA(FA,FB) is faithful for all objects A and B in B. Thus we have to checkwhether the map AHomB

2 (f, g) → Mod(Repf ,Repg) is injective, where Mod denotes the classof all modifications from Repf to Repg. Take any 2-cell α : f ⇒ g in B. If we apply the pseudofunctor Rep, we obtain the modification Repα : Repf ; Repg. By defintion Repα consists ofa family of natural transformations

α•X− : f •X − ⇒ g •X −


indexed by the 0-cells X ∈ B, where f •X − and g •X − denote functors from the categoryHom(X,B) to Hom(X,A).

Consider now α, β ∈ AHomB2 (f, g) and suppose that Repα = Repβ . Then we find in

particular by taking X = B above two times the same natural transformation

α•B− = β•B− : f •B − ⇒ g •B −.

Evaluating this equality in the identity morphism on B, we obtain α•BB = β•BB.Consider now the natural isomorphism ρ (2) from the definition of a bicategory. The naturality

of ρ implies the commutativity of the following diagram

fρAB(f) //

α

f •B B

α•BB

g

ρAB(g) // g •B B

Since ρ is an isomorphism, we thus obtain

α = ρ−1AB(g) (α•BB) ρAB(f)

= ρ−1AB(g) (β•BB) ρAB(f) = β

We can conlude that Rep is locally faithful.

As a final step in our construction, we consider the sub-2-category Rep(B) of [B,Cat(B)]consisting of the full image of the pseudo functor Rep. This means Rep(B) can be describes asfollows

- The 0-cells are pseudo functors of the form RepA, where A is a 0-cell in B;- the 1-cells are pseudo natural transformations of the form Repa : RepB ⇒ RepA, wherea ∈ Hom(A,B) is a 1-cell in B

- the 2-cells are modifications of the form Repα : Repa → Repa′ , where α ∈ AHomB2 (a, a′) is

a 2-cell in B.

Theorem 1.5. Let B be any bicategory and Rep(B) the 2-category associated to B con-structed as above. Then B and Rep(B)op are bi-equivalent.

Proof. By the construction of Rep(B), the pseudofunctor Rep can be corestricted to apseudo functor Bop → Rep(B). Since Rep is locally faithful, the corestriction is locally faithfulas well. Moreover, by construction, the corestricted functor is locally full and surjective on 0-cells.We conclude that Bop and Rep(B) are bi-equivalent, and thus B and Rep(B)op are bi-equivalentas well.

Corollary 1.6. All calculations that hold in 2-categories can be repeated in bicategories.In particular, all diagrams that are constructed out of the associativity and identity isomor-

phisms commute in any bicategory. (This last statement is known as the Coherence Theorem.)

Proof. The first statement is a consequence of Proposition 1.3 and Theorem 1.5.The second statement follows from the first one and the fact that diagrams built out of

associativity and identity isomorphisms commute obviously in a 2-category.

Consequently, from now on, we will rely on the 2-categorical calculus when we are dealing withbicategories.

Remark 1.7. The associativity of the composition of 1-cells and the horizontal compositionof 2-cells in Rep(B)op should be understood as follows. We relate to any 1-cell f ∈ Hom1(A,B)in B a functor Repf = f • − and functors associate in a trivial way. This trivial associativity canbe thought of as “pre-ordening all brackets in B”. Indeed :

((Reph Repg) Repf )(a) = (Reph (Repg Repf ))(a) = h •C (g •B (f •A a))

for all 1-cells f ∈ Hom1(A,B), g ∈ Hom1(B,C), h ∈ Hom1(C,D) and a ∈ Hom1(Ω, A) inB.


1.1.3. Monoidal categories. Consider a bicategory B with only one 0-cell A. Then B iscompletely determined by the Hom-category Hom(A,A) and the value of the functors c and 11 atA. This leads to the following defintion.

We define a monoidal category as a bicategory with only one 0-cell. By the observationabove, a monoidal category is a triple (C,⊗, I) consisting of a category C together with a bifunctor⊗ : C × C → C and an object I ∈ C, such that there exist isomorphisms

(13) (X ⊗ Y )⊗ Z ∼= X ⊗ (Y ⊗ Z) and X ⊗ I ∼= X ∼= I ⊗X,

which are natural in all X,Y, Z ∈ C. We omit to write down the other compatibility relations,which boil down to the fact that all possible diagrams, composed of the natural isomorphisms (13)commute. For more details we refer to e.g. [26, section 6.1] or [88, section VII.1]. We call ⊗ thetensor product of C and I the unit object for ⊗.

A 2-category with one object will be called a strict monoidal category. This is a monoidalcategory such that the natural isomorphisms (13) are identities. A well-known result tells thatevery monoidal category is equivalent to a strict monoidal category, consequently we can proveany theorem for monoidal categories without dealing explicitly with the isomorphisms (13). Theseare consequences of Theorem 1.5 and Corollary 1.6. An explicit proof can be found in [81, SectionXI.5].

One can now easily observe the following

Theorem 1.8. For any 0-cell A in a bicategory B, the Hom-category Hom(A,A) is a monoidalcategory.

Consider now a bicategory B with two objects, A and B, such that Hom(B,B) is the discretecategory with one object and Hom(B,A) is empty. We say that B is a left ghost-object. We knowHom(A,A) is a monoidal category. Clearly Hom(A,B) is not a monoidal category, but we havea functor

cAAB : Hom(A,A)×Hom(A,B) → Hom(A,B).

This leads to the following notion.A left monoidal category on a monoidal category C is a couple (M,⊗M), where M is a

category and ⊗M : C ×M →M is a bifunctor, such that we can construct a bicategory B withone 0-cell A and a left ghost object B after we make the following identifications

Hom(A,A) = C, Hom(A,B) = M, cAAA = ⊗, cAAB = ⊗M, 11A = I.

This means that for all X,Y ∈ C and M ∈ M, the following isomorphisms exists and arenatural in all arguments

(X ⊗C Y )⊗MM ∼= X ⊗M (Y ⊗MM) and I ⊗MM ∼= M ;

where we denoted ⊗C for the tensor product of C and I for the unit object of C. The othercompatibility conditions imply the commutativity of all possible diagrams that can be composedout of the natural isomorphisms introduced in the definition of the monoidal structure on M andC.

Similarly, one defines a right monoidal category on a monoidal category. A study of left andright monoidal categories on C, without their relation with bicategories was done in [98], wherethey are called C-categories.

Theorem 1.9. For any two 0-cells A and B in a bicategory B, the Hom-category Hom(A,B)is a left monoidal category on the monoidal category Hom(A,A) and a left monoidal category onHom(B,B).

Consider again a bicategory with two 0-cells A, B. We know that Hom(A,A) and Hom(B,B)are monoidal categories and Hom(A,B) is a left monoidal category on Hom(A,A) and a rightmonoidal category on Hom(B,B). Moreover, for any triple of 1-cells f ∈ Hom1(A,A), g ∈Hom(A,B) and h ∈ Hom(B,B) we have the following isomorphism in Hom(A,B)

αAABB : (f •A g) •B h→ f •A (g •B h).

We obtain the following definition.


Let (C,⊗C , IC) and (D,⊗D, ID) be two monoidal categories. We say that the triple (M,⊗`,⊗r)is a bimonoidal category on (C,D), if (M,⊗`) is a left monoidal category on C and (M,⊗r) is aright monoidal category on D, such that we can construct a bicategory B with two 0-cells A andB after we make the following identifications

Hom(A,A) = C, Hom(A,B) = M, Hom(B,B) = D, 11A = IC , 11B = ID

cAAA = ⊗C , cABB = ⊗`, cABB = ⊗r, cBBB = ⊗D.Hom(B,A) and the other bifunctors are trivial.

This definition assures that all possible diagrams with tensor products commute up to isomor-phism. In particular,

C ⊗l (M ⊗r D) ∼= (C ⊗lM)⊗r Dfor all C ∈ C, M ∈M and D ∈ D.

Theorem 1.10. For any two 0-cells A and B in a bicategory B, the Hom-category Hom(A,B)is a bimonoidal category on (Hom(A,A),Hom(B,B)).

Consider now a bicategory with three 0-cells A,B and C. We know that Hom(B,B) is amonoidal category, Hom(A,B) is a right monoidal category on Hom(B,B) and Hom(B,C) is aleft monoidal category on Hom(B,B). For any triple of 1-cells f ∈ Hom1(A,B), g ∈ Hom(B,B)and h ∈ Hom(B,C) we have the following isomorphism in Hom(A,C)

αABBC : (f •B g) •B h→ f •B (g •B h).This leads again to a new defintion.

We say that (M,N , ⊗) is a compatible pair of monoidal categories on (C,A) if (M,⊗M) is aright monoidal category on (C,⊗) and (N ,⊗N ) is a left monoidal category on (C,⊗) and we canfind a category A and a the bifunctor ⊗ : M×N → A, such that we can construct a bicategoryB with three 0-cells A,B,C after we make the following identifications

Hom(B,B) = C, Hom(A,B) = M, Hom(B,C) = N , Hom(A,C) = AcAAA = ⊗, cABB = ⊗M, cBBC = ⊗N , cABC = ⊗, 11A = I.

and all other Hom-categories and functors are trivial.This definition implies again that all possible diagrams with tensor products commute. In

particular, we obtain the following isomorphisms

(M ⊗M C)⊗N ∼= M⊗(C ⊗N N)

for all M ∈M, C ∈ C and N ∈ N .

Theorem 1.11. Consider three 0-cells A,B and C in a bicategory B. Then Hom-categoriesHom(A,B) and Hom(B,C) define a compatible pair of monoidal categories on

(Hom(B,B),Hom(A,C)).

Example 1.12. First of all, it is obvious that every monoidal category is also a left and rightmonoidal category on itself.

Example 1.13. Consider the category Cat, where the objects are small categories and themorphisms are functors. Using the cartesian product as tensor product and the discrete one-objectcategory as unit object, Cat becomes a monoidal Hom-Set category.

Similarly we can consider CAT as the monoidal Hom-Class category whose objects are Hom-Setcategories.

Example 1.14. LetA be any category, then we denote by Fun(A,A) the category of covariantfunctors F : A → A and natural transformations between those functors. As mentioned before,Fun(A,A) is in general a Hom-Class category. If, for example, A is a small category, or A isa Grothendieck category and we consider only (right) continuous functors, then Fun(A,A) is aHom-Set category. This category is a monoidal category with the composition as tensor product onthe objects (i.e. functors) and Godemond product as tensor product between the morphisms (i.e.natural transformations). Let B be any other category, then Fun(A,B) is a right monoidal categoryon Fun(A,A) and Fun(B,A) is a left monoidal category on Fun(A,A), the tensor product is in

1.2. MONADS AND ALGEBRAS 17

both cases given by the composition. Finally, A itself is a left monoidal category on Fun(A,A), ifwe define F ⊗A X = F (X) for all F ∈ Fun(A,A) and X ∈ A. Remark that A can be identifiedwith Fun(1,A), where 1 is the one-object category.

Let A, B and C be three categories, then (Fun(A, C),Fun(B,A), ⊗) is a compatible pair on(Fun(A,A),Fun(B, C)). This follows by the above observations if we define

⊗ : Fun(A, C)× Fun(B,A) → Fun(B, C), F ⊗G = F G.

Example 1.15. If A is an associative ring with unit, then the category of A-bimodules AMA

(we introduce rings and modules explicitly in the next section) is well-known to be a monoidalcategory, moreover, the category of right A-modules MA is a right monoidal category on AMA.Similarly, AM is a left monoidal category on AMA. Furthermore, (MA,AM,⊗A) is a compatibleon (AMA,Ab), since the A-tensor product defines a bifunctor MA × AM→ Ab.

1.2. Monads and algebras

1.2.1. Monads, algebras and modules. Let B be any bicategory, with notation as in Sec-tion 1.1. A monad a = (R, a, µ, η) consists of a 0-cell R, a 1-cell a ∈ Hom1(R,R) and two 2-cellsµa ∈ RHomR

2 (a •R a, a) and ηa ∈ RHomR2 (11R, a) such that the following diagrams commute.

a •R aµa

||yyyy

yyyy

y(a •R a) •R a

µa•Raoo

αRRRR

a

a •R aµa

bbEEEEEEEEEa •R (a •R a)a•Rµa

oo

aρRR //

λRR

a •R 11Ra•Rηa

11R •R a ηa•Ra

// a •R a

µa

iiRRRRRRRRRRRRRRRRR

A right module of Ω-type over a monad a or a Ω-a-module is a couple (m, ρm) where m ∈Hom1(Ω, R) for some 0-cell Ω in B and ρm ∈ ΩHomR

2 (m •R a,m) such that ρm (ρm•Ra) =ρm (m•Rµa) αΩRRR and ρm (m•Rηa) ρΩR = m. In a similar way one introduces a-Ωmodules.

A morphism of Ω-a modules ψ : (m, ρm) → (n, ρn) consists of a 2-cell ψ ∈ ΩHomR2 (m,n)

such that ρn (ψ•Ra) = ψ ρm.Let a = (R, a, µa, ηa) and b = (S, b, µb, ηb) be two monads. An a-b bimodule is a triple

(m,λm, ρm) where (m,λ) is a a-S module and (m, ρm) is a R-b module, such that λm(a•Rρm)αRRSS = ρm (λm•Sb). In particular, m ∈ Hom1(R,S).

We can construct a category whose objects are all right modules of Ω-type over C, and whosemorphisms are morphisms of comodules between them. This category is denoted as Rmod(Ω, a).In the same way, Lmod(a,Ω) is the category of all left modules of Ω-type and Bicom(a, b) is thecategory of all a-b bimodules.

Remark 1.16. In the definition of a monad, only one 0-cell R of the bicategory B wasused. This means that the notion of a monad makes sense in the monoidal category Hom(R,R).To construct a right module over a monad, two objects in the bicategory are used, R and Ω,which implies that we can reformulate this notion in the framework of a right monoidal categoryHom(Ω, R) on a given monoidal category Hom(R,R). Similarly, bimodules make use of bimonoidalcategories. Finally if we consider at the same time left and right modules over a single monad weuse three objects in the bicategory B, so this can be formulated over a compatible pair of monoidalcategories. Starting from these observations, we will now introduce algebras and their modulesover monoidal categories.

An algebra (or a monoid) in a monoidal category (A,⊗, I) is a three-tuple A = (A,µ, η),where A is object in A, µ : A⊗A→ A and η : I → A are morphisms in A, such that the following


diagrams commute.

A⊗ (A⊗A)A⊗µ //

∼=

A⊗A

µ

(A⊗A)⊗A

µ⊗A

A⊗A µ// A

A∼= //

∼=

A⊗ I

A⊗η

I ⊗Aη⊗A

// A⊗A

µddJJJJJJJJJJ

The maps µ and η are respectively called the multiplication and the unit of the algebra A. Wewill refer to this algebra both with A as just with the object A.

A right module or a representation of the algebra A in a right monoidal category M on A isa pair M = (M,ρ), where M is an object in M and ρ : X ⊗M A→M is morphism in M, suchthat the following conditions hold

M ⊗M (A⊗A)M⊗Mµ//

∼=

M ⊗M A

ρ

(M ⊗M A)⊗M A

ρ⊗MA

M ⊗M A ρ

// M

M∼= // M ⊗M I

M⊗Mη

M ⊗M A

ρ

ddIIIIIIIIII

Let M = (M,ρM ) and N = (N, ρN ) be two right A-modules, with both M,N ∈ M, then wedefine a A-module morphism between M and N as a map f : M → N in M that is right A-linear,i.e.

M ⊗M Af⊗MA//

ρM

N ⊗M A

ρN

M

f// N

The category of right A-modules in M together with A module morphisms between them will bedenoted by ModM-A.

Similarly one can introduce the category of left A modules A-ModM. Consider algebras Ain C and B in D. If M is a bimonoidal category on (C,D), then we can consider the categoryA-ModM-B of A-B bimodules in M.

Example 1.17. Consider the monoidal category CAT from Example 1.13. Algebras in CATare the strict monoidal categories. Let C be a monoidal category, left modules of C in CAT arethen (strict) left monoidal categories on C. If C and D are (strict) monoidal categories, then aC-D bimodule is a bimonoidal category on (C,D).

Example 1.18 (Monads and their algebras). The algebras in the monoidal category of Exam-ple 1.14 are usually called triples, other terminologies are monads and triads. A triple F = (F,Υ, ζ)on a category A, being an algebra in Fun(A,A), consists of a (covariant) functor F : A → A andtwo natural transformations Υ : F 2 → F and ζ : A → F such that the following relations holdtrue

Υ (FΥ) = Υ (ΥF );Υ (Fζ) = F = Υ (ζF ).

As explained in Example 1.14, A is a left monoidal category on Fun(A,A). A representation ofa triple F on A, in the monoidal category A on Fun(A,A), is usually called an algebra for F.To avoid confusion, we will call the elements of the category F-AMod modules of F in A. The


category is denoted as AF. A module of F in A consists of an element X ∈ A, together with amorphism PX : F (X) → X, such that

PX (PXF ) = PX (FΥX) PX (FζX) ∼= X.

Example 1.19. Consider (Ab,⊗Z,Z) as a monoidal category, an algebra R in Ab is noth-ing else than an associative ring with unit. Moreover, ModAb-R and R-ModAb coincide with

respectively MR and RM.

Example 1.20. Consider a commutative ring k. As a next step, we can start from themonoidal category (Mk,⊗k, k). An Algebra A in this monoidal category are just k-algebras andfor the modules we find ModMk

-A and A-ModMkcoincide with respectively with MA and

AM.

Example 1.21. Let us compute the algebras and modules in the monoidal categories ofExample 1.15. Let A be an associative ring with unit (i.e. a k-algebra as constructed in theprevious example) and A = AMA. An algebra R in A will be called an A-ring. This is nothingelse than a usual ring together with a ring morphism η : A→ R, which plays the role of the unitmorphism.

The right R-modules are the usual right R-modules that also admit a right A-module structure.Moreover, ModMA

−R and MR coincide.

There is an interesting connection between monads in a bicategory, algebras in a monoidalcategory and algebras in Fun, i.e. triples.

Theorem 1.22. Let B be a bicategory. Consider a quadruple a = (R, a, µ, η) in B consistingof a 0-cell R, a 1-cell a : R → R and 2-cells µ : a • a ⇒ a and η : R ⇒ a. The followingstatements are equivalent.

(i) a is a monad in B;(ii) (a, µ, η) is an algebra in the monoidal category Hom(R,R);(iii) RepΩ(a) = (RepΩ(R),RepΩ(a),RepΩ(µ),RepΩ(η)) is a monad in CAT (or Rep(B)) for

all Ω ∈ B;(iv) RepR(a) is a monad in CAT (or Rep(B));(v) (RepΩ(a),RepΩ(µ),RepΩ(η)) is an algebra in the monoidal category of functors and nat-

ural transformations Fun(Hom(Ω, R),Hom(Ω, R)) for all Ω ∈ B;(vi) (RepR(a),RepR(µ),RepR(η)) is an algebra in Fun(Hom(R,R),Hom(R,R)).

Proof. (i) ⇔ (ii). Follows directly from the definition.

(ii) ⇒ (iii). Follows directly from the from the fact that RepΩ is a pseudo functor.

(iii) ⇒ (iv). Trivial.

(iv) ⇒ (ii). Verify that RepR(a) ∼= a, RepR(µ) ∼= µ and RepR(η) ∼= η.

(iii) ⇔ (v) and (iv) ⇔ (vi) are an application of (i) ⇔ (ii).

1.2.2. Comonads, coalgebras and comodules. Let B be a bicategory and use notation asbefore. Comonads, comodules over comonads and morphisms between them in B are defined asmonads, modules and their morphisms in the bicategory Bco. Let us give their explicit definition,since they play a prominent role in this book.

A comonad C = (A, c,∆c, εc) consists of a 0-cell A, a 1-cell c ∈ Hom1(A,A) and two 2-cells∆c ∈ AHomA

2 (c, c •A c) and εc ∈ AHomA2 (c, 11A) that satisfy the following conditions

c •A c∆c•Ac// (c •A c) •A c

αAAAA

c

∆c

<<zzzzzzzzz

∆c ""DDD

DDDD

DD

c •A cc•A∆c

// c •A (c •A c)

c∆c

))RRRRRRRRRRRRRRRRRρAA //

λAA

c •A 11A

11A •A c c •A cεc•Acoo

c•Aεc

OO


A Morphism of Comonads, ϕ : C → D between comonads C = (A, c,∆c, εc) and (A, d,∆d, εd) isa 2 cell ϕ ∈ AHomA

2 (c, d) such that ∆d ϕ = (ϕ•Aϕ) ∆c and εd ϕ = εc.A right comodule of Ω-type over a comonad C or an Ω-C comodule is a couple (m, ρm) where

m ∈ Hom1(Ω, A) for some 0-cell Ω in B and ρm ∈ ΩHomA2 (m,m•Ac) such that αΩAAA(ρm•Ac)

ρm = (m•A∆c) ρm and (m•Aεc) ρm = ρΩA. Left C-Ω comodules (m,λm) are defined in asymmetric way. A Morphism of right Ω-C comodules, ψ : (m, ρm) → (n, ρn) consists of a 2-cellψ ∈ ΩHomA

2 (m,n) such that ρn ψ = (ψ•Ac)ρm. Let C = (A, c,∆c, εc) and D = (B, d,∆d, εd)be two comonads. A C-D bicomodule is a triple (m,λm, ρm) where (m,λm) is a C-B comoduleand (m, ρm) is an A-D comodule, such that αAABB (λm•Bd)ρm = (c•Aρm)λm. In particular,m ∈ Hom1(A,B).

All right comodules of Ω-type over C, with morphisms of comodules between them form acategory that is denoted as Rcom(Ω,C). Similarly one introduces Lcom(C,Ω) as the category ofall left comodules of Ω-type and Bicom(C,D) as the category of all C-D bicomodules.

As for monads and modules, it makes sense to define comonads and comodules in the frameworkof monoidal categories, which gives rise to the concept of coalgebras.

A coalgebra (or comonoid) in a monoidal category (A,⊗, I) is an algebra in the dual category(Aop,⊗, I). More specifically, a coalgebra is a three-tuple C = (C,∆, ε), consisting of an objectC ∈ A, and two morphisms ∆C : C → C ⊗C and εC : C → I in A such that following diagramscommute

C∆C //

∆C

C ⊗ C

∆C⊗C

(C ⊗ C)⊗ C

∼=

C ⊗ CC⊗∆C

// C ⊗ (C ⊗ C)

C∆C

((QQQQQQQQQQQQQQQ C ⊗A∼=oo

A⊗ C

∼=

OO

C ⊗ Cε⊗C

oo

C⊗εC

OO

Similarly one introduces comodules over a coalgebra as the dual notion of modules over an algebra.Let (M,⊗M) be a right monoidal category on A. The pair M = (M,ρM ) is a right comoduleof C in M, if M is an object in M and ρM : M → M ⊗M C is a morphism in M such thatfollowing diagrams commute.

MρM

//

ρ

M ⊗M C

M⊗M∆

M ⊗M (C ⊗ C)

∼=

M ⊗M CρM⊗MC

// (M ⊗M C)⊗M C

M∼= //

ρM $$IIIIIIIIII M ⊗M I

M ⊗M C

M⊗Mε

OO

A morphism C-comodule morphism between the C-comodules M = (M,ρM ) and N = (N, ρN )is a morphism f : M → N in M that makes the following diagram commutative

Mf //

ρM

N

ρN

M ⊗M C

f⊗MC// N ⊗M C

We say that f is a right C-colinear map. The category of all right C comodules in M and C-colinear maps is denoted by ComodM-C. Similarly, one introduces the categories D-ComodMand D-ComodM-C, where D denotes a coalgebra in a monoidal category B such that M is aleft monoidal category on B, respectively M is a bimonoidal category on (B,A).

Example 1.23 (cotriples and their coalgebras). Consider the monoidal category Example 1.14.Coalgebras in this category are called cotriples (or comonads, cotriads, standard constructions). Let


A be any category, then a cotriple F = (F, δ, ε) on A, consists of a covariant functor F : A → Aand two natural transformations δ : F → F 2 = F F , and ε : F → 11A subjected to the followingconditions

• (δF ) δ = (Fδ) δ;• (εF ) δ = (Fε) δ = F ;

We will refer to a cotriple both with F = (F, δ, ε) or only with the cotriple functor F : A → A.We fix now a cotriple F = (F, δ, ε) on A. As explained above, F is a coalgebra in Fun(A,A)

and from Example 1.14, we know that A is a left monoidal category on Fun(A,A), so we canconsider the comodules over F in A. In the literature, these comodules are often called coalgebras.

The category F -AComod is usually denoted by AF . An element X ∈ AF consists of anelement X ∈ A and a morphism %X : X → F (X) in A, such that the following is satisfied:

F (%X) %X = δX %X(14)

εF (X) %X = X(15)

A morphism f : X → Y in AF between consists of a morphism f : X → Y in A satisfying%Y f = F (f) %X .

Example 1.24 (corings and comodules). Let A be a (not necessarily commutative) ring. Withan A-coring1 we mean a coalgebra in the monoidal category AMA, i.e. a three-tuple (C,∆C, εC)such that the following diagrams commute in AMA

C∆C //

∆C

C⊗A C

∆C⊗AC

C⊗A CC⊗A∆C

// C⊗A C⊗A C

C∆C

((RRRRRRRRRRRRRRRR C⊗A A∼=oo

A⊗A C

∼=

OO

C⊗A Cε⊗AC

oo

C⊗AεC

OO

The map ∆C is called the comultiplication and εC the counit of the coring C. We will make use ofthe Sweedler notation, for any c ∈ C, ∆(c) = c(1) ⊗A c(2). This way, the coassociativity conditioncan be formulated in the follwowing way

(C⊗A ∆C) ∆(c) = c(1)(1) ⊗A c(1)(2) ⊗A c(2)= (∆C ⊗A C) ∆(c) = c(1) ⊗A c(2)(1) ⊗A c(2)(2)

= c(1) ⊗A c(2) ⊗A c(3).

The counit property becomes in this way c = c(1)εC(c(2)) = εC(c(1))c(2).A right C-comodule is the name for an object in ComodMA

-C, i.e. a right A-module M thathas an additional right A-linear structure map ρ : M →M ⊗C, such that the usual coassociativityand counit conditions are satisfied. If we make use of the Sweedler notation for comodules, i.e.for any m ∈M , ρ(m) = m[0] ⊗A m[1], then the coassociativity and counit condition read as

(ρ⊗A C) ρ(m) = (M ⊗A ∆) ρ(m)= m[0][0] ⊗A m[0][1] ⊗A m[1] = m[0] ⊗A m[1](1) ⊗A m[1](2)

= m[0] ⊗A m[1] ⊗A m[2]

(M ⊗A ε) ρ(m) = m[0]ε(m[1]) = m.

The category of right comodules is denoted by MC. Let B be another ring and D a B-coring,then one introduces the categories CMD and CMB as respectively C-Comod

AMB-D and C-

ComodAMB

-B. Where we consider B as a trivial B-coring. Objects in CMD are named C-Dbicomodules and objects in CMB are refered to as C-B bicomodules.

Example 1.25 (coalgebras). If k is a commutative ring, then (Mk,⊗k, k) is a monoidalcategory, so we can compute their coalgebras. We call them k-coalgebras. Obviously k-coalgebrasprovide examples of k-corings. However, remark that not every k-coring is a k-coalgebra, since fora k-coring C, the left and right k-module action not nessecarily coincide.

We also obtain the dual version of Theorem 1.22

1a.k.a. a Mannahmannah

http://cage.ugent.be/~jvrcruys/konijnmetpruimen/Mahnahmahna.mpg


Theorem 1.26. Let B be a bicategory. Consider a quadruple C = (A, c,∆, ε) in B consistingof a 0-cell A, a 1-cell c : A→ A and 2-cells ∆ : c⇒ c•c and ε : A⇒ c. The following statementsare equivalent.

(i) C is a comonad in B;(ii) (c,∆, ε) is a coalgebra in the monoidal category Hom(A,A);(iii) RepΩ(C) = (RepΩ(A),RepΩ(c),RepΩ(∆),RepΩ(ε)) is a comonad in CAT (or Rep(B))

for all Ω ∈ B;(iv) RepA(C) is a comonad in CAT (or Rep(B));(v) (RepΩ(c),RepΩ(∆),RepΩ(ε)) is a coalgebra in the monoidal category of functors and

natural transformations Fun(Hom(Ω, A),Hom(Ω, A)) for all Ω ∈ B;(vi) (RepA(c),RepA(∆),RepA(ε)) is a coalgebra in Fun(Hom(A,A),Hom(A,A)).

1.3. Bicategories of bimodules

In this section we give several constructions to construct new bicategories out of a givenbicategory. By application of Theorem 1.5 and Corollary 1.6, we trust in the 2-category calculusto prove all statements, i.e. we omit writing the associativity and unit isomorphisms.

1.3.1. Adjoint pairs and Morita contexts. Let us recall the following definitions.A Morita context in a bicategory B is a sextuple (A,B, p, q, µ, τ), where A and B are 0-cells,

p ∈ Hom1(A,B), q ∈ Hom1(B,A), µ ∈ AHomA2 (p•B q, A) and τ ∈ BHomB

2 (q •A p,B) such thatq•Aµ = τ•Bq and p•Bτ = µ•Ap.

A Takeuchi context in B is a sextuple (A,B, p, q, σ, ν), where A and B are 0-cells, p ∈Hom1(A,B), q ∈ Hom1(B,A), σ ∈ AHomA

2 (A, p •B q) and ν ∈ BHomB2 (B, q •A p), such that

q•Aσ = ν•Bq and p•Bν = σ•Ap.An adjoint pair in B is a sextuple (A,B, p, q, µ, ν), where A and B are 0-cells, p ∈ Hom1(A,B),

q ∈ Hom1(B,A), µ ∈ AHomA2 (p•Bq, A) and ν ∈ BHomB

2 (B, q•Ap), such that (µ•Ap)(p•Bν) =p and (q•Aµ) (ν•Aq) = q.

Examples 1.27. (i) Morita contexts are studied in particular in bicategories of bimodulesover a monoidal category A with coequalizers, which we introduce in more detail below.Then they take the form (A,B, P,Q, µ, τ), where A and B are algebras in A, P is an A-Bbimodule, Q is a B-A bimodule and µ and τ are bilinear maps. We say that a Morita contextis strict if the connecting maps µ and τ are isomorphisms. If A = Mk, then a sufficient (butnot necessary) condition for this to be satisfied is the existence of a pair of invertible elements(p, q). Such a pair consists of an element p ∈ P and q ∈ Q such that µ(p ⊗B q) = 1A andτ(q ⊗A p) = 1B.

(ii) A Takeuchi context has a natural interpretation in a bicategory of bicomodules. It is alsocalled a Morita-Takeuchi context.

(iii) If we consider an adjoint pair in CAT, then we obtain the usual definition of a pair of adjointfunctors. Let C and D be two categories and F : C → D and G : D → C two functorsbetween these categories. We call F a left adjoint of G, G a right adjoint of F or (F,G) apair of adjoint functors if and only if there exist natural isomorphisms

(16) θC,D : HomD(FC,D) ∼= HomC(C,GD).

This is equivalent to the existence of natural transformations η ∈ Nat(11C , GF ) and ε ∈Nat(FG, 11D), such that

εFC F (ηC) = FC, ∀C ∈ C,(17)

G(εD) ηGD = GD, ∀D ∈ D.(18)

From these last conditions we see that (D, C, G, F, ε, η) is an adjoint pair in the categoryCAT in the above sense.

The following theorem is straightforward

Theorem 1.28. (i) (A,B, p, q, µ, ν) is an adjoint pair such that µ is an isomorphism in thecategory Hom(A,A) if and only if (A,B, p, q, σ, ν) is a Takeuchi context with σ = µ−1.

1.3. BICATEGORIES OF BIMODULES 23

(ii) (A,B, p, q, µ, ν) is an adjoint pair such that ν is an isomorphism in the category Hom(B,B)if and only if (A,B, p, q, µ, τ) is a Morita context with τ = ν−1.

(iii) (A,B, p, q, µ, ν) is an adjoint pair such that µ is an isomorphism in the category Hom(A,A)and ν is an isomorphism in the category Hom(B,B) if and only if (A,B, p, q, σ, ν) is aTakeuchi context such that σ = µ−1 and ν is an isomorphism if and only if (A,B, p, q, µ, τ)is a Morita context such that τ = ν−1 and µ is an isomorphism.

In the same way as for monads and comonads, one proves the following theorem that connectsadjoint pairs in B with adjoint pairs in CAT.

Theorem 1.29. Consider a sextuple p = (A,B, p, q, µ, η) in a bicategory B, where A and Bare 0-cells, p ∈ Hom1(A,B), q ∈ Hom1(B,A), µ ∈ AHomA

2 (p•Bq, A) and η ∈ BHomB2 (B, q•Ap).

Then the following statements are equivalent.

(i) p is an adjoint pair in B;(ii) RepΩ(p) is an adjoint pair in CAT for every 0-cell Ω in B;(iii) RepA(p) and RepB(p) are adjoint pairs in CAT.

The proof of the following well-known theorem is also straightforward.

Theorem 1.30. Let p = (A,B, p, q, µ, η) be an adjoint pair in B. Then we associate thefollowing monad and comonad to p

a(p) = (B, q •A p, q•Aµ•Ap, η), C(p) = (A, p •B q, q•Bη•Bp, µ)

For an adjoint pair p = (A,B, p, q, µ, η) and a comonad D = (B, d,∆d, εd), we can construct anew comonad

(A, p •B d •B q, (p•Bd•Bη•Bd•Bq) p•B∆d•Bq, µ (p•Bεd•Bq)).

We will discuss this construction in more detail in Section 3.4, especially in the case whereB = Mk.

1.3.2. The bicategories of Eilenberg-Moore objects.

Definition 1.31. Let C = (A, c,∆c, εc) and D = (B, d,∆d, εd) be two comonads in a bicat-egory B. A right comonad morphism from C to D is a pair (q, α), consisting of q ∈ Hom1(B,A)and α ∈ BHomA

2 (d •B q, q •A c) such that the following diagrams commute

d •B qα //

∆d•Bq

q •A c

q•A∆c

d •B d •B q

d•Bα// d •B q •A c α•Ac

// q •A c •A c

d •B q

εd•Bq ""EEE

EEEE

EEα // q •A c

q•Aεc||zz

zzzz

zzz

q

In a symmetric way, a left comonad morphism from C to D is a pair (p, β), where p ∈ Hom1(A,B)and β ∈ AHomB

2 (p •B d, c •A p), such that the following diagrams commute

p •B dβ //

p•B∆d

c •A p

∆c•Ap

p •B d •B d

β•Bp// c •A p •B d

c•Aβ// c •A c •A p

p •B d

p•Bεd""E

EEEE

EEEE

β // c •A p

εc•Ap||yyyy

yyyy

y

p

From the definition one can immediately see that a right comonad morphism (m, ρ) from Bto C, where B = (B,B,B,B) is the trivial B-comonad, is nothing else then a right C-comoduleof B-type. Similarly, left comonad morphisms from B to C are exactly left C-comodules. Moregenerally, we have the following theorem.

Lemma 1.32. Let C = (A, c,∆c, εc) and D = (B, d,∆d, εd) be two comonads in a bicategoryB. Then the following statements hold

(i) (q, α) is a right comonad morphism from D to C, if and only if d •B q is a D-C bicomodule.(ii) (p, β) is a left comonad morphism from D to C, if and only if p •B d is a C-D bicomodule.


Proof. Suppose first that (q, α) and (p, β) are comonad morphisms. The coactions on d•B qand p •B d are given by the following formulas,

ρd•Bq : d •B q∆d•Bq// d •B d •B q

d•Bα // d •B q •A c

λd•Bq : d •B q∆d•Bq// d •B d •B q

ρp•Bd : p •B dp•B∆d// p •B d •B d

λp•Bd : p •B dp•B∆d// p •B d •B d

β•Bd // c •A p •B d

We will check only the coassociativity of ρd•Bq and leave other verifications (the coassociativityand counit conditions, as well as the compatibility between left and right coaction) to the reader,since they are all similar easy computations. Consider the following diagram, of which the outerquadrangle expresses the coassociativity of ρd•Bq.

d •B q∆d•Bq //

∆d•Bq

d •B d •B qd•Bα // d •B q •A c

∆d•Bq•Ac

d •B d •B q

∆d•Bd•Bq

d•B∆d•Bq//

d•Bα

d •B d •B d •B qd•Bd•Bα // d •B d •B q •A c

d•Bα•Ac

d •B q •A c

d•Bq•A∆c

// d •B q •A c •A c

The lower quadrangle of this diagram commutes by the definition of a right comonad morphism,and the upper quadrangle by application of (4).

Now suppose that d •B q is a D-C bicomodule. We will proof that (q, α) is a comonadmorphism, where

α : d •B qd•Bρ // d •B q •A c

εd•Bq•Ac // q •A c.

We have to check the commutativity of the outer quadrangle of the following diagram.

d •B q

d•Bρ

∆d•Bq // d •B d •B q

d•Bρ

d •B d •B q •A c

d•Bεd•Bq•Ac

d •B q •A c

∆d•Bq•Ac66mmmmmmmmmmmmm

d•Bq•A∆c ((QQQQQQQQQQQQQ

εd•Bq•Ac

d •B q •A c

d•ρ

d •B q •A c •A cεd•Bq•Ac•Ac

q •A c

q•A∆c

// q •A c •A c

The upper inner quadrangle commutes since d•B q is a bicomodule. The upper triangle commutesby the counit property of the comonad D. The lower triangle commutes on the image of theincoming arrow d•Bρ on the left as an application of the coassociativity condition on d •B q as aright C-comodule. Finally, applying (4), we find that the lower quadrangle commutes.

The proof for p •B d is similar.

Following (a dualization of) [107], we introduce now a bicategory of comonads in a givenbicategory B.


By RCOM(B) we denote the right bicategory of comonads in B. This bicategory is defined asfollows.

• A 0-cell in RCOM(B) is a comonad in B;• a 1-cell in RCOM(B) is a right comonad morphism;• a 2-cell in RCOM(B) between two right comonad morphisms (q, α) and (q′, α′) from D

to C is a 2-cell σ : q → q′ in B that makes the following diagram commutative

d •B qd•Bσ //

α

d •B q′

α′

q •A c σ•Ac

// q′ •A c

In a similar way, considering left comonad morphisms, one obtains the left bicategory of comonadsLCOM(B).

In [86], an alternative bicategory of monads is proposed, which containes the same 0-cellsand 1-cells, but different 2-cells. For our purposes, this second type of bicategories will be moreuseful, so we will give a more detailed description of their 2-cells. In [86, p249] there were giventwo equivalent descriptions of the 2-cells, called the reduced and the unreduced form. In [31,Proposition 2.2], where the dual case B = Bim, a bicategory of bimodules over algebras with acommutative base ring was considered (see later in this section for the construction of Bim), theunreduced form is being interpreted as a morphism of bicomodules. However, we believe that thetreatment of 2-cells as bicomodules corresponds to a different description of the 2-cells than theones that can be found in [86], as the following Lemma explains.

Lemma 1.33. Consider a bicategory B. Let (q, α) and (q′, α′) be two right comonad morphismsfrom D to C in B. There exists a bijective correspondence between the following objects

(i) a 2-cell σ : d •B q → q′ in B making the following diagram commutative


∆d•Bq

d •B d •B qd•Bσ // d •B q′

α′

d •B d •B q

d•Bα// d •B q •A c σ•Ac

// q′ •A c

(ii) a 2-cell σ : d •B q → q′ •A c in B making the following two diagrams commutative

d •B qeσ //

∆d•Bq

q′ •A c

q′•Ac

d •B d •B q

d•Bα// d •B q •A c eσ•Ac

// q′ •A c •A c

d •B qeσ //

∆d•Bq

q′ •A c

q′•Ac

d •B d •B q

d•B eσ // d •B q •A c α′•Ac// q′ •A c •A c

(iii) a D-C bicomodule morphism σ : d •B q → d •B q′, where the D-C bicomodule structure ond •B q and d •B q′ is obtained using Lemma 1.32.

Proof. Let us just give the corresponding formulas. The remaining part of the proof is justa computation. The equivalence of (i) and (ii) can also be deduced as a dualization of [86, page249], and the equivalence of (i) and (iii) as a formalisation of [31, Proposition 2.2]. Take σ as instatement (i), then define

σ = (σ•Ac) (d•Bα) (∆d•Bq), σ = (d•Bσ) (∆d•Bq)Conversely, if σ or σ are given, we can define

σ = (εd•Bq′) σ, σ = (q′•Aεc) σ.


Remark 1.34. As mentioned before, the objects considered in the previous Lemma, will con-stitute the 2-cells of a new bicategory of comonads. We call these 2-cells comonad transformations.We will slightly adapt the terminology started in [86]. We call a σ subjected to condition (i) ofLemma 1.33 the reduced form of such a 2-cell, a σ that satisfies condition (ii) will be refered to asthe first unreduced form of the 2-cell and σ that satisfies (iii) will be named the second unreducedform.

We can now introduce two new bicategories of comonads.Let B be any bicategory. We denote by REM(B) the bicategory whose 0-cells and 1-cells are

precisely those of RCOM(B) and whose 2-cells are the objects described in Lemma 1.33. similarlyone introduces the left-handed version LEM(B).

1.3.3. Locally finite duality. Altough the definition of a comonad is perfectly left-right sym-metric, we obtained a two different possibilities (a left and a right) for the definition of the bicat-egory of comonads, both in the orriginal (LCOM and RCOM) and in the extended case (LEM andREM). This was due to the assymmetry in the definition of comonad morphisms. The followingtheorem shows that there exists a ‘local’ duality between the left and right versions, however ingeneral no duality can be obtained for the whole bicategories. For this reason, it might be betterto restrict ourselves to a (full) subbicategory of the bicategory of comonads, in order to obtainsuch a duality.

This duality makes it also possible to regard the 1-cells in this bicategory as some kind ofpre-Galois objects (cf. Chapter 4)

Proposition 1.35. Suppose p = (A,B, p, q, µ, η) is an adjoint pair, C = (A, c,∆c, εc) andD = (B, d,∆d, εd) are two comonads in a bicategory B. Then the following statements areequivent

(i) There exists a morphism of comonads ϕ ∈ AHomA2 (p •B d •B q, c);

(ii) There exists a α ∈ BHomA2 (d •B q, q •A c), such that (q, α) is a right comonad morphism

from D to C;(iii) There exists a β ∈ AHomB

2 (p •B d, c •A p), such that (p, β) is a left comonad morphism fromD to C.

If these equivalent conditions hold, we say that (p, q, ϕ) is a comonad morphism with adjunctionfrom D to C.

Proof. (i) ⇒ (ii) Suppose there exists a morphism of comonads ϕ as in the statement, thenwe define α : q •B d→ c •A q as follows

α : d •B qη•Bd•Bq // q •A p •B d •B q

q•Aϕ // q •A c

Consider the following diagram.


η•Bd•Bq

d •B d •B q

d•Bη•Bd•Bq

d •B q •A p •B d •B q

d•Bq•Aϕ

q •A p •B d •B q

q•Ap•B∆d•Bq//

q•Aϕ

q •A p •B d •B d •B q

q•A•Bd•Bη•Bd•Bq

d •B q •A c

η•Bd•Bq•Ac

q •A p •B d •B q •A p •B d •B q

q•Aϕ•Aϕ ++WWWWWWWWWWWWWWWWWWWWq •A p •B d •B q •A c

q•Aϕ•Ac

q •A c

q•A∆c

// q •A c •A c


The outer diagram expresses the first condition for α to be a right comonad morphism. The upperpart of the diagram commutes by application of (4), the lower part expresses that ϕ is a morhismof comonads. The second condition can be computed as follows

(q•Aεc) α = (q•Aεc) (q•Aϕ) (η•Bd•Bq)= (q•Aµ) (q•Ap•Bεd•Bq) (η•Bd•Bq)= (q•Aµ) (η•Bq) (εd•Bq)= εd•Bq

Here we used in the second equality the counit condition of the morphism of comonads ϕ, thirdequation is again an application of (4) and the last equation follows from the fact that p is anadjoint pair in B.

(ii) ⇒ (i) Suppose α exists as in the statement, then we define

ϕ : p •B d •A qp•Bα // p •B q •A c

µ•Ac // c

We have to check that ϕ is a morphism of comonads. The following quadrangle expresses thecounit condition on ϕ.

p •B d •B qp•Bα //

p•Bεd•Bq

p •B q •A cµ•Ac //

p•Bq•Aεcwwnnnnnnnnnnnnc

εc

p •B q µ

// A

The inner triangle commutes because of the counit condition on α, while the inner quadranglecommutes by (4). In order to avoid too large diagrams, we verify the coassociativity condition ofϕ by a computation on the maps.

(ϕ•Aϕ) (∆p•Bd•Bq)= (µ•Ac•Aµ•Ac) (p•Bα•Bα) (p•Bd•Bη•Bd•Bq) (p•B∆d•Bq)= (µ•Ac•Ac) (p•Bq•Ac•Aµ•Ac) (p•Bα•Ap•Bq•Ac)

(p•Bd•Bq•Ap•Bα) (p•Bd•Bη•Bd•Bq) (p•B∆d•Bq)= (µ•Ac•Ac) (p•Bα•Ac) (p•Bd•Bq•Aµ•Ac)

(p•Bd•Bη•Bq•Ac) (p•Bd•Bα) (p•B∆d•Bq)= (µ•Ac•Ac) (p•Bα•Ac) (p•Bd•Bα) (p•B∆d•Bq)= (µ•Ac•Ac) (p•Bq•A∆c) (p•Bα)= ∆c (µ•Ac) (p•Bα)= ∆c ϕ

The second, third and penultimate equality are applications of (4), the forth one follows from thecondition on the adjoint pair p, and the fifth equation uses the fact that α is a right comonadmorphism.

The equivalence (i) ⇔ (iii) follows by left-right duality from (i) ⇔ (ii).

Definition 1.36. We introduce now two new bicategories fREM(B) and fLEM(B). Thebicategory fREM(B) has the same 0-cells and 2-cells as REM(B), but a 1-cell from C (over A) toD (over B) in fREM is a right comonad morphism (q, α) with the extra condition that there existsan adjoint pair of the form (A,B, p, q, µ, η) in B. And similar for fLEM(B).

From Proposition 1.35 we then immediately obtain

Proposition 1.37. There is a biequivalence between fREM(B) and fLEM(B)co.

Remark also that by Proposition 1.35 the 1-cells in fREM(B) can be regarded as morphismsof comonads ϕ : p •B d •B q → c.


1.3.4. The bicategory of bi(co)modules. Consider in a bicategory B the 1-cells m,n :A → B and the 2-cells α, β : m ⇒ n. Suppose that the equalizer (e, ε) of α and β exists inHom(A,B) = RepA(B) (Notation as in Example 1.2).

eε // m

α //

β// n

Let a : B → C be any 1-cell in B. We say that the equalizer (e, ε) is right a-pure if and only ifRepA(a) : RepA(B) → RepA(C) preserves this equalizer.

e •B aε•Ba // m •B a

α•Ba //

β•Ba// n •B a

similarly, let b : C → A ba a 1-cell in B. We say that the equalizer (e, ε) is left b-pure if and onlyif BRep(b) : BRep(A) → BRep(C) preserves this equalizer.

b •A eb•Aε // b •A m

b•Aα //

b•Aβ// b •A n

In the same way we say that a coequalizer is right a-pure (respectively left b-pure) if RepA(a)(resp. BRep(b)) preserves this coequalizer.

Consider now a comonad C = (A, c,∆c, εc), a right C-comodule of C-type (m, ρm) and a leftC-comodule of B-type (n, λn). If it exists, we will denote the equalizer of ρm•An and m•Aλn bym •c n and we call this the cotensor product,

(19) m •c n // m • nm•Aλ

n

//ρm•An

// m •A c •A n.

Consider now two other comonads D = (B, d,∆d, εd) and E = (C, e,∆e, εe). Supose that(n, λn, ρn) is a C-D bicomodule and (m,λm, ρm) is an E-C bicomodule.

Lemma 1.38. With notation as above, suppose that the equalizer m •c n exists,

(i) if the cotensor product m •c n is right d •B d-pure, then m •c n is a right D-comodule.(ii) if the cotensor product m •c n is left e •C e-pure, then m •c n is a left E-comodule.(iii) if the cotensor product m •c n is both right d •B d-pure and left e •C e-pure, then m •c n is

a E-D bicomodule.

Proof. We only give a short sketch of the proof of part (i). First remark that if an equalizer

q ε // mf,g // n is right d •B d pure, then this equalizer is also right d-pure. This follows easily

from the following diagram, once we observe that d •B εd ∆d = d

q •B dε•Bd //

q•B∆d

m •B df•Bd //

g•Bd//

m•B∆d

n •B d

∆d•Bn

q •B d •B d

ε•Bd•Bd //

q•Bεd•Bd

OO

m •B d •B df•Bd•Bd //

g•Bd•Bd//

m•Bεd•Bd

OO

n •B d •B d

n•Bεd•Bd

OO

With this knowledge, one can proof that the coaction on m•cn is obtained from the right d-purityand the coassociativity of this coaction follows form the right d •B d-purity.

Lemma 1.39. Let C = (A, c,∆c, εc) be a comonad in B and consider a right C-comodule(m, ρ). Then the equalizer of (m•A∆c) and (ρ•Ac) always exists and is isomorphic to (m, ρ), i.e.m •c c ∼= m.

Proof. Consider the following diagram

m •c c e // m •A cρ•Ac //

m•A∆c

// m •A c •A c

m

f

OO

ρ

99rrrrrrrrrr


By the coassociativity condition of the right C-comodule m, ρ equalizes (m•A∆c) and (ρ•Ac).Consequently we find by the equalizer property of m•c c a unique morphism f such that ef = ρ.We will show that (m•Aεc) e is a two-sided inverse for f . From the counit condition on m weobtain (m•Aεc) e f = (m•Aεc) ρ = m. For the converse recall that e is a monomorphismsince (m •c c, e) is an equalizer. Consequently, to prove that m •c c = f (m•Aεc) e it is enoughto check that

e = e (m •c c) = e f (m•Aεc) e = ρ (m•Aεc) eConsider the following diagram

m •c c e // m •A c

ρ•Ac

m•Aεc // m •A Aρ•AA

∼= // m

ρ

m •A c •A c m•Ac•Aεc

// m •A c •A A ∼=// m •A c

The quadrangle on the right commutes by the naturality of the unit morphism in the bicategoryB. The left quadrangle commutes as an application of (4). Thus we find

ρ (m•Aεc) e = (m•Ac•Aεc) (ρ•Ac) e= (m•Ac•Aεc) (m•A∆c) e= (m•Ac) e = c

where we used in the second equality the fact that e equalizes the maps (m•A∆c) and (ρ•Ac) andthe counit condition on c in the third equality. This ends the proof.

Consider a 2-category B such that for every pentuple of comonads C (over A), D (over B), E(over C), F (over D) and G (over E) and for all E-C bicomodule m and every C-D bicomodulen, the cotensor product m •c n exists and is left p-pure and right q-pure for any F-E bicomodulep and any D-G bicomodule q. We can now construct a new bicategory Bic(B) out of B:

• A 0-cell in Bic(B) is a comonad in B;• a 1-cell C → D in Bic(B) is a C-D bicomodule;• a 2-cell ϕ in Bic(B) between two C-D bicomodules, is a left C- and a right D-colinear

morphism.

Composition of 1-cells is given by the cotensor product, unit elements are the comonads, consideredas bicomodules over themselves.

Remark 1.40. It follows from Lemma 1.38 that the composition of 1-cells in Bic(B) is well-defined. Moreover the composition of 1-cells is also associative (up to isomorphism), as we cansee as follows. Consider four comonads in B: C (over A), D (over B), E (over C), F (over D)together with a C-D bicomodule m, a D-E bicomodule n and a E-F bicomodule p. Then we needan isomorphism of the form

(m •d n) •e p ∼= m •d (n •e p).Consider the following diagram

(m •d n) •e p //

ψ1

(m •B n) •e p //

ψ2

// (m •B d •B n) •e p

ψ3

m •d (n •e p) // m •B (n •e p) //// m •B d •B (n •e p)

The bottom row is exact by the definition of the cotensor product. The exactness of the top rowcan be deduced from the fact that m •d n is right p-, right e- and right e •C p-pure. Furthermorewe know by the left m- and left m •B d-purity of n •e p that ψ2 and ψ3 are isomorphisms, henceψ1 is an isomorphism as well by the univeral property of the equalizer.

An example of a bicategory B as above is given by the bicategory of bimodules over divisionalgebras.

Theorem 1.41. Let B be a bicategory as above. Then there exist pseudo functors

F : REM(B) → Bic(B), G : LEM(B)co → Bic(B)


Proof. Since both the 0-cells in REM(B) and Bic(B) are comonads, we can define F tobe identical on 0-cells. A 1-cell in REM(B) is a right comonad morphism (q, α) : D → C. ByLemma 1.32, d •B q is a D-C bicomodule, thus we define F (q, α) = d •B q. A 2-cell in REM(B)is a comonad transformation σ : (q, α) ⇒ (q′, α′). We know from Lemma 1.33 that the secondunreduced form σ of σ is a bicomodule morphism from F (q, α) = d •B q to F (q′, α′) = d •B q′,thus we define F (σ) = σ.

Finally we have to find natural isomorphisms γ (7) and δ (8). To this end, consider a comonadmorphism (q, α) : E → D and a comonad morphism (q′, α′) : D → C. We need to prove thatF (q) •d F (q′) ∼= F (q •B q′). Indeed:

(e •C q) •d (d •B q′) ∼= ((e •C q) •d d) •B q′ ∼= e •C q •B q′

where the first isomorphism is a consequence from the purity conditions. The property for δ istrivial, the unit objects in both categories are comonads.

All above constructions can be dualized. This gives rise to the left and right bicategories ofMonads LMND(B) and RMND(B), and their restricted versions LEM(B) and REM(B), whoseclass of 2-cells consists of monad-transformations. Let a = (R, a, µ, η) be a monad in B. If theHom(X,Y ) possesses coequalizers, then we can define the tensor product of a right a-module ofX-type (m, ρm) and a left a-module (n, λn) of Y -type as the following coequalizer

(20) m •R a •R nρm•Rn //

m•Rλn

// m •R n // m •a n

If such a coequalizer exists for any pentuple of monads a, b, c, d and e and all a-b bimodules mand b-c bimodules n such that m •b n is left p-pure and right q-pure for all d-a bimodules q andc-e bimodules q, then we can construct the bicategory of bimodules Bim(B), whose 0-cells are allmonads in B, 1-cells are bimodules and 2-cells are bimodule-morphisms. Composition of 1-cells inBim(B) is given by the tensor product (20).

We will now briefly discuss the consequences in monoidal categories.Let (A,⊗, I) be a monoidal category. Then A is by definition a bicategory with one 0-cell.

Consequently we can consider the bicategories REM(A), LEM(A), whose 0-cells are coalgebras inA and REM(A) and LEM(A) whose 0-cells are algebras.

Let (M,N ) be a compatible pair of monoidal categories on (C,A). Consider an algebra A inC, a right A-module M in M and a left A-module N in N . Then we can consider the A-tensorproduct of M and N if the coequalizer M ⊗A N exists in A, defined by the following diagram

(M ⊗M A)⊗Nρ1 //ρ2

// M⊗N // M ⊗A N

where ρ1 and ρ2 are induced by the action of A on M and N .We know that any monoidal categoryA provides a trivial compatible pair of monoidal categories

(A,A) on (A,A). If we can construct the needed equalizers and coequalizers in A and the correctpurity conditions hold, we can construct the bicategories Bic(A) and Bim(A). A sufficient (butnot necessary) condition for Bic(A) to exist is that A is a monoidal category with equalizers. Wesay that A is a monoidal category with equalizers if A possesses equalizers and these equalizersare preserved by the tensor product. By this last condition we mean that if for any equalizer in A

Ee // M

f //g// N

and any X ∈ A, (E ⊗X, e ⊗X) is the equalizer of (f ⊗X) and (g ⊗X) and (X ⊗ E,X ⊗ e)is the equalizer of (X ⊗ f) and (X ⊗ g). Similarly, Bim(A) exists if A is a monoidal categorywith coequalizers. Remark that 0-cells in Bim(A) are all algebras in A, 1-cells are bimodules and2-cells are bimodule-maps. In particular we obtain that for any algebra A in A, the category of allA-bimodules is a monoidal category, with tensor product the tensor product over A.

Example 1.42. Let k be a commutative ring with unit and consider the monoidal category(Mk,⊗k, k). It is well-known that this category has coequalizers and that the tensor product

1.4. ENRICHED BICATEGORIES 31

functor is right exact, so in particular it preserves coequalizers. Consequently, we can constructthe bicategory Bim(k), consisting of k-algebras, bimodules between k-algebras and bilinear maps.

We call a k-module M flat if the functor M ⊗ − : Mk → Ab is an exact functor. Sincefor all N,M ∈ Mk, N ⊗ M ∼= M ⊗ N , we obtain that M is flat if and only if the functor− ⊗M : Mk → Ab is exact. Because of the exactness of the forgetful functor Mk → Ab, wefind that every equalizer in Mk is left and right M -pure if M is flat as k-module. Thus we canconsider the bicategory fBic(k) whose 0-cells are flat k-coalgebras, 1-cells are bicomodules and2-cells are bicomodule maps.

If k is e.g. a field then all modules are projective, so in particular flat and fBic(k) = Bic(k).

Example 1.43. We can repeat the example above, replacing k by a noncommutative ring(k-algebra) A and Mk by AMA.

For any ring (k-algebra) A, we say that a right A-module M is flat if the functor M ⊗A − :MA → Ab is exact. A left A-module N is flat if the functor N ⊗A − : MA → Ab is exact. Foran A-A bimodule both conditions are not equivalent in general.

1.4. Enriched bicategories

Let (V,⊗, I) be a concrete monoidal category, i.e. V is a monoidal category together with afaithful functor V → Set. This faithful functor makes it possible give a meaning to the notation“x ∈ X” where X is an object in V (see [88, page 26]). The definition of a V-enriched bicategoryB is essentially the same as the usual definition of a bicategory, the only difference is that theHom-categories are now V-enriched categories. Explicitly, a V-enriched bicategory B consists of:

- a class of objects A,B, . . . which are called 0-cells;- for every two objects A and B, a V-enriched category Hom(A,B).

By V-enrichedness the horizontal composition of 2-cells can be expressed as

• : AHomB2 (f, g)⊗ BHomC

2 (h, k) → AHomC2 (f •B h, g •B k);

All other notation and compatibility conditions stay the same as in the case of usual bicategories.

Example 1.44. If A is a V-enriched monoidal category with coequalizers, then Bim(A) is aV-enriched bicategory.

The following duality theorem for Bim(A) has important implications for the later chapters.

Theorem 1.45. Let A be a V-enriched monoidal category with coequalizers and considera comonad C = (A, c,∆, ε) in Bim(A). Then we can define two monads ∗C and C∗ in Bim(V)such that there is a functor from the category of right comodules over C to the category of rightmodules over ∗C and a functor from the category of left comodules over C to the category of leftmodules over C∗.

Proof. We will define the monad structure on ∗C, the statement about C∗ follows then by left-right duality. Consider AEnd(A), the object in V consisting of all left A-module endomorphismsof A. Using the composition of morphisms as multiplication, AEnd(A) becomes an algebra in V.In the same way, AEnd(c) is an algebra in V.

For any f ∈ AEnd(A) we define an element f ∈ AEnd(c) as the following composition

f : c∼= // c⊗ I

c⊗ηA // c⊗Ac⊗f // c⊗A

ρc // c

where we denoted by ηA the unit map of A and by ρc the right A-action of A on c. One caneasily check that f is left A-linear. In this way we obtain a morphism AEnd(A) → AEnd(c). The


following diagram expresses that this morphism is a morphism of algebras in V.

c∼=c⊗I

∼=

c⊗ηA // c⊗A

∼=

c⊗g // c⊗Aρc // c∼=c⊗I

c⊗ηA // c⊗Ac⊗f // c⊗A

ρc // c

c⊗I⊗I

∼=

c⊗ηA⊗I// c⊗A⊗I

∼=

c⊗A⊗ηA //

c⊗g⊗I ))TTTTTTTTTTTTTTTT c⊗A⊗Ac⊗g⊗A// c⊗A⊗A

c⊗A⊗f//

ρc⊗A

OO

c⊗µ

%%KKKKKKKKKK c⊗A⊗Ac⊗µ //

ρc⊗A

OO

c⊗A

ρc

OO

c⊗A⊗I

∼=

c⊗A⊗ηA

99ssssssssss ∼= // c⊗Ac⊗f

::uuuuuuuuu

c⊗Ic⊗ηA // c⊗A

c⊗g // c⊗Ac⊗f // c⊗A

If we read this quadrangle from the upper left corner to the upper right corner following the upper

row, we obtain f g; if we follow the path that passes the lower side, we obtain f g. Thecommutativity of the diagram follows by the interchange law (4), the left A-linearity of f and theunit-condition on A.

For any f ∈ AEnd(A) and ϕ ∈ AHom(c, a) we define

f · ϕ : cf // c

ϕ // A

ϕ · f : cϕ // A

f // A

One can easily check this equips AHom(c, A) with an AEnd(A)-bimodule structure.For φ, ψ ∈ AHom(c, A), we define φ ∗ ψ ∈ AHom(c, A) as the 2-cell defined by the following

diagram

c∆ // c⊗A c

c⊗Aφ // c⊗A A ∼= cψ // A

i.e. φ ∗ ψ = ψ (c•Aφ) ∆. Finally, we can define a map η : AEnd(A) → AHom(c, A),η(f) = εc f .

With these definitions we obtain that ∗C = (AEnd(A),AHom(c, A), ∗, η) a monad is in Bim(V).We leave it to the reader to verify the statement about the modules.

Example 1.46. Let k be a commutative ring and consider the bicategory Bim(k) = Bim(Mk),the bicategory of k-algebras, bimodules over k-algebras and bilinear maps. Then this bicategory isenriched over V = Mk, and thus the associated bicategory Bim(V) is again the original bicategoryBim(k).

A comonad in Bim(k) is an A-coring (C,∆, ε). By the previous theorem we know that we canassociate two A-rings to C. The right dual C∗ = HomA(C, A) ∼= EndC(C) is an A-ring with uniti : A→ C∗, i(a)(c) = aεC(c) and multiplication given by

f ∗ g(c) = f(g(c(1))c(2)),

for all f, g ∈ C∗ and c ∈ C. In particular, for all a ∈ A, f ∈ C∗ and c ∈ C we obtain

(21) (i(a) ∗ f)(c) = af(c) (f ∗ i(a))(c) = f(ac).

The left dual ∗C = AHom(C, A) is an A-ring with unit i : A → ∗C, i(a)(c) = εC(c)a andmultiplication

f ∗ g(c) = g(c(1)f(c(2))).In particular, for all a ∈ A, f ∈ ∗C and c ∈ C we obtain

(22) (i(a) ∗ f)(c) = f(ca) (f ∗ i(a))(c) = f(c)a.

In this case we also find a third ring, ∗C∗ = AHomA(C, A) which is a k-algebra with unit εC andmultiplication

f ∗ g(c) = g(c(1))f(c(2)).Remark that if A is commutative and C is an A-coalgebra (i.e. the left and right A-action on Ccoincide), ∗C = C∗ = ∗C∗. Furthermore, every right comodule has a ∗C module structure, given by

m · f = m[0]f(m[1]),

1.4. ENRICHED BICATEGORIES 33

for any m ∈ M ∈ MC and f ∈ ∗C. Analogously, every left C-comodule N becomes a leftC∗-module with action given by

f · n = f(n[−1])n[0],

for any n ∈ N and f ∈ C∗.

In the remaining part of this section we will suppose that B is a V-enriched bicategory andthat V is a monoidal category with coequalizers. This implies that we can consider Bim(V).

Let A and B be two 0-cells in B, p : A → B and q : B → A two 1-cells and considerQ = BHomA

2 (q, q), P = AHomB2 (p, p)op, N = BHomB

2 (B, q •A p) and M = AHomA2 (p •B q, A).

With this notation, the following holds.

Proposition 1.47. (i) Q and P are 0-cells in Bim(V);(ii) N is a 1-cell from Q to P ;(iii) M is a 1-cell from P to Q;(iv) there exist two maps O : N ⊗P M → Q and H : M ⊗Q N → P ;(v) M(p, q) = (Q,P,N,M, O, H) constitutes a Morita context in Bim(V).

Proof. (i). The vertical composition of 2-cells in B defines an associative multiplication

on Q = BHomA2 (q, q) and AHomB

2 (p, p) with units q ∈ Q = BHomA2 (q, q) and p ∈ AHomB

2 (p, p).We will denote ∗ for the opposite multiplication in AHomB

2 (p, p).(ii). Take α ∈ Q, β ∈ P and γ ∈ N . We define α · γ = (α•Ap) γ and γ · β = (q•Aβ) γ. Thenwe find

(α · γ) · β = ((α•Ap) γ) · β= (q•Aβ) ((α•Ap) γ)= (α•Ap) ((q•Aβ) γ)= α · (γ · β).

Here we used (4) in the third equality. Both actions are associative, indeed take two other α′ ∈ Qand β′ ∈ P , and compute

α′ · (α · γ) = α′ · ((α•Ap) γ)= (α′•Ap) ((α•Ap) γ)= ((α′ α)•Ap) γ)= (α′ α) · γ

and

(γ · β) · β′ = ((q•Aβ) γ) · β′

= (q•Aβ′) ((q•Aβ) γ)= (q•A(β′ β)) γ= γ · (β′ β) = γ · (β ∗ β′).

We used (3) in both calculations in the third equality. Obviously, q ∈ Q and p ∈ P act trivially onN .(iii). This is the dual statement of part (ii). We only give the definion of the actions and leave

other verifications to the reader. Take α ∈ Q, β ∈ P and δ ∈ M , then β · δ = δ (β•Bq) andδ · α = δ (p•Bα).(iv). Take γ ∈ N and δ ∈M . Then we define γ O δ = (q•Aδ) (γ•Bq);

γ O δ : q ∼= B •B qγ•Bq // q •A p •B q

q•Aδ // q •A A ∼= q

and δ H γ = (δ•Ap) (p•Bγ);

δ H γ : p ∼= p •B Bp•Bγ // p •B q •A p

δ•Ap // A •A p ∼= p.


Let us check that O is P -balanced. Take β ∈ P , then

(γ · β) O δ = (q•Aδ) ((γ · β)•Bq)= (q•Aδ) (((q•Aβ) γ)•Bq)= (q•Aδ) (q•Aβ•Bq) (γ•Bq)= (q•A(δ (β•Bq))) (γ•Bq)= (q•A(β · δ)) (γ•Bq)= γ O(β · δ).

In the same way, we compute that H is Q-balanced.

δ H(α · γ) = (δ•Ap) (p•B(α · γ))= (δ•Ap) (p•B((α•Ap) γ))= (δ•Ap) (p•Bα•Ap) (p•Bγ)= ((δ (p•Bα))•Ap) (p•Bγ)= ((δ · α)•Ap) (p•Bγ)= (δ · α) H γ,

for all α ∈ Q. (v). Consider γ, γ′ ∈ N and δ, δ′ ∈M . We compute

γ · (δ H γ′) = (q•A(δ H γ′)) γ= (q•A((δ•Ap) (p•Bγ′))) γ= (q•Aδ•Ap) (q•Ap•Bγ′) γ= (q•Aδ•Ap) γ•Bγ′

= (q•Aδ•Ap) (γ•Bq•Ap) γ′

= (((q•Aδ) (γ•Bq))•Ap) γ′

= ((γ O δ)•Ap) γ′

= (γ O δ) · γ′.With a similar computation, one checks that δ · (γ O δ′) = (δ H γ) · δ′ and thus we obtain a Moritacontext.

Theorem 1.48. Consider the Morita context M(p, q) from Proposition 1.47. If there existinvertible elements µ ∈M and ν ∈ N for M(p, q) then (A,B, p, q, µ, ν) is an adjoint pair in B.

Conversely, every adjoint pair (A,B, p, q, µ, ν) leads to a Morita context M(p, q) for which(µ, ν) is a pair of invertible elements (in particular M(p, q) is strict).

Proof. This follows directly from the definition of O and H.

References

Most results of this chapter can be found (maybe up to some translation and interpretation)in any textbook in (categorical) algebra. Our presentation was partially inspired by [25] and [87].

This chapter reflect the author’s vision on monads in bicategories and their relation with(monoidal) categories. The new results of Section 1.3.2 and Section 1.3.3 are taken from the jointpaper with J. Gomez-Torrecillas [72].

The results of Section 1.3.1 are taken from the joint paper with M. Iovanov [77].

Chapter 2Local Projectivity versus local

algebraic Structures

This Chapter is mainly devoted to ring theory. We develop several tools that will be essentialto construct new types of corings in Chapter 3 and to develop a Galois theory for comodules inPart II. To this end, we have to extend the notion of a (finitely generated and) projective module,by means of non-unital rings.

In the first Section, we introduce some notation and give preliminary results on colimits and splitdirect systems. In Section 2.2 we study different types of non-unital rings for which certain desiredproperties of unital rings are preserved: firm rings, rings with local units and rings with idempotentlocal units. We also discuss some properties of the M -adic and finite topology. In Section 2.3, weinvestigate several variations of projectivity and by constructing elementary algebras, we link thesenotions with the different notions of non-unital rings introduced in Section 2.2. In the final Section,we discuss dual versions of these constructions, leading to corings with local comultiplications,corings with local counits and rings with local multiplications.

2.1. Colimits and split direct systems

Let F : Z → A be a covariant functor. Recall (see for example [25]) that a cocone on F is acouple (M,m) where M ∈ A and mZ : F (Z) → M is a morphism in A, for every Z ∈ Z, suchthat

(23) mZ = mZ′ F (f),

for every f : Z → Z ′ in Z. The colimit of F is a cocone (C, c) on F satisfying the followinguniversal property: if (M,m) is a cocone on F , then there exists a unique morphism f : C →Min A such that

(24) f cZ = mZ ,

for every Z ∈ Z. If the colimit exists, then it is unique up to isomorphism. We then writecolimF = colimF (Z) = (C, c).The colimit (C, c) has the following property: if f, g : C →M are two morphisms in A such thatf cZ = g cZ , for all Z ∈ Z, then f = g. Indeed, (M,f c = g c) is a cocone on F , and f = gfollows from the uniqueness in the definition of colimit.

In this section, we consider a special kind of colimits. To a partially ordered set (I,≤), wecan associate a category Z. The objects of Z are the elements of I, and HomZ(i, j) = ajiis a singleton if i ≤ j and empty otherwise. Recall that a partially ordered set (I,≤) is calledemphdirected if every finite subset of I has an upper bound.

35

36 CHAPTER 2. LOCAL PROJECTIVITY VERSUS LOCAL ALGEBRAIC STRUCTURES

Let A be a category and Z a category associated to a directed partially ordered set (I,≤). Afunctor M : Z → A will be called a direct system with values in A. To A, we associate a newcategory As. The objects of A and As are the same. A morphism M → N in As is a couple(µ, ν), with µ : M → N and ν : N → M in A such that ν µ = M , that is, ν is a leftinverse of µ. A direct system M : Z → A will be called split, if it can be extended to a functorM s : Z → As, such that M = F M s, where F denotes the forgetful functor As → A. We willadopt the following notation, for all i ≤ j ∈ I:

M s(i) = Mi, Ms(aji) = (µji, νij).

Then µji : Mi →Mj , νij : Mj →Mi, and

(25) νij µji = Mi.

With these notation the forgetful functor F : As → A takes the explicit form F (M) = M ,F (µ, ν) = µ. Remark that the existence of colimM = (M,µ) implies in particular that we havemorphisms µi : Mi →M such that

(26) µi = µj µji.

In the remaining part of this chapter, we will identify a (split) direct system M with its value in acategory A. By this we mean that a (split) direct system M : Z → A will sometimes be denotedas (Mi)i∈I , where I is the partially orded set associated to Z and Mi = M(i).

Proposition 2.1. Let M : Z → A be a direct system, and assume that colimM = (M,µ)exists. Then M is split if and only if there exist unique morphisms νi : M →Mi in A such that

νi µi = Mi(27)

νi = νi µj νj ,(28)

for all i ≤ j in I.

Proof. Suppose first that M is a split direct system. For a fixed i ∈ I, we have a cocone(Mi, u

i) on M defined as follows: for every k ∈ I, uik : Mk →Mi is the composition

νil µlk : Mk →Ml →Mi,

where l ≥ i, k. We have to show that this definition is independent of the choice of l. Takej ≥ i, k, and m ≥ l, j. Then

νim µmk = νil νlm µml µlk = νil Ml µlk = νil µlk,

and, in a similar way,

νim µmk = νij µjk.Then (Mi, u

i) is a cocone on M . Indeed, take k ≥ j in I, and l ≥ i, k; then

uik µkj = νil µlk µkj = νil µlj = uij .

From the universal property of the colimit, it follows that there exists a unique νi : M →Mi suchthat

(29) uij = νi µj ,

for all j ∈ I. In particular,

νi µi = uii = νil µli = Mi.

Let us show that νi = νij νj if i ≤ j. To this end, it suffices to show that

νi µk = νij νj µk,

for all k ∈ I. We take l ≥ j, k and compute

νi µk = uik = νil µlk = νij νjl µlk = νij ujk = νij νj µk.

From this we obtain for all i ≤ j,

νi = νij νj = νij νj µj νj = νi µj νj ,

2.1. COLIMITS AND SPLIT DIRECT SYSTEMS 37

where we used (27) for second equality. We finally prove the uniqueness. Assume that ν ′i : M →Mi satisfies (27). Let i, j ∈ I, and take k ≥ i, j. Then

ν ′i µj(26,27)

= νik ν ′k µk µkj(27)= νik Mk µkj = νik µkj = uij

so the ν ′i satisfy (29). By the uniqueness in the definition of colimit, it follows that ν ′i = νi, for alli ∈ I.

Conversely, define νij = νi µj . Then we find

νij νjk = νi µj νj µk = νi µk = νik,

andνij µji = νi µj µji = νi µi = Ci.

This ends the proof.

Remark 2.2. By classical arguments we know that the colimit of M exists whenever A is anabelian category.

From the proof it is clear that under the equivalent conditions of the previous proposition, (28)means exactly

νi = νij νj ,for every i ≤ j.

We will compute most colimits in module categories. To finish this section, we focus on someduality properties that occure in this particular setting.

Lemma 2.3. Let A and B be two rings and P s : Z ⊗ (BMA)s a split direct system. Thenwe can define a second split direct system

P ∗s : Z → (AMB)s, P ∗s(i) = P ∗i = HomA(Pi, A), P ∗s(aji) = (τ∗ij , σ∗ji),

where for j ≥ i, we have defined the maps

σ∗ji : P ∗j → P ∗i , σ∗ji(ϕj) = ϕj σji,τ∗ij : P ∗i → P ∗j , τ∗ij(ϕi) = ϕi τij .

Proof. It is straightforward to check that P ∗s is a split direct system. Since Pi is a (B,A)-bimodule, P ∗i is a (A,B)-bimodule.

We hold the notation as in Lemma 2.3. Suppose now that colimP = (P, σ) exists then wewill show that colimP ∗ = (P †, τ †) exists as well.

Lemma 2.4. Let i ∈ I and ϕ ∈ P ∗ = HomA(P,A). There exists ϕi ∈ P ∗i such that

ϕ = ϕi τiif and only if

ϕ = ϕ σi τi.In this situation, ϕi is unique, and is given by the formula ϕi = ϕ σi; furthermore, for everyj ≥ i, ϕ = ϕj τj , with ϕj = ϕi τij .

Proof. If ϕ = ϕi τi, then ϕ σi τi = ϕi τi σi τi = ϕi τi = ϕ. The converse isobvious. If ϕ = ψ τi, then ϕ σi = ψ τi σi = ψ. If j ≥ i, then ϕi τij τj = ϕi τi = ϕ.

Let P † = ϕ ∈ P ∗ | ∃i ∈ I : ϕ = ϕ σi τi. For every i ∈ I, we have a map

τ∗i : P ∗i → P †, τ∗i (ϕi) = ϕi τi.

Proposition 2.5. With notation as above, colimP ∗ = (P †, τ∗).

Proof. First, (P †, τ∗) is a cocone on Q∗ since, for all i ≤ j and ϕi ∈ P ∗i , we have

(τ∗j τ∗ij)(ϕi) = ϕi τij τj = ϕ τi = τ∗i .

Let (M,m) be another cocone on Q∗. This means that mi : P ∗i →M and mj τ∗ij = mi if i ≤ j.

We then define f : P † → M as follows: f(ϕi τi) = mi(ϕi), for every i ∈ I and ϕi ∈ P ∗i . Letus show that f is well-defined. Assume that

ϕ = ϕi τi = ϕj τj .


Take k ≥ i, j. Then ϕ = ϕk τk with ϕk = ϕi τik (see Lemma 2.4). Then

mk(ϕk) = mk(ϕi τik) = (mk τ∗ik)(ϕi) = mi(ϕi).

In a similar way, we have that mj(ϕj) = mk(ϕk), and it follows that f is well-defined. Finally,(f τ∗i )(ϕi) = f(ϕi τi) = mi(ϕi).

2.2. Non-unital rings

2.2.1. The Dorroh extension. Consider a ring B and let R be a B-ring without unit andwith multiplication µR : R ⊗B R → R. The Dorroh overring of R is a unital B-ring R = R × B

containing R as a two-sided ideal by the canonical injection ι : R → R, ι(r) = (r, 0). The

is ring structure on R is given as follows. The actions of the B-bimodule R are defined asb′(r, b)b′′ = (b′rb′′, b′bb′′), i.e. R = R ⊕ B in BMB. The map η : B → R × B, η(b) = (0, b)is a B-bimodule map. In this way, R becomes a B-ring with unit map η and multiplication(r, b) · (r′, b′) = (rr′ + rb′ + br′, bb′). We refer to e.g. to [114, section 1.5] in the case whereB = Z.

We define the category MR as the category of all right B-modules M for which there existsa right B-linear map

µM : M ⊗B R→M

such that µM (µM ⊗B R) = µM (M ⊗B µR), i.e. µM is associative with respect to the

multiplication in R. For any M ∈ MR, we can define a unital right R-action via

m(r, b) = mr +mb,

for all m ∈ M , r ∈ R and b ∈ B. Clearly this action is unital with respect to the unit map η. In

this way, we obtain an isomorphism between the categories MR and M bR.One can construct as well the Dorroh extension of a semigroup (in fact, Dorroh extensions

exist in every monoidal category with (finite) coproducts). Let us state this construction for sake ofcompleteness, together with the relation with the Dorroh extension of rings by means of grouprings.

Proposition 2.6. If G is a semigroup, then there exists a monoid G with the followingproperties

• There exists an injective morphism of semigroups ι : G→ G;• for all g ∈ G and g ∈ G, g · g ∈ ι(G), as calculated in G;

• G is commutative if and only if G is commutative.

We call G the Dorroh monoid associated to G.

Proof. Put G = G∪e where we define the multiplication as follows : g · e = g = e · g

Remark that the construction of the Dorroh extension changes the multiplication in the ringcase and not in the semigroup case.

Proposition 2.7. LetG be a semigroup without unit and B any ring, then there is a canonical

ring isomorphism between the Dorroh extension BG of the semigroupring BG and the groupringBG where G is the Dorroh monoid associated to G.

Proof. By the construction of Dorroh extensions of rings, the B-bimodule structure of BG isgiven by BG⊕B. Obviously this is isomorphic to the B-bimodule structure of BG, which is givenby B(G ∪ e). Explicitly, the isomorphism for (

∑rigi, r) ∈ BG⊕ B is given by ϕ(

∑rigi, r) =∑

rigi + re ∈ BG. Finally, we check that this is a ring morphism. The multiplication in BG

is given by (∑rigi, r) · (

∑sjhj , s) = (

∑risjgihj + risgi + rsjhj , rs). In BG we calculate

(∑rigi+re)·(

∑sjhj+se) =

∑risjgihj+risgie+rsjhje+rse2 =

∑risjgihj+risgi+rsjhj+rs.

We find the same result after applying the isomorphism ϕ.

2.2. NON-UNITAL RINGS 39

2.2.2. Firm rings. Let R be as in the previous section a non-unital B-ring. Then RMR is amonoidal category, since it is isomorphic to bRM bR. The monoidal unit in this category is R andthe tensor product is ⊗ bR. The following lemma is an easy observation.

Lemma 2.8. For all M ∈ MR and N ∈ RM, the following tensor products are isormorphicsets in Ab,

M ⊗R N ∼= M ⊗ bR N.Now the natural question arrises wheter we can find a full subcategory of MR that is still a

monoidal category with tensor product ⊗R, and with monoidal unit R, i.e. without introducing

an artificial unit in R or, equivalently, an artificial monoidal unit for the category MR. This leadsus to the following definitions.

A right R-module satisfying the property that the multiplication map

µM,R : M ⊗R R→M, µM,R(m⊗R r) = mr,

establishes an isomorphism will be named a firm right R-module. In this case, we denote theinverse map by

dM,R : M →M ⊗R R, dM,R(m) = mr ⊗ r.

We denote by MR the full subcategory of MR whose objects are all firm right R-modules. If R isa ring with unit then one can easily check that the category MR of firm right R-modules coincideswith our previous definition of MR as the category of unital right R-modules. This terminologyis due to Quillen [99]. Firm modules are called regular modules in [74], where their Morita theoryis developed. Similarly one introduces the category RM as the category of firm left R-modules.

Recall that a diagram in a category A of the form

Mf // N

g // P

is called a sequence if gf = 0. If A is abelian, then we call a sequence in A exact if ker g = Im f .A functor F : A → B is called exact if it preserves exact sequences. We call M ∈ RM flat provided

that the functor −⊗RM : MR → Ab is exact. We will characterize flat modules over firm ringslater in this Section.

In the case where M = R, the maps µM,R = µR,M = µR. Consequently, we obtain thatbijectivity of one of these maps implies bijectivity of the other maps. Therefore, R ∈ MR isequivalent to R ∈ RM. In this situation we say that R is a firm ring and we will denote thestructure maps of R by µR and dR.

Clearly, RMR is a monoidal category with monoidal unit R if R is a firm ring. Moreover, wecan introduce the bicategory with 0-cells firm B-rings, 1-cells firm bimodules over firm rings and2-cells bilinear maps. We denote this bicategory by Frm(B).

Let R and S be two firm rings. A right firm morphism ι : R → S is a non-unital ringmorphism such that the restriction of scalar functor induces a functor MS →MR. If ι is a rightfirm morphism then S becomes a firm S-R bimodule.

In [90] a generator for MR is constructed. If R ∈ MR, we can easily see that R itself is agenerator for MR.

Lemma 2.9. Let R be a firm ring, then the regular right R-module is a generator for MR.

Proof. Take any M ∈MR. Since M is firm as a right R-module we can write any m ∈Mas m =

∑mrr. Consequently we can find a set of elements mi ∈M, i ∈ I that generate M as

a right R-module (i.e. take a set of representants of mr for all m ∈M , in the worst case we canjust take the whole of M). Applying firmness a second time, we find mi = mr

i r =∑ni

ji=1mijirjiwhere in the second equation we just have choosen a set of representants for the element mr

i⊗Rr ∈M ⊗RR. Now construct a map π :

∐i∈I R

ni →M defined on the element indexed by i and ji asπiji(r) = mijir. Then all elements mi are in the image of π, so π is surjective onto M and R isa generator for MR.

It is known ([90], [99]) that the category MR is a Grothendieck category provided thatR ∈MR. In both cases, the proof is indirect, namely, MR is shown to be equivalent to a certainabelian category. In particular, due essentially to the lack of left exactness of − ⊗R R, kernels


cannot be computed in general in Ab. The situation is simpler if, in addition, R is assumed to beflat as a left R-module: MR is then easily shown to be a Grothendieck category, where coproducts,kernels and cokernels are already computed in Ab, i.e. the forgetful functor U : MR → Ab isexact. In fact these are consequences from the following approach to discribe firm rings and firmmodules as corings and comodules over corings.

Different from the case of MR, there exists no unital ring S such that MR is isomorphicto MS , even if R is firm. However, as mentioned above, we can describe MR as a category ofcomodules over a coring (with a unital base ring).

Lemma 2.10. Let R be a non-unital ring. Then R is a firm ring if and only if there exists abimodule map ∆ : R→ R⊗R R ∼= R⊗ bR R such that (R,∆, ι) is an R-coring.

Proof. Suppose R is a firm ring. Take ∆ = dR. The coassociativity of ∆ follows by theassociativity of µR, since they are two-sided inverses in MR. Denote ∆(r) = r(1) ⊗ bR r(2), thenι(r(1))r(2) = r(1)ι(r(2)) = r(1)r(2) = µR dR(r) = r, so the counit property holds as well.

Conversely, if R is an R-coring, one can easily check

(µR ∆)(r) = r(1)r(2) = r(1)ι(r(2)) = r

(∆ µR)(∑i

ri ⊗R r′i) = ∆(∑i

rir′i) =

∑i

∆(ri)r′i

=∑i

ri(1) ⊗R ri(2)r′i =∑i

ri(1)ri(2) ⊗R r′i

=∑i

ri ⊗R r′i

so ∆ is a two-sided inverse for µR and R is a firm ring.

Theorem 2.11. Assume that R is a firm ring and let R = (R,∆, ι) be the associated R-coringfrom Lemma 2.10. Then the categories MR and MR are isomorphic.

Proof. Take (M,ρ) ∈ MR. Then M is in particular a right R-module, so a non-unitalR-module. Let dM,R be the composition M → M ⊗ bR R ∼= M ⊗R R. As in Lemma 2.10 we caneasily check that µM,R and dM,R are mutual inverses, so M is firm as a right R-module.

Conversely, let M be a firm right R-module and define ρ as the composition

MdM,R // M ⊗R R

∼= // M ⊗ bR RThe coassociativity of ρ follows by the coassociativity of µM,R. Finally, for any m ∈M ,

(M ⊗R ι) ρ(m) = mrι(r) = mrr = m,

so the counit property is satisfied as well.

Examples of firm rings are provided by rings with (idempotent) local units, that we will studyin the following Sections. Another class of examples of firm rings was introduced in [33], we willbriefly discuss this example.

Example 2.12. An A-coring C is called coseparable if the comultiplication ∆C : C → C ⊗AC has a left inverse π in CMC. We will study coseparable corings extensively in Chapter 7.Coseparability can be defined as well for corings without counit. It follows from [33, Theorem 2.6and Proposition 2.7] that C is a firm A-ring with multiplication π and that all (right) C-comodulesare firm (right) C-modules. In particular, let ι : B → A be a ring extension with left inverseE : A→ B. Then we can construct the Sweedler coring A⊗B A associated to ι (see Example 3.3in Chapter 3). This coring is a coseparable coring, and thus A ⊗B A is a firm A-ring. Structuremaps are given by

π(a⊗B a′ ⊗A a′′ ⊗B a′′′) = a⊗B E(a′a′′)a′′′, ∆(a⊗B a′) = a⊗B 1⊗A 1⊗B a′.


The Eilenberg-Watts Theorem for firm rings. The classical Eilenberg-Watts Theorem for ad-joint functors between categories of modules over unital rings can be generalized to firm modulesover firm rings. Remark that the formulation of Theorem 2.13 improves [74, Proposition 1.6].

Theorem 2.13. Let A and B be two firm rings. For a functor F : MB →MA, the followingassertions are equivalent:

(i) F has a right adjoint G : MA →MB;(ii) F is right exact and preserves direct sums;(iii) F ∼= −⊗B P for some firm B-A bimodule P .

Moreover, when (i)-(iii) hold, then the right adjoint G is given by

G = HomA(P,−)⊗B B,where P = F (B) and G is unique up to isomorphism.

Proof. (ii) ⇒ (iii). Consider the functor F : MB →MA that is right exact and preserves

direct sums, then F preserves colimits. Denote as usual by A the Dorroh extension of A, which isa unital ring. If we apply [36, Theorem 39.3] (where we take A = MA, B = MB and T = A),then we find a natural isomorphism

Ψ−,A : −⊗ bA F (A) → F (−⊗ bA A)

of functors M bA → MB. If we compose these functors with the inclusion functor J : MA →MA

∼= M bA then the resulting functors MA → MB are again naturally isomorphic. Hence wefind for any M ∈MA the following isomorphisms

M ⊗A F (A) ∼= M ⊗ bA F (A) ∼= F (M ⊗ bA A) ∼= F (M ⊗A A) ∼= F (M)

Here we used Lemma 2.8 that the tensor product over A is isomorphic to the tensor productover A if it is computed with a firm A-module. In particular, if we take M = A, we find thatA⊗A F (A) = F (A), so F (A) is a firm left A-module and consequently a firm A-B bimodule.

(iii) ⇒ (i). Put G = HomB(P,−) ⊗B B, then G is a right adjoint for F . Let us just stateunit η and counit ε for the adjunction. For N ∈MA and M ∈MB, we define

ηN : N → HomA(P,N ⊗A P )⊗B B, ηN (n) = ϕnb ⊗B b,εM : HomA(P,M)⊗A P →M, εM (ϕ⊗A p) = ϕ(p),

where ϕn(p) = n⊗A p for all n ∈ N and p ∈ P = F (B).(i) ⇒ (ii). Classical.

Firm rings as ideals in unital rings. Let R be a firm ring. Consider a non-unital ring morphismR → T , where T a ring with unit. Such a morphism exists for every firm ring R, since we canchoose T = R, in that case R is even an ideal in T . We would like to characterize this situationfor general T .

We have a functor T ⊗R − : RM→ TM, and, for every N ∈ RM, a homomorphism of leftR-modules vN : N → T ⊗R N defined as vN (n) = 1⊗R n.

Lemma 2.14. The following statements are equivalent.

(i) vR : R→ T ⊗R R is an isomorphism;(ii) R is a left ideal of T ;(iii) vN : N → T ⊗R N is an isomorphism for every N ∈ RM.

Proof. (i) ⇒ (ii) For t ∈ T and r ∈ R put s = v−1R (t ⊗R r) ∈ R. Then 1 ⊗R s = t ⊗R r

which implies, after multiplication, that s = tr.(ii) ⇒ (i) Define f : T ⊗R R → R by f(t ⊗R r) = tr. Obviously, f vR = R. For the othercomposition,

(vR f)(t⊗R r) = vR(tr) = vR(tsrs) = tsvR(rs) = ts⊗R rs = t⊗R srs = t⊗R rwhich shows that f = v−1

R .(i) ⇒ (iii) With RN firm we have

T ⊗R N ∼= T ⊗R R⊗R N ∼= R⊗R N ∼= N,


isomorphism which is explicitly given by

(30) t⊗R n 7→ t⊗R r ⊗R nr 7→ tr ⊗R nr 7→ (tr)nr

This isomorphism is the inverse of vN since, for n ∈ N ,

n 7→ 1⊗R n 7→ 1⊗R r ⊗R nr 7→ r ⊗R nr 7→ rnr = n

(iii) ⇒ (i) is obvious.

Lemma 2.15. If R is a left ideal of T then every firm left R-module N is a left T -modulewith the action tn = (tr)nr for t ∈ T and n ∈ N . Moreover, we have an isomorphism M ⊗T N ∼=M ⊗R N for every M ∈MT .

Proof. The isomorphism vN transfers the structure of left T–module from T ⊗R N to Naccordingly to (30). The isomorphism M ⊗T N ∼= M ⊗R N is then obtained, with the help ofLemma 2.14, as the composite

M ⊗T N ∼= M ⊗T (T ⊗R N) ∼= (M ⊗T T )⊗R N ∼= M ⊗R N

Flat and injective modules over firm rings. From now on, let R be a firm ring. Our aim is togeneralize some well-known facts about flat and injective modules over unital rings to the situationof firm rings. Our treatment is an adaptation of the one given in [106, Chapter I, Section 10].

Recall that we defined a right R-module F flat if and only if the functor F⊗R− : RM− → Abis exact.

Lemma 2.16. Let R be a firm ring. Then the following assertions are equivalent:

(ai) R is flat as a left R-module;

(aii) the functor J : MR → MR is exact;(aiii) the forgetful functor U : MR → Ab is exact.

Let F be a left R-module and consider the following statements

(bi) −⊗R F : MR → Ab is (left) exact (i.e. F is flat as a left R-module);

(bii) F is flat as a left R-module.(biii) −⊗R F : MR → Ab is (left) exact;

Then (bi) is equivalent with (bii) and follows by (biii). If R is flat as left R-module, then thethree statements are equivalent.

Proof. (a) Consider the following diagram of functors

(31) MR

−⊗RR //

−⊗RR

""EEE

EEEE

EAb

Hom(R,−)⊗RR

MR

J

bbEEEEEEEE

U==

Remark that we have an isomorphism of functors U ' − ⊗R R : MR → Ab. One can easily

check that (J,−⊗RR) is a pair of adjoint functors between the categories MR and MR (see also

Corollary 3.9 (ii)). Moreover, −⊗R R : MR →MR has also a right adjoint Hom(R,−)⊗R R by

Theorem 2.13. Therefore, − ⊗R R : MR → MR is both left and right exact, hence exact. Wealso know that (ai) implies (aiii).(ai) ⇒ (aii). The functor J is right exact since it has a right adjoint. Moreover since MR

∼= M bRis a module category over a unital ring, the forgetful functor U : MR → Ab is faithful, hence thestatement follows since (ai) implies (aiii).(aiii) ⇒ (ai). This follows from the commutativity of diagram (31) the previous observations.

(aii) ⇒ (ai). The functor −⊗R R : MR → Ab can be considered as the following composition

MR

−⊗RR//MRJ // MR

eU // Ab


since all functors in this composition are exact, we find that R is flat as a left R-module.(b) The equivalence between left exactness and exactness follows from the fact that the functor

−⊗RF has a right adjoint by the Eilenberg-Watts Theorem (whether over the firm ring R or over

the unital ring R).

(bi) ⇔ (bii). By Lemma 2.8 tensor products over R and R are isomorphic. This implies that thefunctors −⊗R F and −⊗ bR F are isomorphic. Hence F is flat as a left R-module if and only if it

is flat as a left R-module.(biii) ⇒ (bi). Consider the following commutative diagram of functors

(32) MR

−⊗RF //

−⊗RR

""EEE

EEEE

EAb

MR

J

bbEEEEEEEE −⊗RF

==

Since the functor −⊗RR : MR →MR is exact (see part (a)), we find that −⊗R F : MR → Abis exact if −⊗R F : MR → Ab is exact.(bi) ⇒ (biii). If R is flat as left R-module, then J is exact by part (a). Hence it follows from the

commutative diagram (32) that −⊗RF : MR → Ab is exact if −⊗RF : MR → Ab is exact.

Since for a unital ring the forgetful functor MR → Ab is exact, Lemma 2.16 implies that thatthe notion of a flat module over a unital ring coincides with the notion of a flat module over afirm ring, as introduced in the beginning of this section.

Recall that M is called an injective right R-module if and only if M is an injective object inthe category MR, which means that for every injective morphism i : N → N ′ of right R-modules,the induced map HomR(N ′,M) → HomR(N,M) must be surjective.

Proposition 2.17. A right R-module E is injective if and only if for every right ideal I ofR, the morphism HomR(R,E) → HomR(I, E) is surjective.

Proof. In one direction the statement is trivial. For the other way, we start from a monomor-phism α : L→M and any morphism ϕ : L→ E. Consider the set

M = ϕ′ : L′ → E | L ⊂ L′ ⊂M, s.t. ϕ′ extends ϕ.We can equip M with a partial order, defining ϕ′ ≤ ϕ′′ if and only if ϕ′′ is a further extension.

Take a totally ordered T ⊂M and put L =∑

L′∈T L′, and define ϕ : L→ E by the condition

ϕ(l) = ϕ′(l) for l ∈ L′ ∈ T . In this way we find an upper bound for every totally ordered T ⊂Mand by the Zorn’s Lemma, this implies the existence of a maximal element ϕ0 : L0 → E in M.We have finished if we can prove that L0 = M .

Suppose there exists a x ∈M , x 6∈ L0, we show it is possible to extend ϕ0 to ψ : L0+xR→ E,from which the contradiction follows. Put I = a ∈ R | xa ∈ L0, then I is a right ideal in R.Define β : I → E, β(a) = ϕ0(xa). By hypothesis, there exists a ϕR : R → E, extending β. Forall z ∈ L0 and a ∈ R, we define ψ(z + xa) = ϕ0(z) + ϕR(a). To see that ψ is well defined,suppose z + xa = z′ + xa′ then z − z′ = x(a′ − a) and we find a′ − a ∈ I. By definition of ϕRwe get then ϕ0(z − z′) = ϕ0(x(a′ − a)) = ϕR(a′ − a) and thus ψ(z + xa) = ϕ0(z) + ϕR(a) =ϕ0(z′) + ϕR(a′) = ψ(z′ + xa′).

Finally, ψ is right R-linear and clearly it extends ϕ0, which ends the proof.

Proposition 2.18. Consider F an R−B–bimodule, where B is a unital ring, and let E bean injective cogenerator for MB. Then F is flat as a left R-module if and only if HomB(F,E) isinjective as a right R-module

Proof. The proof is identical as in the classical case, since it makes no use of units at all(see e.g. [106]).

Corollary 2.19. A right R-module F is flat if and only if I(F ) = HomZ(F,Q/Z) is injectiveas a right R-module.

A right ideal I of R is said to be of finite type if I = a1R+ · · ·+anR for some a1, . . . , an ∈ R.


Proposition 2.20. A firm left R-module F is flat if and only if the multiplication mapµI : I ⊗R F → F is injective for every ideal of finite type I of R.

Proof. If F is flat, the statement is clear, so we only prove the other implication. Supposethat µI is injective for every right ideal I of finite type of R, then it is also the case for all rightideals J of R. Indeed, take ri ⊗ yi ∈ J ⊗R F and let ri = rsi

i si be given by the equality R2 = R.If we denote by I =

∑rsii R, which is of finite type, then the following diagram is commutative

and µI is injective,

I ⊗R FµI //

%%LLLLLLLLLL F

J ⊗R FµJ

;;wwwwwwwww

It follows that µJ(ri ⊗ yi) = 0 ⇔ µI(ri ⊗ yi) = 0 ⇔ ri ⊗ yi = 0, so µJ is injective as well.Now we will construct a commutative diagram

HomR(R, I(F ))γ //

∼= α

HomR(J, I(F ))

∼=β

I(F )

δ// I(J ⊗R F ),

where, for any M , we define I(M) = HomZ(M,Q/Z). For all Φ ∈ HomR(R, I(F )) we definedefine α(Φ)(y) = Φ(r)(yr) for all y ∈ F . Conversely, we define α−1(Ψ)(r)(y) = Ψ(ry) for allΨ ∈ I(F ), r ∈ R, y ∈ F . One easily computes

α α−1(Ψ)(y) = α−1(Ψ)(r)(yr)= Ψ(ryr) = Ψ(y)

α−1 α(Φ)(r)(y) = α(Φ)(ry)= Φ(s)((ry)s) = Φ(r)(y).

In the last step, we used the R-linearity of Φ together with s⊗R (ry)s = r ⊗R y, which is basedon the unique expression of the element ry mapped trough the isomorphism dM : M → R⊗RM .

Analogously, define β(a⊗R y) = Φ(ar)(yr) for every Φ ∈ HomR

(J, I(F )

)and a⊗R y ∈ J⊗RF ,

which is shown to be an isomorphism analogously to the case of α.Since µJ : J ⊗ F → F is mono and Q/Z is injective, δ, which is the dual morphism of µJ

must be an epimorphism and by the diagram, γ is also epi. It follows from Proposition 2.17 thatI(F ) is injective and thus F is flat by Corollary 2.19.

Let R be any firm ring and F a firm left R-module. If for all N ∈MR, the relation N⊗RF = 0implies N = 0, then we call F totally faithful as a left R-module. We say that F is faithful if forall r ∈ R the relation rf = 0 for all f ∈ F implies r = 0.

Recall that a left R-linear morphism f : M → N is said to be pure, if and only if for all rightR-modules P we have an exact sequence

(33) 0 // P ⊗R ker f // P ⊗RMP⊗Rf // P ⊗R N // P ⊗R Im f // 0 .

Proposition 2.21. Let R be a firm ring and F a firm left R-module. If RF is totally faithfulthen F is faithful as a left R module and the canonical morphism ` : R→ EndA(F ) is a pure leftR-linear morphism.

Proof. Remark first that by the Eilenberg-Watts theorem we only have to check the leftexactness of the sequence (33). Take any N ∈ MR and consider the map N ⊗R ` : N ∼=N ⊗R R→ N ⊗R EndA(F ). Let K be the kernel of this morphism and consider K ⊗R F . Thenwe can compute for all n⊗R f ∈ K ⊗R F

n⊗R f = nr ⊗R rf = 0.

We find K ⊗R F = 0 and so K = 0 as F is totally faithful. In particular, we find in case N = Rthat ` is injective. Since it follows from our argument that N ⊗R ` is also injective for all choicesof N ∈MR, we conclude that ` is a pure as a left R-linear morphism.


We will later encounter some sufficient conditions under which Proposition 2.21 can be reversed.In the unital case, we can give a further characterization of totally faithful modules.

Proposition 2.22. Let B be a ring with unit and F a unital left B-module. Then thefollowing statements are equivalent

(i) For all N ∈MB and n ∈ N , n⊗ f = 0 for all f ∈ F implies n = 0;(ii) For all cyclic N ∈MB, the relation N ⊗B F = 0 implies N = 0;(iii) For all N ∈MB, the relation N ⊗B F = 0 implies N = 0.

Proof. (iii) ⇒ (ii) and (i) ⇒ (iii) are obvious.

(ii) ⇒ (i). Consider any N ∈MB and n ∈ N such that n⊗B f = 0 for all f ∈ F . Put M = nBthe cyclic right B-module generated by n, then we find that M ⊗B F = 0, and consequentlyM = 0. Since n = n1B ∈M , we find n = 0.

Remark 2.23. If B is a ring with local units (as defined in the next Section), then thecharacterization of totally faithfhul modules as in Proposition 2.22 still remains valid.

A left R-module is called faithfully flat if F is flat as a left R-module and the functor −⊗RF :MR → Ab reflects exact sequences.

Proposition 2.24. Let B be a firm ring and F a firm left R-module. The following state-ments are equivalent

(i) F is faithfully flat;(ii) F is flat and totally faithful as a left R-module;(iii) F is flat as a left R-module and for all proper right ideals I ⊂ R, we have (R/I)⊗R F 6= 0

(i.e. IF 6= F ).

Proof. This proof is an adaption of [114, 12.17].(i) ⇒ (ii). Take any N ∈ MR and consider the sequence 0 → N → 0 in MR. If N ⊗R F = 0,

then the sequence 0 → N ⊗R F → 0 is exact in Ab. By the property of part (i) we obtain that0 → N → 0 also has to be exact. Consequently N = 0.

(ii) ⇒ (i). Consider any sequence Kf // L

g // N in MR and suppose that the correspondigsequence

(34) 0 → K ⊗R Ff⊗RF // L⊗R F

g⊗RF // N ⊗R F → 0

in Ab is exact. Consider the canonical sequence

0 → Im f ⊗R F // ker g ⊗R F // ker g/Im f ⊗R F

Then the exactness of the sequence (34) implies that Im f ⊗R F ∼= ker g ⊗R F . Therefore,ker g/Im f ⊗R F = 0. Hence, we obtain from (ii) that ker g = Im f , i.e. the given sequence isexact.(ii) → (iii). By part (ii), (R/I) ⊗R F = 0 would imply R/I = 0, i.e. R = I. For the laststatement, consider the following exact row in MR,

0 // I // R // R/I // 0

since F is flat as a left R-module, we obtain the following commutative diagram in Ab with exactrows

0 // I ⊗R FµI,F

// R⊗R FµR,F

// (R/I)⊗R F //

γ

0

0 // IF // F // F/IF // 0Since F is firm as a left R-module, µR,F is an isomorphism. From the properties of the diagramwe immediately obtain that γ is surjective. We can also see that µI,F is surjective. Take anyif ∈ IF , then if = µI,F (ir ⊗R rf), where we used the firmness of F and the fact that ir ∈ Isince I is a right ideal in R. The surjectivity of µI,F implies that γ is injective.(iii) ⇒ (ii). Consider any N ∈MR. We have to show that N ⊗R F = 0 implies that N = 0, or


equivalently N 6= 0 implies that N ⊗R F 6= 0. Suppose there exists an element 0 6= n ∈ N . Thenwe can construct a right ideal in R as follows In = r ∈ R | nr = 0. One can easily observe thatR/Ir is isomorphic to the cyclic right R-module nR. By (iii) we find

0 6= R/Im ⊗R F ∼= nR⊗R F ⊂ N ⊗R F,

the inclusion is a consequence of the flatness of F as a left R-module.

2.2.3. Rings with local units.

Definition 2.25. Let B be a ring with unit and R a B-ring. Let M be a right moduleover the B-ring R (this implies M is a right B-module by definition). A right unit map on Mis a B-bimodule map ηM : B → R, such that the following diagram of right B-modules iscommutative

M ⊗B B

IM⊗BηM

∼=

$$IIIIIIIII

M ⊗B R µM

// M

Left unit maps are defined in a similar way. If M is a bimodule and ηM a left and right unit map,then we call ηM just a unit map.

Lemma 2.26. Let R be a B-ring and M a right R-module. The following statements areequivalent.

(i) For every m ∈ M there exists an element e ∈ RB := r ∈ R | br = rb, for all b ∈ B suchthat m · e = m;

(ii) for every finitely generated B-submodule T of M , there exists an element e ∈ RB such thatt · e = t for all t ∈ T ;

(iii) there exists a unit map ηT on every finitely generated B-submodule T of M .

In this case we say that R has a right local units on M , and we call e a right local unit and ηT aright local unit map on T .

Proof. (i) ⇒ (ii) Let t1, . . . , tk be a set of generators for T . We proceed by induction

on k. If k = 1 then T = tB and from the first statement, we find an e ∈ RB such that te = t.Consequently tbe = teb = tb. If k > 1, we can find a right local unit e′ for t1. From the inducionhypothesis, we can also find a right local unit e′′ for the k − 1 elements ti − tie

′, i = 2, . . . , k.Now e := e′ + e′′ − e′e′′ ∈ RB and

t1e = t1e′ + t1e

′′ − t1e′e′′ = t1 + t1e

′′ − t1e′′ = t1;

tie = tie′ + tie

′′ − tie′e′′ = tie

′ + (ti − tie′)e′′ = tie

′ + ti − tie′ = ti,

and we find that e is a right local unit for all ti (1 ≤ i ≤ k) and consequently for every element inT .(ii) ⇒ (i) is trivial.

(ii) ⇔ (iii) follows from the fact that RB ∼= BHomB(B,R) as Z-modules.

Definition 2.27. A B-ring R has a right local units, if it has a right local units as a rightmodule over itself. In the same way, we define rings with left and two-sided local units. By localunits, we will always mean two-sided local units.If R has a right local units, then a right R-module M on which R has a right local units will betermed a right unital R-module.

Lemma 2.28. Let R be a B-ring, M a right R-module and N a left R-module. If there exista right local unit e′ ∈ R on m ∈M and a left local unit e′′ ∈ R on n ∈ N , then there exists alsoan element e ∈ R which is at the same time a right local unit on m ∈M and a left local unit onn ∈ N .

Proof. Take e := e′ + e′′ − e′e′′.


Corollary 2.29. If R has a left and right local units on a bimodule M , then it also hastwo-sided local units on M . In particular, if R is a B-ring with left and right local units, then Ris a B-ring with two-sided local units.

Remark 2.30. Recall from [111] that a ring R is called left (resp. right) s-unital if u ∈ Ru(resp. ∈ uR), for every u ∈ R. R is called s-unital if u ∈ Ru ∩ uR, for every u ∈ R. It followsimmediately from Lemmas 2.26 and 2.28 and Corollary 2.29 that a Z-ring R has a left (resp. right,two-sided) local units if and only if it is left s-unital (resp. right s-unital, s-unital).

Lemma 2.31. Let R be a B-ring with right local units. A right R-module M has right localunits if and only if it is firm. In particular, R is a firm B-ring.

Proof. Assume that M has a right local units; the map ψ : M →M⊗RR, ψ(m) := m⊗Re,with e a right local unit on m, is well-defined. Let e′ be another right local unit on m, and choosea right local unit e′′ on e and e′. We find m ⊗R e = m ⊗R ee′′ = me ⊗R e′′ = m ⊗R e′′ =me′ ⊗R e′′ = m ⊗R e′e′′ = m ⊗R e′. It is obvious that ψ is inverse to the module structure mapM ⊗R R→M .

Conversely, if M is a firm right R-module, then M ∼= M ⊗R R, and since R has local unitson itself, it also has local units on M .

We call E ⊂ R a complete set of local units in R, if for any element r ∈ R there exists anelement e ∈ E such that e is a local unit for r.Let R and S be two rings with (right) local units, then we say that R→ S is a morphism of ringswith right local units if it is a morphism of non-unital rings such that there exists a complete setof right local units ES in S that is the image of a set of right local units in R.

Lemma 2.32. Let R be a ring with local units and let E be a complete set of right local units.Then for any M ∈ MR and for any m ∈ M there exists an element e ∈ E such that m · e = m.This means that E is a complete set of right local units for M . Consequently, let ι : R→ S be amorphism of rings with right local units, then ι is a right firm morphism.

Proof. Let em be a right local unit for m. Then there exists a right local unit e ∈ E for em.Therefore, we obtain

m = mem = meme = me,

and e is a right local unit for m.

Theorem 2.33. If R is a B-ring with right local units, then every firm right module M overR (considered as a Z-ring) is also a right B-module. Moreover the category of firm right modulesover R as a Z-ring is isomorphic to the category of firm right modules over R as B-ring.

Proof. We know that M ∼= M ⊗R R, and since R is a right B-module, M is also a rightB-module as well. The B-action is given by the formula m · b = m(eb), with e a right local uniton m. A similar argument applies to the morphisms.

Theorem 2.34. Let R be a B-ring (without unit). Let M be a full subcategory of MR.

Then R has local units on all M ∈ M if and only if there exists a full subcategory N of MB,such that every M ∈ M is generated by objects of N as a right B-module, and, for all N ∈ Nand f ∈ Hom(N,M), we can find an e ∈ RB ∼= BHomB(B,R) such that f = fe f , where

fe = µM (M ⊗B e) : M ∼= M ⊗B BM⊗Be // M ⊗B R

µM // M.

In other words, M is generated by B-modules on which there exists a local unit.

Proof. Suppose first that the subcategory N exists. Take any M ∈M. Since N generatesM, we can find a family of right B-modules (Ni)i∈I in N such that there exists a surjective mapπ :

∐i∈I Ni → M . Consequently, for any m ∈ M , we can write m =

∑i∈J π(ni) where J is

a finite subset of I. We show by induction on the cardinality of J that we can find a local unite ∈ RB. If the cardinality of J equals one, then m = π(n) for some n ∈ N . We know thatthere exists an element e ∈ RB such that π = fe π. Consequently π(n)e = fe π(n) = π(n),so e is a unit for π(n) = m. Now suppose m =

∑ki=1 π(ni) with ni ∈ Ni and k > 1. By the


induction hypothesis we can find a local unit e ∈ RB for∑k−1

i=1 π(ni) and a local unit e′ ∈ RB forπ(nk)− π(nk)e. Then e′′ = e+ e′ − ee′ is a local unit for m since

me′′ =

(k∑i=1

π(ni)

)(e+ e′ − ee′) =

(k−1∑i=1

π(ni)

)(e+ e′ − ee′) + π(nk)(e+ e′ − ee′)

=k−1∑i=1

π(ni) +k−1∑i=1

π(ni)e′ −k−1∑i=1

π(ni)e′ + π(nk)e+ (π(nk)− π(nk)e)e′

=k−1∑i=1

π(ni) + π(nk) = m.

Conversely, let M be a subcategory of MR on which R has local units. We define N asthe category consisting of finitely generated B-submodules of modules in M. Then clearly Ngenerates M in MB. Moreover, for any N ∈ N , M ∈M and f ∈ HomB(M,N), we obtain thatIm f is a finitely generated B-submodule of M . By the definition of a module with local units, wecan find a local unit e ∈ RB for Im f . Consequently, fe f(n) = f(n)e = f(n) for all n ∈ N .

2.2.4. Rings with idempotent local units.

Definition 2.35. With notation as in Definition 2.25, a right local unit map ηM on Mis called idempotent if ηM is a morphism of B-rings. This means that the following diagramcommutes:

B ⊗B BηM⊗BηM //

µB

R⊗B RµR

B ηM

// R

Lemma 2.36. Let R be a B-ring, and M a right R-module. The following statements areequivalent.

(i) R has an idempotent right local unit map on every finitely generated right B-submodule Nof M ;

(ii) for every finitely generated right B-submodule N of M , we can find an idempotent e ∈ RBsuch that ne = n for all n ∈ N ;

(iii) for every finitely generated right B-submodule N of M , we can find an idempotent e ∈ RBsuch that N ⊆Me;

In this case we say that R has idempotent right local units on M and we call e an idempotentright local unit.

Proof. The equivalence (i) ⇔ (ii) can be proved in the same way as Lemma 2.26, taking

into account that ring morphisms B → R correspond to idempotents in RB.(iii) ⇔ (ii) is trivial.

It follows from Lemma 2.36 that R is a B-ring with idempotent left (resp. right) local unitsif and only if R is the direct limit of a split direct system of rings with a left (resp. right) unit.

We call E ⊂ R a complete set of idempotent local units in R, if for any element r ∈ R thereexists an element e ∈ E such that e is a local unit for r.Let R and S be two rings with (right) idempotent local units, then we say that R → S is amorphism of rings with idempotent local units if it is a morphism of non-unital rings such thatthere exists a complete set of idempotent local units ES in S that is the image of a set ofidempotent local units in R.

Lemma 2.37. A B-ring R with left and right idempotent local units also has two-sided idem-potent local units.

Proof. For every finite subset r1, . . . , rk ⊂ R, we have to find an idempotent e ∈ RB

such that eri = rie = ri. By assumption, we can find an idempotent left local unit e′ on


r1, . . . , rk, and an idempotent right local unit e′′ on r1, . . . , rk, e′. An easy calculation showsthat e = e′ + e′′ − e′′e′ is an idempotent two-sided unit on r1, . . . , rk.

Remark 2.38. It follows from Lemmas 2.36 and 2.37 that R is a B-ring with two-sided localunits if and only if for every finitely generated B-subbimodule F of R, there exists an idempotente ∈ RB such that F ⊆ eRe. Notice that eRe is a B-ring with unit e. In the case where B = Z,we recover the definition of ring with local units as introduced by Anh and Marki in [7].

We will denote by FB the category of firm B-rings.

Theorem 2.39. The following statements are equivalent.

(i) R is a ring with (right) idempotent local units;(ii) There exists a split direct system Rs : Z → Fs

B such that R = colim (R, β), where βji =R(aji) and Ri is a ring with (right) unit;

(iii) There exists a direct system R : Z → FB such that colimR = R, where βji = R(aji) andRi is a ring with (right) unit.

Proof. (i) ⇒ (ii). On the index set I of the idempotent local units, we define a partial

ordering ≤ as follows: i ≤ j if and only if eiej = ejei = ei (if R has only right local units, wesay i ≤ j if and only if eiej = ei). This partial ordering is directed: for all i, j ∈ I, there existsk ∈ I such that k ≥ i, j. Indeed, by definition of a B-ring with idempotent local units, for thetwo elements ei and ej , we can find an element ek with k ∈ I, such that ek is a (right) localunit for both ei and ej , i.e. k ≥ i, j. Then let Ri = eiRei (if R has only right local units, weput Ri = Rei), for each i ∈ I. If i ≤ j, then Ri is a subalgebra of Rj , and the inclusion mapβji : Ri → Rj is a morphism in FB. Also γij : Rj → Ri, γij(bj) = eibjei is a morphism of firmalgebras (if R has only right local units, we define γij(bj) = bjei). As in Section 2.1, we associatea category Z to the partially ordered directed set (I,≤). Then we have a split direct systemRs : Z → Fs

B, Rs(i) = Ri, Rs(aji) = (βji, γij). Clearly colimR = (R, β), with βi : Ri → R

the inclusion map.(ii) ⇒ (iii) is trivial.

(iii) ⇒ (i). Recall that module categories contain colimits and they can be described asfollows. Let

B = (i, bi) | i ∈ I, bi ∈ Ri.be the disjoint union of the Ri. An equivalence relation ∼ on B is defined as follows: (i, bi) ∼ (j, bj)if and only if there exists k ≥ i, j such that βki(bi) = βkj(bj), where βki = R(aij). Then letR = B/ ∼ and βi : Ri → R, βi(bi) = [(i, bi)]. The elements of the form [(i, 1Ri)] make up a setof idempotent (right) local units for R.

Remark 2.40. Abrams [1] proved the implication (i)⇒(iii) in Theorem 2.39 under the strongerassumption that R is a ring with commuting idempotent local units. Theorem 2.39 tells us thatthe implication still holds if we drop the condition that the idempotents commute, and then weeven have an equivalence. Abrams [1, Lemma 1.5] also shows that firm modules over a ring withcommuting idempotent local units can be written as direct limits. In Lemma 2.36, this property isgeneralized to arbitrary rings with idempotent local units, and it is shown that they are preciselythe ones that can be written as direct limits.

Let R be a B-ring with (right) idempotent local units and A a B-ring with unit. Let I be theindex set of idempotent local units of R and Z the associated category as in Theorem 2.39.

Lemma 2.41. P is a firm (R,A)-bimodule if and only if we can describe P in the followingway. There exists a split direct system P s : Z →MA

s where we denote for all i ≤ j ∈ I:P s(i) = Pi, P

s(aji) = (σji, τij),

and such that the following conditions hold

- for all i ≤ j ∈ I, bi ∈ Ri, pj ∈ Pj :(35) βji(bi)pj = σji(biτij(pj));

- each Pi is a (Ri, A)-bimodule for the B-subring Ri ⊂ R with (right) unit and- colimP = (P, σ).


Proof. Suppose first that P is a firm (R,A) module. For each i ∈ I, we consider Pi = eiP .Then P = ∪i∈IPi. Moreover it is clear that Pi is a left Ri = eiRei-module and a (Ri, A)-bimodule.For i ≤ j ∈ I, we have right A-module maps σji : Pi → Pj (the inclusion map) and τij : Pj → Pi,τij(pej) = pei. This defines a split direct system P s : Z → Ms

A, and colimP = (P, σ), withσi : Pi → P the inclusion map. Finally, we check that (35) holds in this situation. Let i ≤ j, andtake bi = eibei ∈ Ri, pj = ejp ∈ Pj . Then

σji(biτij(pj)) = eibeieiejp = eibejp = biejp = βji(bi)pj ,

as needed.For the converse, the construction of the colimit is done using arguments similar to the ones

in the proof of Theorem 2.39. From Proposition 2.1, we know that σi : Pi → P has a left inverseτi : P → Pi. Take p ∈ P , b ∈ R. Making use of the characterization of R given in Theorem 2.39,we can find i, j ∈ I such that p = σi(pi), b = βj(bj), with pi ∈ Pi, bj ∈ Rj . Take k ≥ i, j, anddefine

(36) bp = σk(βkj(bj)σki(pi)).

To prove that this is a well-defined action of R on P , we have to show that (36) is independentof the choise of the index k. Suppose l ≥ i, j and consider σl(βlj(bj)σli(pi)). Take any m ≥ l, kthen we compute

σl(βlj(bj)σli(pi)) = σm σml(βlj(bj)τlmσmlσli(pi))(35)= σm(βmlβlj(bj)σmlσli(pi))= σm(βmj(bj)σmi(pi))

In a similar way, we prove that we can replace k by m in (36) and by this the left R actionof P is independent of the choice of the index k. Finally, P is firm as a left R-module: takep = σi(pi) ∈ P ; then βi(1Ri)p = p.

One can easily adapt the proof in case R has only right local units.

Lemma 2.42. If P satisfies the equivalent conditions of Lemma 2.41, the following formulashold, for all i ≤ j ∈ I, pi ∈ Pi, pj ∈ Pj , ϕi ∈ P ∗i , ϕj ∈ P ∗j and bi ∈ Ri.

(σji τij)(pj) = βji(1Ri)pj ;(37)

σji(bipi) = βji(bi)σji(pi);(38)

ϕjβji(bi) = ((ϕj σji)bi) τij = τ∗ij(σ∗ji(ϕj)bi);(39)

ϕjβji(1Ri) = ϕj σji τij = τ∗ij(σ∗ji(ϕj));(40)

τij(βji(bi)pj) = biτij(pj);(41)

τ∗ij(ϕibi) = τ∗ij(ϕi)βji(bi).(42)

Proof. (37) follows after we take bi = 1Ri in (35). (38) can be shown as follows:

βji(bi)σji(pi)(35)= σji(bi(τij σji)(pi)) = σji(bipi).

We next prove (39). Take any pj ∈ Pj ,(ϕjβji(bi))(pj) = ϕj(βji(bi)pj)

(35)= (((ϕj σji)bi) τij)(pj)

Then (40) follows after we take bi = 1Ri in (39), and (41) also follows easily:

τij(βji(bi)pj)(39)= (τij σji)(biτij(pj)) = biτij(pj).

(42) follows immediately from (41).

Recall from Proposition 2.5 the construction of the dual split direct system P ∗s of a directsystem P s and the ‘finite’ dual colimit P †.

Lemma 2.43. If P satisfies the equivalent conditions of Lemma 2.41, then colimP ∗ = (P †, τ †)exists and P † is a firm (A,R)-module.

Proof. Since Pi is a unital (Ri, A)-bimodule, P ∗i is a unital (A,Ri)-bimodule. The statementfollows by Lemma 2.41 using left-right duality.


Using the characterizations Theorem 2.39 and Lemma 2.41 we get an explicit description forP †

P † = ϕ ∈ P ∗ | ∃i ∈ I : ϕ = ϕ σi τi= ϕ ∈ P ∗ | ∃i ∈ I : ϕ(p) = ϕ(eip), for all p ∈ P.(43)

The right R-action on P † can be described as follows: take ϕ = ϕi τi ∈ P † and b = βj(bj) ∈R. For k ≥ i, j, we have

(44) ϕb = ((ϕi τik)βkj(bj)) τk.In particular, we have, for ϕi ∈ P ∗i and bi ∈ Ri:(45) (ϕi τi)βi(bi) = (ϕibi) τi.In explicit form this means (ϕb)(p) = ϕi(eiejbejp) or just (ϕb)(p) = ϕ(bp). Having this in mind,we can give an alternative characterization of P †,

Lemma 2.44. If P satisfies the equivalent conditions of Lemma 2.41, then the firm (A,R)-module P † constructed in Lemma 2.43 is isomorphic to P ∗ ⊗R R

Proof. Let us construct an isomorphism ω : P ∗ ⊗R R → P †, ω(φ ⊗R r) = φr. For theconverse, remark that the description of P † in (43) means exactly that every ϕ ∈ P † is of the formϕeϕ for some idempotent eϕ ∈ R. By this observation we define $ : P † → P ∗ ⊗R R, $(ϕ) =ϕ ⊗R eϕ. Let us check that ω and $ are mutual inverses. Let e be an idempotent local unit forr ∈ R, then φr = φre.

$ω(φ⊗R r) = $(φr) = φr ⊗R e= φ⊗R re = ϕ⊗R r

ω$(ϕ) = ω$(ϕeϕ) = ω(ϕ⊗R e)= ϕeϕ = ϕ

Lemma 2.45. If P satisfies the equivalent conditions of Lemma 2.41, then we have for alli ∈ I, bi ∈ Ri, p ∈ P and ϕ ∈ P †,

βi(bi)p = σi(biτi(p));(46)

ϕβi(bi) = (ϕ σi)bi τi.(47)

Proof. By the characterization of Theorem 2.39 and Lemma 2.41, we can write bi = eibeiand τi(p) = eip, where ei is an idempotent in R. Moreover the maps βi and σi are injections.With this information in hand we easily find

βi(bi)p = eibeip

= σi(biτi(p)) = eibeieip.

The other equation follows by

ϕβi(bi)(p) = ϕ(βi(bi)p) = ϕ(σi(biτi(p))) = (ϕ σi)bi τi(p),where we used (46) in the second equality.

2.2.5. The M-adic and finite topology. Let A be any ring (not necessary with unit) andM a right A-module. Recall that the M -adic topology on A is the topology generated by thebasis of open sets

Q(a,m1, . . . ,mk) = b ∈ A | mia = mibConsider subsets R ⊂ A and N ⊂ M . In [48], an element a ∈ A was said to be multiplicativelyapproximated from the right by R on N , if for every finite subset n1, · · · , nk ⊂ N , there existsr ∈ R such that ni · a = ni · r, for all i ∈ 1, · · · , k. If every a ∈ A can be multiplicativelyapproximated by R on N , then R is a right multiplicative approximation of A on N . From thesedefinitions it is clear that R ⊂ A is a multiplicative aproximation of A on M if and only if R isdense in the M -adic topology on A. If A has a unit and R ⊂ A, then R has a right local units ifand only if 1A can be multiplicatively approximated from the right by R on R.


Proposition 2.46. Let A be a ring with unit, and consider an additive subset R ⊆ A. Thenthe following assertions are equivalent:

(i) R is dense in the R-adic topology on A;(ii) R is a right ideal of A and has a right local units.

If these conditions hold, then R is dense in the M -adic toplogy for any right A-module M onwhich R has a right local units.

Proof. (i) ⇒ (ii) For every finite set of elements r1, · · · , rn ⊂ R ⊂ A and a ∈ A, wehave an element r ∈ R such that ria = rir ∈ R. This means R is a right ideal, and taking a = 1we find that R has right local units.(ii) ⇒ (i) R has a right local units, so for all r1, · · · , rn ∈ R, we can find e ∈ R such that rie = ri.For every a ∈ A, we then have riea = ria, and ea ∈ R, since R is a right ideal.The proof of the final statement is similar.

If R is any ring with with right local units and R is the Dorroh extension of R, then we canalways take A = R in Proposition 2.46.

Recall that for any right A-module M , the finite topology on M∗ is the topology generatedby the basis of open sets

O(f, p1, . . . , pn) = g ∈M∗ | g(pi) = f(pi), 1 ≤ i ≤ nwhere f ∈M∗ and p1, . . . , pn ∈M .A subset R ⊂ M∗ is dense with respect to this topology if and only if for every f ∈ M∗ andp1, . . . , pn ∈M , we can find a g ∈ R such that g(pi) = f(pi), for 1 ≤ i ≤ n.Let C be an A-coring, and consider its dual ring ∗C (see Example 1.46). Then we can consider on∗C both a finite topology and an M -adic topology for any right ∗C-module M . It is very naturalto compare both topologies. Recall from Example 1.46 that any right C-comodule has a naturalright ∗C-module action.

Proposition 2.47. Let C be an A-coring, and R ⊂ ∗C a subring.

(i) If R is dense in the finite topology on ∗C then R is dense in the M -adic topology on ∗C, forany right C-comodule M . In particular R has local units on every C-comodule.

(ii) If R is dense in the C-adic topology on ∗C, then R is dense in the finite topology on ∗C.

Proof. (i) Take M ∈ MC. For every m ∈ M and f ∈ ∗C, we have m · f = m[0]f(m[1]).Now, by the density of R, there exists a g ∈ R such that f(m[1]) = g(m[1]), and so m · f = m · g.(ii) For every finite c1, · · · , cn ⊂ C, and f ∈ ∗C, there exists g ∈ R such that ci(1)f(ci(2)) =ci · f = ci · g = ci(1)g(ci(2)), for all i. Applying εC to both sides, we find that f(ci) = g(ci) for alli, which means exactly that R is dense in the finite topology on ∗C.

Corollary 2.48. If R ⊂ ∗C is a right C-comodule, then R is dense in the finite topology on∗C if and only if R is an ideal in ∗C, R has a right local units and C is firm as a right R-module.

Proof. If R is dense in the finite topology on ∗C, then R is dense in the R-adic topology on∗C, so by Proposition 2.46 R is a ring with local units and an ideal in ∗C. Since by Proposition 2.47R has also local units on C we know from Lemma 2.31, that C is firm as a right R-module.

Conversely, we find by Proposition 2.46 that R is dense in the C-adic topology on ∗C, andhence Proposition 2.47 (ii) tells us that R is dense in the finite topology on ∗C.

2.3. Projectivity

2.3.1. Firmly projective modules. Let A and B be rings with unit. Following [37, Sec.1.2], we call a triple M = (M,M †, µ) a dual pair , whenever M ∈ BMA, M † ∈ AMB andµ : M † ⊗B M → A is an A-bilinear map.

Recall that M∗ = HomA(M,A) is an (A,B)-bimodule, with structure given by

(afb)(m) = af(bm),

for all a ∈ A, b ∈ B, f ∈M∗ andm ∈M . Similarly the left dual ofM †, i.e. ∗M † = AHom(M †, A)is a (B,A)-bimodule. Then (M,M∗, µ), with µ(f ⊗B m) = f(m) is a dual pair. Also recall that

2.3. PROJECTIVITY 53

the adjunction between tensor product an the Hom functor gives us the following isomorphisms ofZ-modules:

(48) AHomA(M † ⊗B M,A)ζ−→ AHomB(M †,M∗)

ξ−→ BHomA(M, ∗(M †)).

For later use, we give the explicit description of the connecting maps:

ζ(µ)(m′)(m) = µ(m′ ⊗B m), ζ−1(ϕ)(m′ ⊗B m) = ϕ(m′)(m),

for all µ ∈ AHomA(M † ⊗B M,A), ϕ ∈ AHomB(M †,M∗), m ∈ M and m′ ∈ M †. We canconsider ϕ∗ ∈ BHomA(∗(M∗), ∗(M †)), and ξ(ϕ) = ϕ∗ ι, with ι : M → ∗(M∗) the canonicalmorphism. The proof of Lemma 2.49 is straightforward.

Lemma 2.49. Let A and B be rings with unit, and M = (M,M †, µ) a dual pair. ThenZ† := M ⊗AM † is a (possibly non-unital) B-ring, with multiplication map M ⊗A µ⊗AM †, andwe have morphisms of B-rings Φ : Z† → EndA(M) and Ψ : Z† → AEnd(M †), given by

Φ(m⊗A m′)(n) = mµ(m′ ⊗A n) and Ψ(m⊗A m′)(n′) = µ(n′ ⊗A m)m′.

This makes M into a Z†-B bimodule, and M † into a B-Z† bimodule with actions given by

µZ†,M = M ⊗A µ, and µM†,Z† = µ⊗AM †.

Z† is called the elementary B-ring associated to M .

It is straightforward that µ(m′z⊗Bm) = µ(m′⊗B zm) for all m′ ∈M †, m ∈M and z ∈ Z†.This means that the map

(49) µ : M † ⊗Z† M → A, µ(m′ ⊗Z† m) = µ(m′ ⊗B m)

is well-defined. Consider a ring morphism ι : R → M ⊗A M †, ι(r) = er ⊗A fr (summationunderstood). Then ι induces an R-module structure on M and M † by restriction of scalars.Explicitly, the R-action is given by the formula

(50) rm = erµ(fr ⊗B m).

In this way, the A-bimodule map (49) induces a homomorphism of A-bimodules

µ : M † ⊗RM → A, µ(m′ ⊗R m) = µ(m′ ⊗Z† m).

Therefore the equality

(51) rm = erµ(fr ⊗R m)

is deduced from (50). In the same way,

(52) m′r = µ(m′ ⊗R er)fr.The above reasoning justifies that from now on, we will no longer make explicit distinction

between µ, µ and µ.We say that (M,M †, µ) is an R-firm dual pair if there exists a firm ring R together with a

ring morphism ι : R → M ⊗A M † such that M and M † are firm as left, respectively firm rightR-modules, where the R-action is induced by ι.

A firm dual pair is a particular case of the previous situation, taking R = Z† and ι the identity.Consider the dual pair (M,M∗, ev), where ev : M∗ ⊗B M → A is the evaluation map. We

will denote the associated B-ring by Z, the multiplication in Z by µZ , and the actions of Z on Mand M∗ by µZ,M and µM∗,Z .

We will say that M is firmly projective as a right A-module if and only if the elementary B-ringZ = M ⊗AM∗ is a firm ring and M is firm as a left Z-module.

Similarly, for a firm B-ring R, M will be named R-firmly projective as a right A-module if andonly if there exists a ring morphism ι : R→ Z, ι(r) = er ⊗A fr and M is a firm R-module underthe R-module structure induced by ι.

By definition M is Z-firmly projective if it is firmly projective. In the following Proposition wehave collected elementary results about R-firmly projective modules and R-dual pairs.

Proposition 2.50. (i) If M is R-firmly projective for any firm B-ring R, then M is firmlyprojective.


(ii) If M is R-firmly projective, then (M,M †, µ) is an R-firm dual pair, with M † = M∗ ⊗R Rand µ = ev (M∗ ⊗R µZ,M ).

(iii) If (M,M †, µ) is an R-firm dual pair, then M is R-firmly projective and M † ∼= M∗ ⊗R R asA-R bimodule.

(iv) There is a bijective correspondence between the following objects(i) B-A bimodules M that are R-firmly projective as a right A-module;(ii) A-B bimodules M † that are R-firmly projective as a left A-module;(iii) R-firm dual pairs (M,M †, µ).

Proof. (i). First remark that the multiplication R⊗BM →M is a right A-linear (and thus

an R-A bilinear) map, as it is the composition of right A-linear maps: (M⊗A ev) (ι⊗BM). Thisimplies that if M is R-firmly projective and, a fortiori, firm as a left R-module, the isomorphismR⊗RM ∼= M holds as an isomorphism of R-A bimodules. This isomorphism induces a well-definedmap d as the following composition

M ⊗AM∗ ∼= R⊗RM ⊗AM∗ ι⊗RZ−→ M ⊗AM∗ ⊗RM ⊗AM∗ π−→M ⊗AM∗ ⊗Z M ⊗AM∗

or d(u⊗A ϕ) = er ⊗A fr ⊗Z ur ⊗A ϕ for all u⊗A ϕ ∈ M ⊗AM∗. Obviously, µZ d = Z, sinceerfr(ur) ⊗A ϕ = u ⊗A ϕ. Conversely, consider an element u ⊗A ϕ ⊗Z v ⊗A ψ ∈ M ⊗AM∗ ⊗ZM ⊗AM∗ (summation understood). Then we compute that

(d µZ)(u⊗A ϕ⊗Z v ⊗A ψ) = er ⊗A fr ⊗Z (uϕ(v))r ⊗A ψ= er ⊗A fr ⊗Z urϕ(v)⊗A ψ= er ⊗A fr ⊗Z (ur ⊗A ϕ) · v ⊗A ψ= er ⊗A fr · (ur ⊗A ϕ)⊗Z v ⊗A ψ= er ⊗A fr(ur)ϕ⊗Z v ⊗A ψ= erfr(ur)⊗A ϕ⊗Z v ⊗A ψ= u⊗A ϕ⊗Z v ⊗A ψ,

so d µZ = Z and Z is a firm ring. In a similar way, we find that M is a firm Z-module.(ii). Follows immediately since X ⊗R R is a firm right R-module for any right R-module X.

(iii). Consider the morphism ζ : M † → M∗, ζ(ϕ)(u) = µ(ϕ ⊗B u) for any ϕ ∈ M † and

u ∈ M . Recall that we have a morphism ι : R → M ⊗A M †, ι(r) = er ⊗R fr. Then we canconsider the morphisms

α : M † →M∗ ⊗R R, α(ϕ) = ζ(ϕr)⊗R rβ : M∗ ⊗R R→M †, β(ψ ⊗R r) = ψr = ψ(er)fr

A similar calculation as in part (i) shows that α and β are inverses.(iv). From part (ii) and (iii) we obtain the bijectivity between R-firmly projective right A-

modules and R-firm dual pairs. The bijectivity with R-firmly projective left A-modules follows byleft-right duality.

Examples of firmly projective modules can be obtained from projective modules and locallyprojective modules that are studied in the next Section.

Let k be a commutative ring and consider the bicategory Frm(k) introduced in Section 2.2.2.An adjoint pair in Frm(k) is a sextuple (R,A,M,M †, η, ε), consisting of two firm rings A and R,two firm bimodules M ∈ RMA and M † ∈ AMR and two bilinear maps η : R → M ⊗AM † andε : M † ⊗RM → A, rendering the following diagrams commutative.

(53) M∼= //

∼=

R⊗RM

η⊗RM

M ⊗A A M ⊗AM † ⊗RMM⊗Aεoo

M †∼= //

∼=

M † ⊗R R

M†⊗Rη

A⊗AM † M † ⊗RM ⊗AM †ε⊗AM

†oo

Following the terminology introduced in [32], we call this adjoint pair a comatrix coring context(in [32] A and R were rings with unit).


Applying Theorem 1.30, we can construct an R-ring Z† = M ⊗A M † with unit η and mul-tiplication M ⊗A ε ⊗A M † and an A-coring D = M † ⊗R M with counit ε and comultiplicationM † ⊗B η ⊗B M . Moreover M is a left Z†-module and a right D-comodule, M † is a right Z†-module and a left D-comodule and (M,M †, ε) is a dual pair. Actions and coactions are given bythe following formula

ρM,D = (η ⊗B M) dB,M , µZ†,M = µM,A (M ⊗A ε);ρD,M† = (M † ⊗B η) dM†,B, µM†,Z† = µA,M† (ε⊗AM †).

We will return in a more profound way to the construction of these comatrix corings in Section 3.4.

Theorem 2.51. Let A and R be firm rings, M ∈ AMR and M † ∈ RMA. The followingstatements are equivalent

(i) (M,M †, µ) is an R-firm dual pair;(ii) M is R-firmly projective as a right A-module and M † ∼= M∗ ⊗R R;(iii) M † is R-firmly projective as a left A-module and M ∼= R⊗R ∗(M †);(iv) there exists a comatrix coring context (R,A,M,M †, η, ε);(v) (−⊗RM,−⊗AM †) is a pair of adjoint functors

(54) MR

−⊗RM //MA,−⊗AM

†oo

(vi) there exists a natural isomorphism α : HomA(M,−)⊗R R ' −⊗AM †;(vii) (M † ⊗R −,M ⊗A −) is a pair of adjoint functors

(55) RMM†⊗R− //MA,M⊗A−

oo ;

(viii) there exists a natural isomorphism β : R⊗R AHom(M †,−) 'M ⊗A −;(ix) there exists a “triple” of adjoint funtors (F,G,H) between the categories MR and MA, i.e.

(F,G) is a pair of adjoint functors between MR and MA and (G,H) is a pair of adjointfunctors between MA and MR, such that M = F (R) and M † = G(A);

(x) there exists a “triple” of adjoint funtors (F †, G†,H†) between the categories RM and AM,such that M † = F †(R) and M = G†(A).

If any of the above equivalent conditions is satisfied, then M is flat as a right A-module if R isflat as a right R-module and M † is flat as a left A-module if R is flat as a left R-module.

Proof. (i) ⇔ (ii). Follows by Proposition 2.50 part (ii) and (iii).

(i) ⇔ (iii). In the same way, by left-right symmetry.

(i) ⇒ (iv). Suppose (M,M †, µ) is an R-firm dual pair, then there exists a ring morphism η : R→M ⊗AM †. One easily checks that (R,A,M,M †, ι, µ) is a comarix coring context.(iv) ⇒ (i). The definition of a comatrix coring context implies that M and M † are firm as R-

module. If we denote η(r) = er ⊗A fr for r ∈ R, then the commutativity of the left diagram of(53) means that r · u = erfr(u), which means exactly that the R-module structure of M can beseen as induced by η, as in the definition of an R-firm dual pair. The condition on the compatibilityof the R-module structure on M † follows in the same way from the left diagram of (53).(iv) ⇒ (v). Follows from Theorem 1.29 (i) ⇒ (ii), taking Ω = k. Let us give the unit and counitof the adjunction.

αN : N → N ⊗RM ⊗AM †; α(n) = nr ⊗R er ⊗A fr;βP : P ⊗AM † ⊗RM → P ; β(p⊗A ϕ⊗R u) = pε(ϕ⊗ u);

for any N ∈MR and M ∈MA.(v) ⇒ (iv). Denote by α and β the unit and counit of the adjunction. The unit evaluated in Rinduces a right R-linear map

αR : R→M ⊗AM †,


which is also left R-linear by naturality. Moreover, the counit evaluated in A provides us with aright A-linear map

βA : M † ⊗RM → A,

this map becomes left A-linear by naturality. We prove first that for any N ∈MR,

(56) αN ∼= N ⊗R αR,i.e. the following diagram commutes.

NαN //

∼=

N ⊗RM ⊗AM †

N ⊗R RN⊗RαR

44jjjjjjjjjjjjjjjj

Take any n ∈ N , and consider the following map in MR, fn : R→ N , fn(r) = nr. By naturalityof α, the following diagram commutes,

RαR //

fn

R⊗RM ⊗AM †

fn⊗RM⊗AM†

N

αN // N ⊗RM ⊗AM †

and (56) follows. Similarly we prove that for M ∈ MA, βM ∼= M ⊗A βA. The condition that(F,G) is an adjoint pair means that βFM F (αM ) = F (M) and G(βM ) αGM = G(M). If weevaluate these conditions in M and M †, we obtain the following commutative diagrams,

M ∼= R⊗RMαM

∼=αR⊗RM //

UUUUUUUUUUUUUUUUUU

UUUUUUUUUUUUUUUUUU M ⊗AM † ⊗RM

M⊗AβA

M ∼= M ⊗A A

M † ∼= A⊗AM †

VVVVVVVVVVVVVVVVV

VVVVVVVVVVVVVVVVVM † ⊗RM ⊗AM †

βM†∼=βA⊗AM

†oo

M † ∼= M † ⊗R R

M†⊗RαR

OO

This means exactly that (R,A,M,M †, αR, βA) is a comatrix coring context.(v) ⇔ (vi). Follows by Theorem 2.13 and the uniqueness of the adjoint functor.

(v) ⇔ (ix). This follows immediately from the Eilenberg-Watts Theorem (see Theorem 2.13).

(iv) ⇔ (vii) ⇔ (viii) ⇔ (x). Follow by left-right symmetry.The last statement can be proved as follows. The functor M ⊗A − : AM → RM has a rightadjoint, being R⊗R AHom(M,−), hence M ⊗A − is right exact. By condition (vii), this functorhas as well a left adjoint, and hence this functor is also left exact, thus exact. If R is flat as a rightR-module, then the functor R ⊗R − : RM → Ab is exact. Therefore, M ⊗A − : AM → Ab,which is the composition of M ⊗A− : AM→ RM and R⊗R− : RM→ Ab is exact as well, i.e.

M is flat as a right A-module. The statement about M † is proven in the same way.

We end this section with a special case of the previous situation. Let (M,M ′, µ) be a dualpair over the rings B and A and consider as well the canonical dual pair (M,M∗, ev) associatedto M . Then the following diagram commutes in a trivial way

M ′ ⊗B M

µ$$IIIIIIIIII

ϕµ⊗BM // M∗ ⊗B M

evzztttttttttt

A

where we donote ϕµ = ζ(µ), the morphism induced by (48). One could say that there exists amorphism of dual pairs (M,M ′, µ) → (M,M∗, ev). Let R be a firm B-ring and consider a ring


morphism ι′ : R→ Z ′ = M ⊗AM ′, then we obtain a ring morhism ι : R→ Z = M ⊗AM∗ suchthat the following diagram commutes

(57) Rι′

~~~~~~

~~~

ι

???

????

Z ′M⊗Aϕµ

// Z

Furthermore ι′ induces an R-module structure on M . If M is firm as R-module under this action,then clearly M is R-firmly projective. However, the converse is not true: the fact that M isR-firmly projective implies the existence of the map ι but there is no reason why it should factorizeover Z ′. This problem is solved in Proposition 2.53. Let us first prove the following lemma. Wekeep the notation that we have just introduced.

Lemma 2.52. Let ι′ : R→ Z ′ be a non-unital ring morphism. If M is firm as a left R-moduleunder the action induced by ι′ then ι is a left firm ring morphism.

Proof. Let P be any firm left Z ′-module. We have to show that P is a firm left R-moduleunder restricted action. This follows by the following isomorphisms

P ∼= Z ′ ⊗Z′ P = M ⊗AM ′ ⊗Z′ P ∼= R⊗RM ⊗AM ′ ⊗Z′ P ∼= R⊗R P

Proposition 2.53. Let (M,M ′, µ) be a dual pair over the rings B and A, and let R be anyfirm B-ring. Then the following statements are equivalent.

(i) There exists a non-unital ring morphism ι′ : R→ Z ′ such that M is firm as a left R-moduleunder the action induced by ι′;

(ii) M is R-firmly projective and there exists a ι′ : R→ Z ′ such that diagram (57) commutes;(iii) M is Z ′-firmly projective and there exists a right firm ring morphism ι′ : R→ Z ′;(iv) M ′ is a (not necessary firm) right R-module and (M,M ′ ⊗R R, µ) is an R-firm dual pair,

where µ = (M ′ ⊗R µR,M ) µ.

Proof. (i) ⇔ (ii). Follow immeadiately from the previous observations.

(i) ⇒ (iii). By Lemma 2.52.

(iii) ⇒ (i). We know that M is firm as a left Z ′-module. Since ι′ is a left firm morphism M is

also firm as a left R-module, so (i) holds as well.

(i) ⇒ (iv). Define ι = (ι′ ⊗R R) dR : R→M ⊗AM ′ ⊗R R. One can easily check that M and

M ′ ⊗R R become a firm left, respectively right R-module with the the R-action induced by ι.(iv) ⇒ (i). Now we may suppose that there exists a non-unital ring morphism ι : R→M⊗AM ′⊗RR that induces firm R-module structures on M and M ′ ⊗R R. Then define ι′ = M ⊗R µM ′,R :R→M ⊗RM ′.

2.3.2. Weakly locally projective modules. In order to be able to distinguish different notionsof local projectivity, we introduce “weak” and “strong” local projectivity. In the literature, bothnotions are termed “local projectivity”.

Definition 2.54. With notation as above, let R be an additive subgroup of M∗, and N asubset ofM . A dual (M,R)-basis ofN is a finite set (ui, fi) ⊂M×R such that n =

∑i uifi(n),

for every n ∈ N . The basis (ui, fi) is called (left) B-linear if∑

i uifi(bm) =∑

i buifi(m), forall m ∈M .

We call M weakly R-locally projective as a (B,A)-bimodule if every finite subset N of M hasa B-linear dual (M,R)-basis. In case B = Z, we say that M is weakly R-locally projective as aright A-module. If moreover R = M∗ then we just say that M is right weakly locally projectiveas a right A-module.

Remark 2.55. It is clear that projective modules are weakly locally projective and that weaklyR-locally projective modules are weakly locally projective.

Obviously, if M is finitely generated as an A-module, then weak local projectivity of M isequivalent to projectivity. However, it is not so hard to prove that the same property holds if M is


generated by a countable subset. A proof of this last property can be found in [11]. If A is rightperfect, then local projectivity is equivalent to projectivity for general A-modules.

Lemma 2.56. Let M be a (B,A)-bimodule. M is weakly R-locally projective as (B,A)-bimodule if and only if every finitely generated (B,A)-subbimodule of M has a B-linear dual(M,R)-basis.

Proof. This follows by the (B,A)-linearity of the dual bases, where the A-linearity is satisfiedsince R ⊂ HomA(M,A).

Take M ∈MA and R ⊂M∗ as above and consider P ∈ AM. We can define a map

αP,R : M ⊗A P → HomZ(R, P ), αM,R(m⊗A p)(r) = r(m) · pIf R is a left A-submodule of M∗, then Im (αP,R) ⊂ AHom(R, P ) and αP,R factors as thecomposition of a map

α′P,R : M ⊗A P → AHom(R, P )followed by the natural inclusion AHom(R, P ) ⊂ HomZ(R, P ).

We say that M satisfies the α-condition for R iff αN,R is injective for all N ∈ AM. If Msatisfies the α-condition for M∗ we just say that M satisfies the α-condition.

More specifically, the α-condition for R is equivalent to the following statement: For anyelement

∑imi ⊗A pi ∈M ⊗A P ,

∑i f(mi)pi = 0 for all f ∈ R implies that

∑imi ⊗A pi = 0.

Lemma 2.57. If S is an abelian subgroup of R, then the α-condition for S implies the α-condition for R.

Proof. This follows directly from the fact that kerαP,R ⊂ kerαP,S and ImαP,S ⊂ ImαP,R.

Example 2.58 (extension of scalars). Let B → A be a ring morphism, and Q ∈ MB. ThenM = Q⊗B A ∈MA, and we have a morphism of left B-modules

γ : Q∗ = HomB(Q,B) →M∗ = HomA(M,A), γ(f)(q ⊗B a) = f(q)a

If Q satisfies the α-condition for Q∗, then M also satisfies the α-condition for M∗ (since M⊗AP ∼=Q⊗B P for every left A-module P ).

The following Theorem is a generalization of some results of [68] and [119]; we recover theresults of [68] and [119] if we take R = M∗.

Theorem 2.59. Let M be a right A-module, R ⊂ M∗ an additive subgroup and S = ARthe left A-submodule of M∗ generated by R. Then the following statements are equivalent :

(i) M satisfies the α-condition for R;(i)’ M satisfies the α-condition for S;(ii) αN,R is injective for every cyclic left A-module N ;(ii)’ αN,S is injective for every cyclic left A-module N ;(iii) There exists a dual (M,R)-basis for every m ∈M ;(iii)’ There exists a dual (M,S)-basis for every m ∈M ;(iv) M is weakly R-locally projective as a right A-module(iv)’ M is weakly S-locally projective as a right A-module(v) M satisfies the α-condition and S is dense with respect to the finite topology on M∗.

Proof. (i) ⇒ (i)′, (i)′ ⇒ (ii)′, (i) ⇒ (ii) and (ii) ⇒ (ii)′ are trivial.

(ii)′ ⇒ (iii)′.

S(m) = f(m) | f ∈ S = ∑i

aifi(m) | fi ∈ R, ai ∈ A

is a left ideal of A, and A/S(m) is a cyclic left A-module. For all f ∈ S, f(m)1 = 0 in A/S(m),hence m⊗A 1 = m = 0 in M ⊗A A/S(m) ∼= M/MS(m), and m ∈MS(m).(iii)′ ⇒ (iii) We know that m =

∑i eisi(m), with si ∈ S and ei ∈ M . Since S is generated by

R, we can write si =∑

jiajirji and we find m =

∑i

∑jiei(ajirji)(m) =

∑i

∑ji(eiaji)rji(m)

and so m ∈MR(m).


(iii) ⇒ (iv) We prove by induction on k that every finite set n1, · · · , nk has a finite dual basis.

If k = 1, this follows immediately form (iii). Now suppose we have a dual basis (ei, ri) | i ∈ Ifor n1, . . . , nk−1, and consider then nk −

∑i eiri(nk) ∈ P and take a dual basis (e′j , r′j) | j ∈ J

for this element. An easy calculation shows that (e′j , r′j) | j ∈ J∪(ei−∑

j e′jr′j(ei), ri) | i ∈ I

is a dual basis for n1, · · · , nk.(iv) ⇒ (iv)′ is trivial.

(iv)′ ⇒ (iv) is similar to (iii)′ ⇒ (iii).(iv) ⇒ (i) LetN be a left A-module, take

∑imi⊗Ani ∈M⊗AN and suppose that

∑i f(mi)ni =

0 for all f ∈ R. From (iv) we know we kan find a dual basis ej , rj for the elements mi.∑i

mi ⊗A ni =∑i,j

ejrj(mi)⊗A ni =∑i,j

ej ⊗A rj(mi)ni = 0

(i) ⇒ (v). It follows from Lemma 2.57 that M satisfies the α-condition. For every f ∈ M∗ and

m1, . . . ,m` ∈ M , we have to find g ∈ S such that f(mi) = g(mi). From (iv)’, we know thatm1, · · · ,mn has a dual basis (ej , sj) | j ∈ J ⊂M × S. We find

f(mi) = f(∑j

ejsj(mi)) =∑j

f(ej)sj(mi) =∑j

(f(ej)sj)(mi)

and our statement follows since∑

j f(ej)sj ∈ S.

(v) ⇒ (iv)′ The α-condition for M , gives us a dual basis (ei, fi) | i ∈ I ⊂ M ×M∗ for anyfinite subset N ⊂ M . Since S is dense in M∗, we can find elements si ∈ S, such that fi and sihave the same action on N . Then (ei, si) | i ∈ I ⊂M ×S is the dual basis that we are lookingfor.

Remark 2.60. Note that the equivalent conditions of Theorem 2.59 do not imply that R isdense in the finite topology. If one takes P = A⊗B Q as in Example 2.58, then ∗Q is never densein the finite topology on ∗P if B → A is a proper ring extension.

Finally, let us remark that M is weakly M∗-locally projective as a right A-module if and onlyif M is locally projective in the sense of Zimmermann-Huisgen [119]. This is equivalent to thefollowing condition (see [68]): for any commutative diagram with exact rows in the category ofright A-modules of the form

0 // Fi // M

g

N ′

f// N // 0

with F finitely generated, there exists a right A-linear map h : M → N ′ such that g i = f h i.

2.3.3. Strongly locally projective modules.

Definition 2.61. Let M be a B-A bimodule and R ⊂M∗ an additive subset. Take any sub-set N ⊂M . We say that a dual (M,R)-basis (ui, fi) of N is idempotent if ui =

∑j ujfj(ui).

We call M strongly R-locally projective as a (B,A)-bimodule if every finite subset N of Mhas an idempotent B-linear (M,R)-dual basis.

If B = Z, we say that M is strongly R-locally projective as a right A-module. If moreoverR = M∗, we say that M is strongly locally projective a right A-module.

It is clear thatM is stronglyR-locally projective as a (B,A)-bimodule if every finitely generated(A,B)-subbimodule N of M has an idempotent B-linear (M,R)-dual basis.

It is clear that strongly locally projective modules are weakly locally projective. The converseimplication is not true in general, since projective modules are not necessarily strongly locallyprojective.

In [7], Anh and Marki define that a right A-module P is locally projective if there exists asplit direct system P s : Z → (MA)s, such that colimP = (P, σ) and Pi is finitely generated andprojective for all i ∈ Z.

Theorem 2.62. The following statements are equivalent:


(i) P is locally projective in the sense of Anh and Marki [7];(ii) every finitely generated submodule F of P is contained in a finitely generated projective

submodule PF of P , which is a direct summand of P ;(iii) P is strongly locally projective as a right A-module.

Proof. (i) ⇒ (ii). Let p1, . . . , pn be a set of generators of F . From the way we calculate

colimits in MA we know that there exists a k ∈ I such that p1, . . . , pn ⊂ Pk, hence F ⊂ Pk. Weknow Pk is finitely generated projective and by Proposition 2.1 we know that there exist morphismsνk : P → Pk and µk : Pk → P such that νk µk = Pk, therefore Pk is a direct summand of P asa right A-module.

(ii) ⇒ (iii). Let (uk, fk) be a dual basis of PF as a right A-module. Since PF is a directsummand of P , the maps fk : PF → A can be extended to fk : P → A.

(iii) ⇒ (i). Let pii∈I be a set of generators of P and let J be the set of all finite subsets

of I. For every J ∈ J , let (uJk , fJk ) be a finite dual basis of the module generated by pjj∈J .We define PJ as the finitely generated and projective submodule of P generated by the uJk andψJ : P → PJ , ψ(p) =

∑ukfk(p). Clearly these are projections and consequently the PJ are direct

summands of P . Furthermore, we define J ≤ J ′ iff (uJ ′k′ , fJ′

k′ ) is a dual basis for the elements

uJk . Since the dual bases are idempotent we have J ≤ J and moreover J is directed. Define Zas the category associated to (J ,≤). Then we can consider the functor PZ →MA, P (J) = PJand for J ≤ J ′ P (aij) : PJ → PJ ′ is the inclusion. Than one can easily verify that the morphismsψJ together with the canonical inclusions PJ → P satisfy the conditions of Proposition 2.1, suchthat P is split.

2.3.4. Local projectivity versus local units. In this section, M = (M,M ′, µ) will be a fixeddual pair and S the associated elementary B-ring constructed as in Lemma 2.49. Consider alsothe ismorphisms ζ and ξ as in (48).

A finite dual basis in M is a set (ui, fi) | 1 ≤ i ≤ n ⊂M ×M∗, such that m =∑

i uifi(m)for all m ∈ M . If we consider the map Φ from Lemma 2.49, then this dual basis can also beregarded as an element e =

∑i ui ⊗A fi ∈M ⊗AM∗, such that Φ(e) = M , the identity on M .

Lemma 2.63. If there exists∑

i u′i ⊗B ui ∈ (M ′ ⊗B M)A such that µ(

∑i u

′i ⊗B ui) = 1A,

then M is a firm left S-module, and M ′ is a firm right S-module.

Proof. For every m ∈ M , we have that m = m1A = mµ(u′i ⊗B ui) = Φ(m ⊗A u′i)(ui) =(m⊗A u′i) · ui, and this shows that the canonical map S ⊗S M →M is surjective. We still haveto show that this map is injective. To this end, it suffices to show that the sequence

S ⊗B S ⊗B Mλ−→ S ⊗B M

ψ−→M → 0,

with λ(s ⊗B t ⊗B m) = st ⊗B m − s ⊗B tm and ψ(t ⊗B m) = tm, is exact. It is clear thatψ λ = 0. Take

x =∑j

mj ⊗A m′j ⊗B nj ∈ Kerψ,

this means that∑

jmjµ(m′j ⊗B nj) = 0. Then

λ(∑i,j

mj ⊗A u′i ⊗B ui ⊗A m′j ⊗B nj

)=

∑j

mj ⊗A m′j ⊗B nj −

∑i,j

mj ⊗A u′i ⊗B uiµ(m′j ⊗B nj)

=∑j

mj ⊗A m′j ⊗B nj −

∑i,j

mj ⊗A µ(m′j ⊗B nj)u′i ⊗B ui

=∑j

mj ⊗A m′j ⊗B nj = x ∈ Im (λ).


The following Theorem specialises Theorem 2.51 to the case where both rings are unital. Itshows that for a ring with unit B and a right A-module M , B-firm projectivity means exactly thatM is finitely generated and projective as a right A-module

Theorem 2.64. Let A and B be rings with unit, and M = (M,M ′, µ) be a dual pair. Withnotation as above, the following statements are equivalent

(i) S is a B-ring with unit and M and M ′ are firm, respectively as a left and right S-module;(ii) M is finitely generated and projective as a right A-module and ζ(µ) = ϕ : M ′ → M∗ is

bijective;(iii) M ′ is finitely generated and projective as a left A-module and ξ(ζ(µ)) = ψ : M → ∗M ′ is

bijective;(iv) the map Φ of Lemma 2.49 is an isomorphism;(v) the map Ψ of Lemma 2.49 is an isomorphism;(vi) there exists a B-bimodule map η : B → M ⊗A M ′ such that (A,B,M ′,M, µ, η) is a

comatrix coring context;(vii) M is B-firmly projective.

Proof. (i).⇒ (ii). Let e =∑

i ui⊗u′i ∈M ⊗AM ′ = S be the unit element of S. It follows

from Lemma 2.31 that M and M ′ are unital, resp. as a left and right S-module. For all m ∈M ,we have that

m = em =∑i

uiµ(u′i ⊗B m) =∑i

uiϕ(u′i)(m),

so (ui, ϕ(u′i)) is a finite dual basis of M as a right A-module. Since ϕ is left A-linear, we havefor every f ∈M∗ that

f =∑i

f(ui)ϕ(u′i) = ϕ(∑i

f(ui)u′i) ∈ Im (ϕ),

so ϕ is surjective. If ϕ(m′) = 0, then µ(m′ ⊗ m) = 0, for all m ∈ M , hence m′ = m′u =µ(m′ ⊗ ui)u′i = 0, and it follows that ϕ is injective.(ii).⇒ (iv). and (iv).⇒ (i). are trivial.

The equivalence of (i), (iii) and (v) can be proven in a similar way.(i) ⇒ (vi) Take η : B →M ⊗AM ′, η(b) = be, with e the unit of S.

(vi) ⇒ (i) η(1) is a unit on S, and acts trivially on M and M ′.

(vi) ⇔ (vii) Follows from Theorem 2.51.

We will now discuss how properties of local units can be translated into properties of localprojectivity. A first result is the following.

Theorem 2.65. We keep the notation introduced above, and letR := Im ζ(µ). The followingstatements are equivalent:

(i) M is a weakly (resp. strongly) R-locally projective right A-module;(ii) S is a Z-ring with left local units (resp. left idempotent local units) and M is a firm left

S-module;(iii) S has local units (resp. idempotent local units) on M as Z-ring.

In this situation, ξ(ζ(µ)) = ψ and Ψ are injective.

Proof. (i) ⇒ (ii) We first show that S has a left local units on M . Take a finite subset

F = m1, · · · ,mn ⊂M ; then there exists a dual (M,R)-basis (ui, ϕµ(u′i)) of the submoduleof M generated by F . Let e =

∑i ui ⊗A u′i ∈ S; then we have, for all j ∈ 1, · · · , n, that

emj =∑

i uiϕµ(u′i)(mj) = mj .

A similar argument shows that S has a left local units: take G = s1, · · · , sm ⊂ S, and writesj =

∑kmjk ⊗A m′

jk. There exists a dual (M,R)-basis (ui, ϕµ(u′i)) of the submodule of M

generated by the mjk, and∑

i(ui ⊗A u′i)sj = sj , for all j.It then follows from Lemma 2.31 that M is firm as a left S-module.If M is strongly R-locally projective, then we can choose an idempotent dual (M,R)-basis


(ui, ϕµ(u′i)) in the above construction. Then

e2 = (∑i

ui ⊗ u′i)(∑j

uj ⊗ u′j) =∑i,j

uiµ(u′i ⊗B uj)⊗A u′j

=∑i,j

uiϕµ(u′i)(uj)⊗A u′j =∑j

uj ⊗A u′j = e,

and the local units are idempotent.(ii) ⇒ (iii) follows from Lemma 2.31.

(iii) ⇒ (i) Take m ∈M , and a local unit e = ui⊗Au′i ∈ S on the submodule of M generatedby m. Then

m = e ·m =∑i

uiµ(u′i ⊗B m) =∑i

uiϕµ(u′i)(m),

so (ui, ϕµ(u′i)) is a dual basis for m.Finally, if m ∈ Kerψ, then we have for all m ∈ M that ψ(m)(m′) = ϕ(m′)(m) = 0. Take

a dual (M,R)-basis (ui, ϕµ(u′i)) of m, then m = uiϕ(u′i)(m) = 0, and it follows that ψ isinjective. Similarly if m⊗Am′ ∈ Ker Ψ then Ψ(m⊗Am′)(n′) = ψ(m)(n′)m′ = ϕ(n′)(m)m′ = 0.Thus we find m⊗A m′ = uiϕ(u′i)(m)⊗A m′ = ui ⊗A ϕ(u′i)(m)m′ = 0 and Ψ is injective.

Corollary 2.66. A right A-module M is R-firmly projective for a ring with (idempotent)local units R if and only M is weakly (resp. strongly) locally projective as a right A-module.

Proof. Suppose that M is weakly locally projective as a right A-module. Then we know fromby Theorem 2.65, 1 ⇒ 2 that this implies that M is S-firmly projective as a right A-module, andS is a ring with local units. So we can take R equal to S or more general R equal to any subringof S containing a complete set of local units for M . Conversely, if M is R-firmly projective, thenthere exists a ringmorhism R → M ⊗A M∗ such that M is firm as R module under the actioninduced by this ring morphism. Since R is a ring with local units, R acts with local units on M .We see that the local units are of the form ι(r) ∈M⊗AM∗, so M is firm as M⊗AM∗-module aswell. We obtain again from Theorem 2.65, 2 ⇒ 1 that M is locally projective as a right A-modulesince we can consider the canonical dual pair (M,M∗, ev) and take S = M ⊗AM∗.

Theorem 2.67. We keep the notation from Theorem 2.65. If Φ is injective, then the followingstatements are equivalent:

(i) M is a weakly (resp. strongly) R-locally projective (B,A)-bimodule;(ii) S is a B-ring with left local units (resp. right idempotent local units) and M is a firm left

S-module;(iii) S has local units (resp. idempotent local units) on M as a B-ring;(iv) M is S-firmly projective and S is a B-ring with left local units.

Proof. (i) ⇒ (ii) We only have to show that the local units e =∑

i ui ⊗A u′i constructed

in the proof of 1.⇒ 2. in Theorem 2.65 can be taken in SB. We can take a B-linear dual(M,R)-basis (ui, ϕ(u′i) of F . We then have for all indices j and all b ∈ B:

Φ(eb)(mj) = Φ(e)(bmj) =∑i

uiϕ(u′i)(bmj) =∑i

buiϕ(u′i)(mj) = Φ(be)(mj),

so Φ(eb) = Φ(be), and e ∈ SB, since Φ is injective.(ii) ⇒ (iii) follows by Lemma 2.31.

(iii) ⇒ (i) The B-linearity of the dual basis is an immediate consequence of the fact that the localunits commute with the elements of B.(ii) ⇔ (iv) If S has a left local units, then S is a firm ring by Lemma 2.31. The equivalencefollows directly from the definition of S-firm projectivity.

Corollary 2.68. Put S = Im (ψ). The following statements are equivalent

(i) M is a weakly (resp. strongly) R-locally projective as a (B,A)-module and M ′ is weakly(resp. strongly) S-locally projective as an (A,B)-module;

(ii) S is a B-ring with (resp. idempotent) local units, M is a firm right S-module and M ′ is afirm left S-module;


(iii) S has a left (resp. idempotent) local units on M and right (resp. idempotent) local units onM ′ as a B-ring.

In the strong/idempotent case, previous assertions are also equivalent the following condition:

(iv) There exists split direct system P s : Z → BMAs, such that all Pi are finitely generated and

projective as a right A-module, colimP = (M,σ) and colimP ∗ = (M ′, τ∗).

Proof. (i) ⇒ (ii) It follows from Theorem 2.65 that Φ and Ψ are injective. Then the impli-

cation (i) ⇒ (ii) follows from Theorems 2.65 and 2.67.

(ii) ⇒ (iii) and (iii) ⇒ (i) follow immediately from Theorem 2.67 (and its analogous version,

with the roles of M and M ′ interchanged), taking into account the fact that the hypothesis thatΦ is injective is only needed in the proof of (i) ⇒ (ii) in Theorem 2.67.

(iv) ⇒ (i). Since ϕ is injective, we can view M ′ as a submodule of M∗, and therefore R = M ′.For a finitely generated submodule N of M , we can find i ∈ I such that N ⊂ Pi. The dual basisof Pi is contained in Pi ⊗A P ∗i ⊂ M ⊗A R, and is an idempotent dual basis on N ; in a similarway, we can find idempotent local bases on finitely generated submodules of M ′.(i) ⇒ (iv). Applying the technique of Theorem 2.62, we find split direct systems M s : Z1 →BMA

s with Mi finitely generated and projective as a right A-module and colimM = (M,σ)and M ′s : Z2 → AMB

s with M ′i finitely generated and projective as a left A-module and

colimM ′ = (M ′, τ ′). Moreover, it follows from the proof of Theorem 2.62 that Mi = eiM ,where ei ∈ S is an idempotent local dual basis for Mi. Analogously, M ′

j = M ′e′j with e′j ∈ S

local dual bases for M ′j . We will construct from these direct systems a new split direct system

that satisfies the desired properties.Let K ⊂ S be the set consisting of all the previously considered dual bases ei and e′j for

respectively Mi and M ′j . For two elements k1, k2 ∈ K we say that k1 ≤ k2 if and only if

k1k2 = k1 in S. Let Z be the category associated to (K,≤). Then similar as in the proof ofTheorem 2.62 ((iii) ⇒ (i)) we can define P s : Z → (BMA)s with Pk = k ·M for all k ∈ K such

that colimP = (P, σ). And in the same way, we find P ′s : Z → AMBs, with P ′k = M ′ · k for all

k ∈ K and colimP ′ = (M ′, τ ′).Let us proof that P ′ = P ∗. Taking the restriction and corestriction of the morphism ϕ : M ′ →

M∗, we find a morphism ϕk : P ′k → P ∗k , ϕk(m′)(m) = µ(m′ ⊗B m) for all m′ ∈ P ′k and m ∈M .Suppose µ(m′ ⊗B m) = 0 for all m ∈ Pk. We know by the construction of Mk that m′ = m′ · k.Also by the construction of Pk and P ′k, we can find elements ui ∈ Pk and fi ∈ P ′i such thatk = ui ⊗A fi. Furthermore, m′ = m′ · k = µ(m′ ⊗B ui)fi = 0. This implies ϕk is injective.Furthermore, ϕk is also surjective. Take f ∈ P ∗k . Then

f(m) = f(k ·m) = f(∑i

uiµ(fi ⊗m)) =∑i

f(ui)µ(fi ⊗m),

for all m ∈ Pk, where we denoted again k =∑

i ui ⊗ fi. We conclude that f = ϕk(f(ui)fi).

Let S be a ring with idempotent local units. Let E be the set of idempotent local units ofS. We can consider S as well as a ring with left idempotent local units and as a ring with rightidempotent local units. It follows from Theorem 2.39 that S can be described as the colimitof unital rings eSe, rings with left unit eS or rings with right unit Se. Let us denote this asS = lim−→Se = lim−→ eS = lim−→ eSe, where the limit is taken over the set of idempotents e of S.In the case where S is as in Corollary 2.68, we have in the same notation that M = lim−→Pi and

M ′ = lim−→P ∗i , where Pi are B-A bimodules that are finitely generated and projective as a rightA-module. Since the tensor product commutes with direct limits, we find

S = M ⊗AM ′ = lim−→Pi ⊗AM ′ = lim−→M ⊗ P ∗i .

Here Pi ⊗AM ′ is a ring with left unit ei, with ei the finite dual basis for Pi, and ei is a right unitof the ring M ⊗ P ∗i .Recall that Pi ⊆ Pj if i ≤ j. Take k ≥ i, j. Then Pi ⊗ P ∗j ⊆ Pk ⊗A P ∗k , and it follows that

S = M ⊗AM ′ = lim−→Pi ⊗A P ∗j = lim−→Pk ⊗A P ∗k ,

with Pk ⊗A P ∗k a ring with unit by Theorem 2.64.


We will now look at the situation where the direct limit is a direct sum. Recall that a ring Rhas enough idempotents if there exists a set eiI of pairwise orthogonal idempotents, such thatevery element in R admits a finite sum of these idempotents as a two-sided unit, or, equivalentlyR =

⊕i∈I eiR =

⊕i∈I Rei. We call eiI a complete family of idempotents for R.

Theorem 2.69. Let M = (M,M ′, µ) be a dual pair over the rings A and B, and S =M ⊗AM ′ the associated elementary B-ring. The following statements are equivalent

(i) S is a ring with enough idempotents and M and M ′ are firm, respectively as a left and rightS-module;

(ii) there exist a family of finitely generated and projective right A-modules Pi | i ∈ I suchthat

M =⊕i∈I

Pi and M ′ =⊕i∈I

P ∗i .

Proof. (i) ⇒ (ii). S has a complete family of idempotents ei | i ∈ I, so it has idempotent

local units, and, by Corollary 2.68, M is strongly ϕµ(M ′)-locally projective and ϕµ is injective.For every i ∈ I, eiM is now a finitely generated and projective direct summand of M (seeTheorem 2.62). We claim M =

⊕i∈I eiM . Indeed, if eim = ejm

′ with i 6= j, then eim = e2im =eiejm

′ = 0.In a similar way, we find that M ′ =

⊕i∈IM

′ei. As in the proof of Corollary 2.68, we showthat M ′ei ∼= (eiM)∗.

(ii) ⇒ (i). Clearly, M and M ′ are strongly locally projective, so S is a ring with idempotent

local units and M and M ′ are firm S-modules. We are done if we can construct a complete familyof idempotents. Let ei =

∑j u

ji ⊗ f ji ∈ Pi ⊗ P ∗i be the finite dual basis for Pi. f

ji (e

`k) = 0 for all

j, ` if i 6= k, so eiek = 0, and the ei form a complete set of idempotents. We then have

S =⊕i∈I

eiPi ⊗M ′ and S =⊕i∈I

M ⊗ P ∗i ei.

The following is the generalization of a well-known result about Morita contexts between ringswith a unit. We keep the same notation as we used before in this section.

Theorem 2.70. (i) If M = (M,M ′, µ) is a dual pair, then we can construct a Moritacontext C(M) = (A,S,M ′,M, µ, S), which is strict if and only if µ is surjective.

(ii) Let C = (A,S,M ′,M, f, g) be any Morita context between a ring A and a B-ring S, thenM(C) = (BM ′

A,AMB, f) is a dual pair, where

f : M ′ ⊗B M −→M ′ ⊗S Mf−→ A.

(iii) For any dual pair M , we have M(C(M)) = M .(iv) For any Morita context C we find a homomorphism of morita contexts ψ : C → C(M(C)).

Furthmore, ψ becomes an isomorphism if the map g from the original context is surjective.(v) Let C = (A,S,M ′,M, f, g) be a Morita context such that g is surjective. If S has (idempo-

tent) local units, then g is bijective and M ′ and M are weakly (strongly) locally projectiveA-modules, if M and M ′ are firm S-modules. This last condition will be satisfied if f is alsosurjective.

Proof. We leave it to the reader to verify the first four statements. The last statement isan immediate consequence of Corollary 2.68 and Lemma 2.63.

2.3.5. Representations of rings with local units. Let k be a commutative field and Aa k-algebra. It is well-known that the category of A-modules that are finite dimensional as ak-vector space is isomorphic to the category of finite dimensional representations of A. Herean n-dimensional representation is defined as a ring morphism ϕ : A → Matn(k). Applyingelementary results from the previous sections, we will generalize this observation first to arbitrarynoncommutative rings A and then to the infinite dimensional case.

2.4. LOCAL STRUCTURE MAPS 65

Let A and B be two non-commutative rings. By AMB,fgp we denote the full subcategory of

AMB whose objects A-B bimodules are finitely generated and projective as a right B-module.Similarly, AMB,wlp is the notation for the full subcategory of AMB whose objects are weak locallyprojective as a right B-module.

A finite representation of A over B is defined as a ring morphism ϕ : A → P ⊗B P ∗, whereP ∈MB,fgp.

More general, a locally finite representation of A over B is a collection of ring morphismsϕe : A → P ⊗B P ∗, indexed by a (complete) set of local units e ∈ P ⊗B P ∗, where P ∈ MB isweakly locally projective as a right B-module.

Proposition 2.71. (i) The category of finite representations of A over B is isomorphic to

AMB,fgp.(ii) The category of locally finite representations of A over B is isomorphic to AMB,wlp.

Proof. (i) Consider a finite representation of A over B. Then we find by definition a moduleP ∈ MB,fgp together with a ring morphism ϕ : A→ P ⊗B P ∗. We know from Lemma 2.49 thatP is a P ⊗B P ∗-B bimodule. By restriction of scalars, ϕ induces an A-B bimodule structure on P .Conversely, suppose that P ∈ AMB,fgp. Then we find by Theorem 2.64 that P ⊗B P ∗ is a unitalA-ring. Consequently, the required ring morphism ϕ is the unit map of this elementary A-ring.

(ii) Is proven in the same way, applying Theorem 2.65.

2.4. Local structure maps

2.4.1. Corings with local comultiplications. Let M = (M,M ′, µ) be a dual pair over therings A and B, and consider the A-bimodule C = M ′ ⊗B M . Write εC = µ : C → A. Inorder to make C into an A-coring, we need a comultiplication. Dualising the construction oflocal units for the B-ring S = M ⊗A M ′, we can start from an element e ∈ SB and define∆e(m′ ⊗B m) = m′ ⊗B e ⊗B m ∈ C ⊗A C. This map is clearly coassociative, but the counitproperty mi ⊗B e · m′

i = c(1)ε(c(2)) = c = ε(c(1))c(2) = mi · e ⊗B m′i holds only for elements

c = m′i ⊗B mi such that e is a local dual basis (local unit) for the mi and m′

i. This leads us toDefinition 2.72.

Definition 2.72. Let A be a ring with unit, C an A-bicomodule and εC : C → A an A-bimodule map. A right ε-comultiplication on D ⊂ C is a coassociative A-bimodule map ∆D : C →C⊗A C such that c = c(1)εC(c(2)) = (C⊗A εC) ∆D(c) for all c ∈ D. A right ε-comultiplication∆D is called idempotent if (C⊗A ε) ∆D is an idempotent in AEndA(C).

If there exists a right ε-comultiplication on C, we say that C is an A-coring with right comul-tiplication.

We call C a coring with weak (resp. strong) right local ε-comultiplications if for every finitelygenerated right A-submodule D of C, we can find a right ε-comultiplication (resp. an idempotentright ε-comultiplication) ∆D on D.

In a similar way, we define corings with weak and strong left local ε-comultiplications.We say that C is a coring with two-sided weak (resp. strong) local ε-comultiplications if for

every finitely generated right A-submodule D ⊂ C there exists a bilinear coassociative map ∆D

which is at the same time a right and a left ε-comultiplication (resp. a strong right and leftε-comultiplication).

A right A-module M is called a weak local right C-comodule if for every finitely generatedA-submodule N ⊂M , there exists a comultiplication ∆N : C → C⊗AC on C and a right A-linearmap δN : M → M ⊗A C such that n[0]ε(n[1]) = n for all n ∈ N and δN (m[0]) ⊗A m[1] =m[0] ⊗A ∆N (m[1]) for every m ∈ M . We call M a strong local right C-comodule if M is a weaklocal right C-comodule and, in addition, (C ⊗A ε) δN is an idempotent in EndA(M), for everyfinitely generated N ⊂M .

We say that two comultiplications ∆ and ∆′ on C coassociate if (C⊗A∆)∆′ = (∆′⊗AC)∆and (C⊗A ∆′) ∆ = (∆⊗A C) ∆′.

Observe that the fact that C is a local comodule over C does not imply that C is a coring withlocal comultiplication, since it is possible that δD 6= ∆D.


Theorem 2.73. If C is an A-coring with (left, right) weak (resp. strong) local comultiplica-tions then AEndA(C) has (left, right) (resp. idempotent) local units on C.

Proof. The local units are of the type (C⊗ε)∆D, where ∆D is a right local comultiplicationon a finitely generated submodule D ⊂ C.

Theorem 2.74. Let C be an A-bimodule and εC : C → A an A-bimodule map. Then thefollowing statements are equivalent.

(i) For every c ∈ C, there exists a right ε-comultiplication ∆c on c, such that ∆c and ∆c′

coassociate, for all c, c′ ∈ C;(ii) there exists a right ε-comultiplication on every A-subbimodule of C generated by a single

element such that two such comultiplications coassociate;(iii) C is a coring with right weak local comultiplications.

Proof. (i) ⇒ (ii) Take c ∈ C and ∆c as in part (i) We only have to prove that the counit

property holds for every element of the form acb, with a, b ∈ A. But ∆c(acb) = ac(1) ⊗A c(2)b, soac(1)ε(c(2)b) = ac(1)ε(c(2))b = acb.

(ii) ⇒ (iii) Let D be the A-subbimodule of C generated by the elements c1, . . . , ck. Weproceed by induction on the number of generators. Let ∆ be the ε-comultiplication on the A-subbimodule generated by c1, and ∆′ the ε-comultiplication on the k− 1 elements ci− ci(1)ε(c

i(2)),

i = 2, . . . , k. By assumption, these comultiplications can be chosen in such a way that theycoassociate. Now ∆′′ = ∆ + ∆′ −∆′ (C⊗A ε) ∆ is a comultiplication on C. It is obvious that∆′′ is a bimodule map; let us sketch the proof of the coassociativity. For bimodule maps ∆ and ε,we always have, without any counit property assumption, that

∆ (ε⊗A C) = (ε⊗A C⊗A C) (C⊗A ∆).

Using this property and the (mixed) coassociativity of ∆ and ∆′, we can show that ∆′′ is coas-sociative. We restrict ourselves to proving the following identity, leaving all other details to thereader.

(C⊗A C⊗A C⊗A ∆) (C⊗A C⊗A ∆′) (C⊗A ∆) ∆′

= (C⊗A C⊗A ∆′ ⊗A C) (C⊗A C⊗A ∆) (C⊗A ∆) ∆′

= (C⊗A C⊗A ∆′ ⊗A C) (C⊗A ∆⊗A C) (C⊗A ∆) ∆′

= (C⊗A ∆⊗A C⊗A C) (C⊗A ∆′ ⊗A C) (C⊗A ∆) ∆′

= (C⊗A ∆⊗A C⊗A C) (C⊗A ∆′ ⊗A C) (∆′ ⊗A C) ∆= (C⊗A ∆⊗A C⊗A C) (∆′ ⊗A C⊗A C) (∆′ ⊗A C) ∆= (∆′ ⊗A C⊗A C⊗A C) (∆⊗A C⊗A C) (∆′ ⊗A C) ∆.

We now check that the counit property holds for all elements in D. We know that (C⊗Aε)∆(c1) =c1, and since ∆′′(c1) = ∆(c1)+∆′(c1)−∆′(c1), this implies that ∆′′ is also an ε-comultiplicationon c1. Furthermore, for every i = 2, . . . , k we have

(I ⊗ ε) ∆′′(ci) = (I ⊗ ε) ∆(ci) + (I ⊗ ε) ∆′ (ci − ci(1)ε(ci(2)))

= ci(1)ε(ci(2)) + ci − ci(1)ε(c

i(2)) = ci,

and we conclude that c = c(1′′)ε(c(2′′)) for all c ∈ D.

(iii) ⇒ (i) We know now that there exists an ε-comultiplication for every c ∈ C. Giventwo elements, we know that there exists a common ε-comultiplication, so the coassociativity isautomatically satisfied.

Theorem 2.75. If ∆ is a left ε-comultiplication on c ∈ C and ∆′ is a right ε-comultiplicationon c′ ∈ C such that ∆ and ∆′ coassociate, then there exists a coassociative A-bimodule map ∆′′

which is left ε-comultiplication on c and a right ε-comultiplication on c′


Proof. Put ∆′′ = ∆ + ∆′ − (C⊗A ε⊗A C) (∆′ ⊗A C) ∆. The coassociativity of ∆′′ canbe proven along the same lines as in the previous theorem. Using the bilinearity of ε and ∆′ wefind the following identity

(ε⊗A C) (C⊗A ε⊗A C) (∆′ ⊗A C) ∆= (ε⊗A C) (ε⊗A C⊗A C) (∆′ ⊗A C) ∆= (ε⊗A C) ∆′ (ε⊗A C) ∆.

We now apply this to prove the left counit property on c.

(ε⊗A C) ∆′′(c) = (ε⊗A C) ∆(c) + (ε⊗A C) ∆′(c)−(ε⊗A C) (C⊗A ε⊗A C) (∆′ ⊗A C) ∆(c)

= c+ (ε⊗A C) ∆′(c)− (ε⊗A C) ∆′ (ε⊗A C) ∆(c)= c+ (ε⊗A C) ∆′(c)− (ε⊗A C) ∆′(c) = c

Using the coassociativity, one proves in a similar way that ∆′′ is a right ε-comulti-plication onc′.

Corollary 2.76. If C is a coring with left and right weak local comultiplications, then C hasalso two-sided weak local comultiplications.

Theorem 2.77. Let C be an A-bimodule and ε : C → A an A-bimodule map. The followingstatements are equivalent

(i) C is an A-coring with right strong local comultiplications;(ii) every finitely generated right A-submodule D ⊂ C is contained in an A-subbimodule E ⊂ C,

which is an A-coring with right comultiplication and a direct summand of C,(iii) there exists a split direct system (Ci)i∈I , where Ci is an A-coring with a right comultiplication,

such that C = lim−→Ci.

Proof. (i) ⇒ (ii) Suppose C has a right strong local comultiplications and let D be a finitely

generated right A-submodule of C. Then we know that ψ = (C ⊗A ε) ∆D is an idempotent in

AEndA(C) and thus a projection. Denote E := Imψ and define a new comultiplication

∆E = (C⊗A ε⊗A C) (∆D ⊗A ∆D) : C → C⊗A C

It is an easy computation to check that this map is coassociative. Furthermore, for e = c(1)ε(c(2)) ∈E , with c ∈ C, we find ∆E(e) = c(1)ε(c(2))⊗A c(3)ε(c(4)) ∈ E ⊗A E . Moreover,

(C⊗A ε) ∆E(e) = (C⊗A ε) (C⊗A ε⊗A C) (∆D ⊗A ∆D) (C⊗A ε) ∆D(c)= (C⊗A ε) ∆D (C⊗A ε) ∆D (C⊗A ε) ∆D(c)= (C⊗A ε) ∆D(c) = e

We can conclude that E is an A-coring with right comultiplication. Finally D ⊂ E , since c = ψ(c)for all c ∈ D.

(ii) ⇒ (iii) Denote by I the set consisting of all A-corings with right comultiplication that

are direct summands of C. This set is partially ordered: for E , E ′ ∈ I we define E ≤ E ′ if ∆E ′ is aright ε-comultiplication on E . In this situation, the projection ψE = (C⊗A ε) ∆E factors troughψE ′ , and the rest follows easily.

(iii) ⇒ (i) For every finitely generated right A-submodule D ⊂ C, we can find a Ci containing

D. Since Ci is a direct summand of C, we can extend the comultiplication on Ci to the whole ofC by making it zero on the complement. This is a right ε-comultiplication on D and this finishesthe proof.

Theorem 2.78. Let C be an A-bimodule and ε : C → A an A-bimodule map. The followingstatements are equivalent

(i) C is an A-coring with two-sided strong local comultiplications;(ii) every finitely generated A-bimodule D ⊂ C is contained in an A-subbimodule E ⊂ C, such

that E is an A-coring and a direct summand of C;(iii) there exists a split direct system (Ci)i∈I of A-corings, such that C = lim−→Ci.


Proof. (i) ⇒ (ii) Let ∆ be the left and right ε-comultiplication of a finitely generated A-

subbimodule D ⊂ C. Since C has two-sided strong local comultiplications, α = (C⊗A ε) ∆ andβ = (ε ⊗A C) ∆ are idempotents. α and β commute, since (α β)(c) = ε(c(1))c(2)ε(c(3)) =(β α)(c), for all c ∈ C. Therefore γ = α β = β α is also an idempotent, and a projectionC → C. Let E be the image of γ, then the restrictions of α and β to E are the identity, sinceα γ = γ and β γ = γ. We obtain that E is an A-coring with comultiplication ∆. The proof ofthe other implications is similar to the one of the corresponding implications in Theorem 2.77.

Remark 2.79. We know from the previous section that there is a strong relationship be-tween local units and local projectivity. Let us study how local projectivity is related to localcomultiplications.

Consider a dual pair (BMA,AM′B, µ), and assume that the associated elementary algebra

S = M ⊗A M ′ is a B-ring with local units on M and M ′ (which is equivalent to M being M ′-locally projective and M ′ being M -locally projective, in such a way that the dual bases can bechosen in SB). Then C = M ′ ⊗BM is an A-coring with weak local comultiplications and M andM ′ are local C-comodules. If the local units can be chosen to be idempotent, then C is a coringwith strong local comultiplications. Observe that we do not have the converse implication: thefact that M and M ′ are local comodules, does not necessarily imply that M and M ′ have localdual bases.

As a special case of this construction, we recover the construction of a comatrix coring (see[63]): let M ∈ BMA be finitely generated and projective as a right A-module, and consider thedual pair (M,M∗, µ) where µ(f ⊗B m) = f(m) for all m ∈ M and f ∈ M∗. We can find anidempotent dual basis e =

∑i ui ⊗A fi ∈ S. Since this is a dual basis for every element in M ,

it is a dual basis for every B-submodule of M . It follows from Corollary 2.68 that e ∈ SB. Thismeans that we can construct a local ε-comultiplication on every finitely generated submodule ofC, and since M itself is finitely generated, we have a usual comultiplication and C is an A-coring.The comultiplication and counit are explicitly defined by

(58) ∆C(f ⊗B m) =∑i

f ⊗B ui ⊗A fi ⊗B m, εC(f ⊗B m) = f(m),

for all f ⊗B m ∈ C. This proves the following Theorem.

Theorem 2.80. The conditions in Theorem 2.64 are equivalent to

(7) There exists e ∈ (M ⊗A M ′)B such that (58) defines an A-coring structure on C =M ′ ⊗B M , and M and M ′ are respectively a right and left C-comodule, with coactions

ρr(m) = e⊗B m and ρl(m′) = m′ ⊗B e.

Recall the notion of a ‘multiple coequalizer’. Let A be any category and A and B be twoobjects in A. If (ai)i∈I are a family of morphisms in HomA(A,B) then the multiple coequalizerof (ai)i∈I is defined as (E, e), where E is an object in A and e : B → E is a morphism in A suchthat e ai = e aj for all i, j ∈ I and such that for all (E′, e′) with the same property, there existsa unique morphism f : E → E′ such that e′ = f e.

Theorem 2.81. Let C be a coring with right local comultiplications. Consider the multiplecoequalizer (C, δ) of the family consisting of the zero-morphism C → C and morphisms (C⊗A ε−ε ⊗A C) ∆c for all choices of ∆c in a complete set of right comultiplications for all elementsc ∈ C. Then C is an A-coring.

Proof. Observe that for any choice of E, we obtain a trivial identity

ε (C⊗A ε) ∆E = ε (ε⊗A C) ∆E.

Consequently by the universal property of the multiple coequalizer, we obtain a map εC : C → Asuch that εC = εC δ.

We define the comultiplication on C. We first define a map ∆ : C → C⊗A C as follows. For anyc ∈ C, we take any right comultiplication ∆c for c and define ∆(c) = (δ⊗Aδ)∆c(c) := c(1)⊗Ac(2).Let us prove that this is indepentent of the choice of the right comultiplication ∆c for c. Let ∆′

c

be any other right comultiplication for c, and denote (δ ⊗A δ) ∆′c(c) := c(1′) ⊗A c(2′). We have

to show that c(1) ⊗A c(2) = c(1′) ⊗A c(2′). To this end, consider a set of representables for the


elements c(1) and c(1′) and let ∆′′ be a right comultiplication for these elements. Then we cancompute

c(1) ⊗A c(2) = c(1)(1′′)ε(c(1)(2′′))⊗A c(2)

= c(1)(1′′) ⊗A ε(c(1)(2′′))c(2)

= c(1)(1′′) ⊗A c(1)(2′′)ε(c(2))

= (c(1)ε(c(2)))(1′′) ⊗A (c(1)ε(c(2)))(2′′)= c(1′′) ⊗A c(2′′)

Here we used in the first equality that ∆′′ is a right comultiplication on the elements c(1), in thethird equality the defining property of the multiple coequalizer, the right A-linearity of ∆′′ in thefourth equality and the fact that ∆c is a right comultiplication for c in the last equality. In thesame way we find c(1′)⊗A c(2′) = c(1′′)⊗A c(2′′), and we conclude that c(1)⊗A c(2) = c(1′)⊗A c(2′).

Let us now check that for all c ∈ C,

(59) ∆ (C⊗A ε) ∆c = ∆ (ε⊗A C) ∆c.

Indeed,

c(1) ⊗A c(2)ε(c(3)) = c(1) ⊗A ε(c(2))c(3)= c(1)ε(c(2))⊗A c(3)= ε(c(1))c(2) ⊗A c(3)

Therefore by the universal property of the multiple coequalizer, the maps (59) induce a map∆C : C → C⊗A C.

We leave it to the reader to verify that ∆ is coassociative and that the counit condition issatisfied.

Theorem 2.81 was the key to construct more general infinite comatrix corings (see Section 3.4and [73]).

2.4.2. Corings with local counits.

Definition 2.82. Let A be a ring with unit, ∆C a coassociative comultiplication on an A-bimodule C, and M a right A-module with a coassociative right A-coaction δM : M →M ⊗A C.A right counit on M is an A-bimodule map εM : C → A such that

(M ⊗A εM ) δM = M.

εM is called idempotent if (εM ⊗A εM ) ∆C = εM .We say that M has (idempotent) right local counits if there exists an (idempotent) right counitεN on every finitely generated right C-subcomodule N ⊂M .If there exist right (idempotent) local counits on C, then we call C a coring with right (idempotent)local counits.Left and two-sided (idempotent) local counits can be introduced in a similar way.

The terminology “idempotent” counit is justified by the following Lemma.

Lemma 2.83. Let εM be a right counit on a right C-comoduleM . The following are equivalent.

(i) εM is an idempotent counit;(ii) εM : C → A is comultiplicative (the comultiplication of A is the canonical isomorphism

A ∼= A⊗A A);(iii) εM is an idempotent in ∗C.

Proof. The equivalence of (i) and (ii) is obvious. In ∗C, we easily compute that (εM ∗εM )(c) = εM (c(1)εM (c(2))) = εM (c(1))εM (c(2)), and we deduce immediately the equivalence of(i) and (iii).

Theorem 2.84. There exists (idempotent) right local counits on a right C-comodule M ifand only if the A-ring ∗C has (idempotent) right local units on M . In particular, C is an A-coringwith (idempotent) right local counits if and only if the A-ring ∗C has (idempotent) right local unitson C.


Proof. The right counits on M are precisely the units in ∗C on M .

The results of Section 2.2.3 can be restated in terms of local counits. In particular, theexistence of a counit on every c ∈ C implies the existence of local counits on C, and if C has (left,right or two-sided) idempotent local counits, then we can write C = lim−→Ci, with (Ci)i∈I is a splitdirect system of corings with a (left, right or two-sided) counit.

We call M ∈MC cofirm if the right coaction δM on M induces an isomorphism M ∼= M ⊗C Cin MC.

Theorem 2.85. If C is an A-coring with right local counits, a right C-comodule M is cofirmif and only if C has a right local counits on M .

Proof. First assume that C has a right local counits on M . Suppose that δM (m) = 0. Takea local counit εm on m; then m = (M ⊗ εm)(δM (m)) = 0, and it follows that δM is injective.Take

∑imi⊗A ci ∈M ⊗C C, and a right local counit ε on the right A-submodule of C generated

by c1, · · · , cn. Then we compute that

δM (∑i

miε(ci)) =∑i

mi[0] ⊗A mi[1]ε(ci) =∑i

mi ⊗A ci(1)ε(ci(2)) =∑i

mi ⊗A ci,

so it follows that δM : M →M ⊗C C is surjective.Conversely, if C is a coring with right local counits, then C has a right local counits on M ⊗C C.If M is cofirm, then M ⊗C C ∼= M , hence C also has a right local counits on M .

As rings with local units have known applications in various fields, corings with one-sidedlocal counits seem to be useful in the study of C∗ algebras. In [97] (see also [96]), O’uchi Motointroduced the notion of an approximate counit for a coring. After correct translation, this notioncorresponds to a coring with local counits.

2.4.3. Rings with local multiplication. Let B be a ring with unit, R a B-bimodule, andη : B → R a B-bimodule map. A right η-multiplication on T ⊂ R is an associative B-bimodulemap µT : R ⊗B R → R such that the triangle in the following diagram commutes on the imageof i:

0 // Ti // R

∼= // R⊗B B

IR⊗Bη

R⊗B R

µT

ccGGGGGGGGG

If R has a right η-multiplication on itself, then we call R an A-ring with right multiplication.We call R a ring with weak right local η-multiplications if there exists a right η-multiplication

on every finitely generated B-subbimodule of R.Left and two-sided η-multiplications are defined in a similar way.We say that R has strong right local η-multiplications if for every finitely generated B-

subbimodule T ⊂ R, there is a B-subbimodule S ⊂ R containing T on which there exists aright η-multiplication µS , such that µ(R⊗B R) ⊆ S.

Let M be a right B-module. We say that R has weak right local multiplications on M if forevery finitely generated B-submodule N ⊂M , there exist a right B-linear map ν : M⊗BR→Mand a B-bimodule map µ : R ⊗B R → R satisfying the usual associativity conditions, and suchthat ν(n⊗B η(1B)) = n for all n ∈ N .

R has strong right local multiplications on M if for every finitely generated B-submoduleN ⊂M , we can find a B-submodule N ′ ⊂M , containing N , together with a right B-linear mapν : M ⊗B R → M and a B-bimodule map µ : R ⊗B R → R satisfying the usual associativityconditions, and such that ν(n⊗B η(1B)) = n for all n ∈ N ′, and, in addition, ν(M ⊗B R) ⊂ N .

We say that two multiplications µ and µ′ on R associate if µ (R⊗B µ′) = µ′ (µ⊗B R) andµ′ (R⊗B µ) = µ (µ′ ⊗B R).

Remark that η is completely determined by η(1) = e ∈ RB, and an associative map µT :R⊗B R→ R is a right η-multiplication on T ⊂ R if and only if µ(r ⊗B e) = r for all r ∈ T .For b ∈ B and r ∈ R, we will use the notation µ(r ⊗ η(b)) = µ(r ⊗ b).


Theorem 2.86. Let R be a B-bimodule and η : B → R a B-bimodule map. The followingstatements are equivalent.

(i) For every r ∈ R, there exists a right η-multiplication µr on r, such that µr and µr′associate, for all r, r′ ∈ R;

(ii) there exists a right η-multiplication on every B-subbimodule of R generated by a singleelement such that the η-multiplications on two such subbimodules associate;

(iii) R is a ring with right weak local η-multiplications.

Proof. (i) ⇒ (ii) Let r be the generator of a B-subbimodule M of R. We know that there

exists a right η-multipication µr on r. If we denote µr(s, t) = s · t, for all s, t ∈ R, we just verifythat arb ·e = ar · (be) = ar · (eb) = a(r ·e)b = arb, for all a, b ∈ B, so µ is a right η-multiplicationon M .

(ii) ⇒ (iii) Let T be a finitely generated B-subbimodule of R and t1, . . . , tk a set of genera-tors for T . We proceed by induction on k. Let µ be a multiplication on the B-bimodule generatedby t2, . . . , tn and denote µ(r, s) = r · s for all r, s ∈ R. Let µ′ be a multiplication on theB-bimodule generated by t1− t1 ·e and choose µ and µ′ in such a way that they associate. Denoteµ′(r, s) = r ∗ s. Now define a new multiplication on R by r s = r · s+ r ∗ s− r · 1 ∗ s. We leaveit to the reader to verify that this is an associative right η-multiplication on T .

(iii) ⇒ (i) is trivial.

Theorem 2.87. If there exists a right η-multiplication on t ∈ R and a left η-multiplicationon s ∈ R, such that µ and µ′ associate, then there exists a multiplication µ′′ on R which is a rightη-multiplication on t and a left η-multiplication on s

Proof. Write µ(r, r′) = r · r′ and µ′(r, r′) = r ∗ r′, and define µ′′(r, r′) = r · r′ + r ∗ r′ − r ·1 ∗ r′.

Corollary 2.88. R is a ring with left and right local weak η-multiplications if and only if Ris a ring with two-sided weak local η-multiplications.

Theorem 2.89. Let R be a B-bimodule and η : B → R a B-bimodule map. The followingstatements are equivalent.

(i) R is a ring with strong right local multiplications;(ii) every finitely generated B-subbimodule of R is contained in a direct summand S of R, which

is a B-ring with right multiplication;(iii) R can be written as the direct limit of a split direct system (Si)i∈I of B-rings with right

multiplication.

Proof. (i) ⇒ (ii). As in the definition, consider for a B-subbimodule T of R a subbimodule

S of R containing T , on which there exists a right η-multiplication with µ(R ⊗B R) ⊂ S. Thenµ(S ⊗B S) ⊂ S. We also find that e = µ(η(1) ⊗B η(1)) ∈ S. Now we define a new B-bilinearmap η′ : B → S by η′(b) = eb = be. It is easy to see that S is a B-ring with right multiplicationand unit e. The map ψS : R → S, ψS(r) = µ(r ⊗ e) is a projection onto S, and S is a directsummand of R.

(ii) ⇒ (iii). Take the split direct system of all B-rings with right multiplication S as con-

structed in the previous part and define S ≤ S′ if µS′ is a right η-multiplication on S.(iii) ⇒ (i). Similar to the proof of Lemma 2.36 and Theorem 2.77.

Let us now describe the connection between local comultiplications and local multiplications.If C is an A-bimodule, and ∆ : C → C⊗A C is an associative multiplication, then ∗∆ given by

(f ∗∆ g)(c) = g(c(1)f(c(2)))

is an associative multiplication on ∗C := AHom(C, A).

Lemma 2.90. If ∆ and ∆′ are two coassociating comultiplications on an A-bimodule C, thenthe corresponding multiplications ∗∆ and ∗∆′ associate.

Proof. If we write ∆(c) = c(1) ⊗ c(2) and ∆′(c) = c(1′) ⊗ c(2′), then the fact that ∆ and ∆′

coassociate means that

c(1) ⊗ c(2)(1′) ⊗ c(2)(2′) = c(1′)(1) ⊗ c(1′)(2) ⊗ c(2′).


An easy computation shows that

(f ∗∆ g) ∗∆′ h(c) = h(c(1)g(c(2)(1′)f(c(2)(2′))));

f ∗∆ (g ∗∆′ h)(c) = h(c(1′)(1)g(c(1′)(2)f(c(2′)))),

and the result follows.

Theorem 2.91. If C is an A-coring with right (resp. left) weak (resp. strong) local comulti-plications, then ∗C has a right (resp. left) weak (resp. strong) local multiplications on C.

Proof. Let D be a finitely generated A-submodule of C and let ∆ be a weak right localε-comultiplication on D. For c ∈ C and f ∈ ∗C we define c ·∆ f = (C ⊗A f) ∆(c). An easycomputation shows that this is a weak local ε-multiplication on C.

Now let ∆ be strong. If we construct E as in Theorem 2.77, then the projection on E is givenexactly by right multiplication with ε.

References

This Chapter contains several new results, completed with results from the author’s paper [113](Sections 2.1, 2.2.3, 2.2.4, 2.3.2, 2.3.3, 2.3.4 and 2.4), the author’s paper [112] (Section 2.3.1)the joint paper with S. Caenepeel and E. De Groot [41] (Sections 2.1, 2.2.4 and 2.3.3), the jointpaper with S. Caenepeel and S. Wang [48] (Theorem 2.59 and Section 2.2.5) the joint paper withM. Iovanov [77] (Theorem 2.81), the joint paper with J. Gomez-Torrecillas [73] (Section 2.2.2).

Chapter 3Corings and Comodules

In this chapter we study the main characters of this book : corings and comodules. Coringscan be considered as ‘representations’ (or ‘corepresentations’) of corings.

After recalling elementary definitions and properties, we study in Section 3.1 some adjointfunctors and construct the Dorroh extension for a coring without counit.

Section 3.2.1 is devoted to the introduction of entwining structures and their relation to corings.It was this construction of corings out of entwining structures that triggered the coring revolutionat the end of the last century. It connects coring theory with the theory of Doi-Koppinen structuresand in this way with a wide variety of algebraic structures studied in the last dacades, striking theimportance of corings. A special type of entwining structures that have a close connection toDoi-Koppinen structures and factorization structures is considered in Section 3.2.2.

In Section 3.3 we have a closer look on the relation between a coring and its dual ring. Thisleads to the study of rational modules, which will be important in later chapters.

We will study in Section 3.4 in more detail the construction of a comatrix coring out of acomatrix coring context, already mentioned in Chapters 1 and 2. This construction makes itpossible to study the internal structure of a coring and its category of comodules and will be ofcentral importance for the Galois theory of corings studied in Part II.

Finally, we provide in the last section tools to construct corings from colimits. It is shown thatthese corings arise as special cases of the corings constructed in Section 3.4.

3.1. Basic properties of corings and comodules

3.1.1. Definitions. Let k be a commutative ring and A any k-algebra. Recall from Exam-ple 1.24 that an A-coring is an A-bimodule C furnished with a coassociative comultiplication∆ = ∆C : C → C ⊗A C, ∆(c) = c(1) ⊗A c(2) and a counit ε = εC : C → A, satisfying

ε(c(1))c(2) = c = c(1)ε(c(2)). A right C-comodule is a pair (M,ρM ), where M is a right A-

module and ρM : M →M ⊗A C, ρM (m) = m[0] ⊗Am[1] is a right A-linear map that satisfies thecoassociativity and right counit condition.

Lemma 3.1. Let C be an A-bimodule possessing a coassocicative comultiplication ∆ (i.e. C isa non-counital coring). If ε : C → A is a right A-linear map such that ε(c(1))c(2) = c for all c ∈ C

(i.e. ε is a left counit for C) and ε′ is a left A-linear map such that c(1)ε′(c(2)) = c for all c ∈ C

(i.e. ε is a right counit for C), then ε(c) = ε′(c) for all c ∈ C and thus C has a unique two-sidedcomultiplication.

73

74 CHAPTER 3. CORINGS AND COMODULES

Proof. This follows from the following computation

ε(c) = ε(c(1)ε′(c(2))) = ε(c(1))ε

′(c(2))

= ε′(ε(c(1))c(2)) = ε′(c)

for arbitrary c ∈ C.

Corollary 3.2. The counit of a coring is unique.

The most fundamental non-trivial (i.e different from an algebra or a coalgebra) example of acoring is the following.

Example 3.3. Let ι : B → A be a ring extension. Then we can associate to ι the followingA-coring D that is known as the canonical Sweedler coring . Put D = A⊗B A and define

∆D : A⊗B A→ (A⊗B A)⊗A (A⊗B A) ∼= A⊗B A⊗B A and εD : A⊗B A→ A

given by

∆D(a⊗B b) = (a⊗B 1)⊗A (1⊗B b) = a⊗B 1⊗B b, εD(a⊗B b) = ab

for all a, b ∈ A. It is easily verified that these maps define a coring structure on D.

An element g ∈ C is called grouplike if ∆C(g) = g ⊗A g and εC(g) = 1. We denote by G(C)the set of all grouplike elements in C.

The following lemma can be found in e.g. [46, Sec 4.8] or [29].

Lemma 3.4. Let C be an A-coring.

G(C) ∼= ρA : A→ A⊗A C ∼= C | ρA makes A into a right C-comodule∼= λA : A→ C⊗A A ∼= C | λA makes A into a left C-comodule.

Proof. For a fixed grouplike element g ∈ C, the associated coactions on A are given by

ρA(a) = ga ; λA(a) = ag.

Conversely, one can check that ρA(1) and λA(1) are grouplike elements for any right C-coactionρA on A and left C-coaction λA on A.

Consider the canonical Sweedler coring D = A⊗B A associated to the ring extension B → A.Then 1⊗B 1 is a grouplike element in D.

Theorem 3.5. There is a bijective correspondence between A-corings (C,∆, ε) and cotriples(F, δ, ε) on MA, such that F is right exact and preserves direct sums (i.e. F preserves colimits).This corresprondence is given by F ∼= −⊗A C and C ∼= F (A).

Proof. By the Eilenberg-Watts theorem (see Theorem 2.13), we know that a functor F :MA →MA is right exact and preserves direct sums if and only if F is of the form −⊗AX, whereX = F (A) is an A-bimodule. The statement follows then directly from Theorem 1.26.

Let C be an A-coring. In general, the category of (right) C-comodules has colimits and thesecan be computed in Ab. Furtheremore,MC is a Grotendieck category if C is flat as a left A-module.

In that case, limits exist as well in MC, and they can be computed in Ab (see [66]).

Let C be an A-coring that is flat as a left A-module. A comodule M ∈ MC is called rightC-coflat if it is flat as a right A-module, and if M ⊗C− : CM→ Ab is exact. A similar definitionapplies to left C-comodules.

Let R be a firm ring, take M ∈ MC and N ∈ CMR. We can consider the cotensor productM ⊗C N , which is defined as the equalizer

0 // M ⊗C N // M ⊗A NρM⊗AN //

M⊗AλN// M ⊗A C⊗A N .

This equalizer is computed in Ab. Recall from Section 2.2.2 that MR is a Grothendieck category.However, the forgetful functor MR → Ab is only exact if R is flat as a left R-module, hence there

is no guarantee that M ⊗C N is firm as a right R-module, so in general M ⊗C N ∈ MR.The proof of the following lemma can be found in [36, 21.4, 21.5], in case where R is a unital

ring. The proof can be easily generalized if R is firm.

3.1. BASIC PROPERTIES OF CORINGS AND COMODULES 75

Lemma 3.6. Let R be a firm ring and C an A-coring. Take M ∈ MC, N ∈ CMR andP ∈ RM, the natural map

f : (M ⊗C N)⊗R P →M ⊗C (N ⊗R P )

is an isomorphism in each of the following situations:

(i) P is flat as a left R-module;(ii) R is flat as a left R-module and M is coflat as a right C-comodule.

Proof. (i). As P is flat as a left R-module, the functor − ⊗R P : MR → Ab is exact andwe obtain a commutative diagram with exact rows

0 // (M ⊗C N)⊗R P //

f

M ⊗A N ⊗R P //

∼=

// M ⊗A C⊗A N ⊗R P∼=

0 // M ⊗C (N ⊗R P ) // M ⊗A N ⊗R P // // M ⊗A C⊗A N ⊗R P

The result follows from the universal property of the equalizer.(ii). Since R is a generator in RM, we can construct an exact sequence

F1g // F2

k // P // 0 ,

where F1 and F2 are free as a left R-module. Since R is flat as a left R-module, F1 and F2 are alsoflat as left R-modules. Then P is exactly the cokernel of g. Since for any X ∈ MR the functorX⊗R− is right exact, and M ∈MC is coflat, we obtain the following commutative diagram withexact rows

(M ⊗C N)⊗R F1

∼=

// (M ⊗C N)⊗R F2

∼=

// (M ⊗C N)⊗R P

f

// 0

M ⊗C (N ⊗R F1) // M ⊗C (N ⊗R F2) // M ⊗C (N ⊗R P ) // 0

The first two vertical arrows are isomorphisms by part (i). Consequently f is an isomorphism aswell by the universal property of the coequalizer.

3.1.2. Adjunctions.

Proposition 3.7. (i) Let C be an A-coring and Σ any right C-comodule, and let T =EndC(Σ). Then (−⊗T Σ,HomC(Σ,−)) is a pair of adjoint functors

MT

−⊗T Σ //MC

HomC(Σ,−)

oo

(ii) Let B → S be a ring morphism between two (unital) rings. Then (−⊗B A,UA) is a pair ofadjoint functors

MB

⊗BA //MAUA

oo

we say that UA is the restriction of scalars (or the forgetful) functor and − ⊗B A is theextension of scalars functor.

(iii) Let C be an A-coring, B a ring and Σ any B-C bicomodule. Then (− ⊗B Σ,HomC(Σ,−))is a pair of adjoint functors

MB

−⊗BΣ //MC

HomC(Σ,−)

oo

Proof. (i). We will only give the unit and counit of the first adjunction, leaving all otherverifications to the reader. For N ∈MT :

νN : N → HomC(Σ, N ⊗T Σ), νN (n)(u) = n⊗T u,


and for M ∈MC:

ζM : HomC(Σ,M)⊗T Σ →M, ζM (ϕ⊗T u) = ϕ(u).

(ii). This follows from part (i) by taking T = B and C = A the trivial A-coring. Then

HomA(A,N) ∼= N for all N ∈ MA, so UA ∼= HomA(A,−). Nevertheless let us give explicitformulas for the unit and counit.

ν ′N : N → N ⊗B A, ν ′N (n) = n⊗B 1,ζ ′M : M ⊗B A→M, ζ ′M (m⊗B a) = ma,

for all N ∈MA and M ∈MB.(iii). Since Σ ∈ BMC, we find a ring morphism ι : B → T = EndC(Σ), ι(b)(u) = bu. Combining

the adjunction of part (i) with the adjunction of part (ii) for the ring extension ι, we obtain theneeded adjunction.

MB

−⊗BT //MT

−⊗T Σ //

UA

oo MC

HomC(Σ,−)

oo

Explicit formulas for unit and counit are

νN : N → HomC(Σ, N ⊗B Σ), νN (n)(u) = n⊗B u,ζM : HomC(Σ,M)⊗B Σ →M, ζM (ϕ⊗B u) = ϕ(u).

For N ∈MT and for M ∈MC.

Proposition 3.8. Let C = (C, δ, ε) be a cotriple (comonad) on the category A and AC thecategory of comodules over C. Then there exists a pair of adjoint functors (FC ,GC),

ACFC

// AGC

oo

where FC is the forgetful functor and GC is the induction functor defined as GC(M) = (CM, δM ).

Proof. Again, we give only the explicit formula for unit and counit

νN : N → CN, νN = ρN ,

ζM : CM →M, ζM = εM .

For (N, ρ) ∈ AC and for M ∈ A.

Corollary 3.9. (i) Let C be an A-coring. Then the induction functor GC is the rightadjoint of the forgetful functor FC : MC → MA and the left adjoint of the functor HC =HomC(C,−).

MC

FC//

HC//MAGC

oo

(ii) Let R be a firm ring and R the Dorroh extension of R. Then (J,−⊗RR) is a pair of adjointfunctors

MR

J // MR−⊗RRoo

where J is the forgetful functor.

Proof. (i). The adjunction (FC,GC) follows from Proposition 3.8 applied to the cotripleC = −⊗A C : MA →MA. Explicit formula of unit and counit are as follows

%M = ρM : M →M ⊗A C, %M (m) = m[0] ⊗A m[1]

εM = M ⊗A εC : M ⊗A C →M, εM (m⊗A c) = mεC(c)

The adjunction (GC,HC) is the special case Σ = C, T = A of part (iii) in Proposition 3.7.

(ii). From Lemma 2.10 we know that R is an R-coring. Furthermore, − ⊗R R ∼= − ⊗ bR R byLemma 2.8. Therefore, the adjunction follows as a special case of the adjunction between theforgetful functor and the induction functor of part (i).


The adjunction between (FC,GC) of Corollary 3.9, part (i) implies the following isomorphismof k-modules (see (16)).

(60) θM,N : HomC(M,N ⊗A C) ∼= HomA(M,N)

for all M ∈MC and N ∈MA. This map is defined as θM,N (f) = (N ⊗A εC) f and θ−1M,N (g) =

(g ⊗A C) ρM for all f ∈ HomC(M,N ⊗A C) g ∈ HomA(M,N).

Corollary 3.10. (i) Let C be an A-coring, R a firm ring and Σ ∈ RMC. Then (GΣ =−⊗R Σ,HΣ = HomC(Σ,−)⊗R R) is a pair of adjoint functors

MR

−⊗RΣ //MC

HomC(Σ,−)⊗RR

oo

(ii) Let A be any ring, R a firm ring and Σ ∈ RMA bimodule. Then we find a pair of adjointfunctors (−⊗R Σ,HomA(Σ,−)⊗R R)

MR

−⊗RΣ //MAHomA(Σ,−)⊗RRoo

(iii) Let C be an A-coring, R a firm ring and Σ ∈ RMC. The following diagram of adjoint functorscommutes, in the sense that the adjunction of arrows in the upper row (i.e. the adjunctionof part (ii)), splits trough MC (i.e. it is the composition of the adjunctions in part (i) andCorollary 3.9, part(ii))

MR

−⊗RΣ //

−⊗RΣ

''OOOOOOOOOOOOOOOOOOOOOOOOOO MAHomA(Σ,−)⊗RR

oo

−⊗AC

MC

FC

OO

HomC(Σ,−)⊗RR

ggOOOOOOOOOOOOOOOOOOOOOOOOOO

Proof. (i). One has to combine the adjunction of Proposition 3.7 part (iii) with the adjunc-

tion of Corollary 3.9 part (ii). In this way we obtain (recall that since Σ is firm as a left R-module,

we can replace the tensor product over R by the tensor product over R by Lemma 2.8)

MR

J //M bR −⊗RΣ //

−⊗RRoo MC

HomC(Σ,−)

oo

For N ∈MR the explicit formula for the unit reads as

νN : N → HomC(Σ, N ⊗R Σ)⊗R R, νN (n) = ϕnr ⊗R r,

where ϕn(u) = n⊗R u for all n ∈ N . For M ∈MC, remark first that HomC(Σ,M)⊗RR⊗RΣ ∼=HomC(Σ,M) ⊗R Σ since Σ is firm as a left R-module. Then we can define the counit of theadjunction as

ζM : HomC(Σ,M)⊗R R⊗R Σ →M, ζM (ϕ⊗R r ⊗R u) = ϕ(ru).

(ii). This adjunction follows immediately from Theorem 2.13, or by considering A as a trivial A-coring this adjunction from the first statement. For N ∈MR and M ∈MA, the explicit formulasfor the unit and counit read as

ηN : N → HomA(Σ, N ⊗R Σ)⊗R R, νN (n) = ϕnr ⊗R r,εM : HomA(Σ,M)⊗R R⊗R Σ →M, ζM (ϕ⊗R r ⊗R u) = ϕ(ru),

where ϕn(u) = n⊗R u for all n ∈ N .(iii). Clearly FC(−⊗RΣ) defines a functor MR →MA. Applying the natural isomorphisms (60),

we obtain for all N ∈ MR that HomC(Σ, N ⊗R Σ⊗A C)⊗R R ∼= HomA(Σ, N ⊗R Σ)⊗R R. Sowe find that the functors of part (ii) are indeed the composition of the functors of part (i) and theforgetful and induction functor. Let us check that the unit and counit of the adjunction of part


(ii) can also be obtained as compostion units and counits of the correct adjunctions. Combiningthe unit of (GΣ,HΣ) with the unit of (FC,GC) we find

ηN : N → HΣGCFCGΣN = HomC(Σ, N ⊗R Σ⊗A C)⊗R R, ηN (n) = ψ(nr)⊗R r

for r ∈ N where ψ(n)(u) = n⊗R u[0]⊗A u[1]. Combining this with the isomorphism θΣ,N⊗RΣ, weobtain exactly the unit from the adjunction between −⊗A Σ and HomA(Σ,−)⊗R R. The prooffor the counit is similar.

3.1.3. The Dorroh coring. Let C be a (counital) A-coring. An A-subbimodule D ⊂ C iscalled a (two-sided) coideal if ∆(D) ⊆ C⊗A D + D⊗A C and εC(D) = 0.

Theorem 3.11. Let C be an A-coring, not necessarily with a counit. There exists a counital

A-coring C with the following properties:

(i) C is isomorphic to a coideal of C;

(ii) there exists a surjective A-coring morphism π : C → C;

(iii) there exists an injective A-coring morphism ι : A→ C;(iv) the category of (not necessarily counital) comodules over C is isomorphic to the category of

counital comodules over C.

Proof. C = C × A is an A-bimodule with left and right A-action given by b · (c, a) · b′ =(bcb′, bab′), for all c ∈ C and a, b, b′ ∈ A, i.e. C = C ⊕ A in AMA. The map ε : C → A,

ε(c, a) = a is an A-bimodule map. It is easy to verify that ∆ : C → C⊗A C, given by the formula

∆(c, a) = (c(1), 0)⊗A (c(2), 0) + (0, 1)⊗A (c, a) + (c, a)⊗A (0, 1)− (0, a)⊗A (0, 1),

is a coassociative A-bimodule map, and that (C, ∆, ε) is a counital coring. The map π is the

canonical surjection C → C, and ι is the canonical injection A → C. All further verifications arestraightforward.

Let (M, δM ) be a right C-comodule. A C-coaction δM on M is defined as follows:

δM : M →M ⊗A C, δM (m) = m[0] ⊗A (m[1], 0) +m⊗A (0, 1),

for all m ∈ M . It is straightforward to show that (M, δM ) is a right C-comodule. Conversely, if

(M, δM ) is a right C-comodule, then (M, δM = (M ⊗A π) δM ) is a right C-comodule. Both con-structions are functorial, and define an isomorphism between the categories of right C-comodules

and unital right C-comodules.

We call the counital coring C constructed in Theorem 3.11 the Dorroh coring associated to

C. Remark that C is not a subcoring of C, since ∆(C) is not included in C⊗A C. This has to becompared to the fact that the quotient map from the Dorroh overring to the original ring is not aring morphism.

If C is a coalgebra over a commutative ring k, the Dorroh coring C is also a coalgebra.Moreover, in this situation, if C is cocommutative, than C is also cocommutative.

We can relate the Dorroh extension of corings with the Dorroh extension of rings.

Proposition 3.12. Let C be an A-coring, not necessary with counit. Then the dual ring ofthe Dorroh extension of this coring is isomorphic to the Dorroh extension of the dual ring of theoriginal coring, i.e.

∗(C) ∼= ∗C

Proof. We know that as A-bimodules, ∗(C) = AHom(C⊕A,A) and ∗C = AHom(C, A)⊕A.

Let us define an isomorphism α : ∗(C) ∼= ∗C by α(f) = (f ′, af ), where f ′(c) = f(c, 0) and

af = f(0, 1). Then we find for all (c, a) ∈ C = C⊕A,

f(c, a) = f ′(c) + aaf


Now take f, g ∈ ∗(C) and compute

f ∗ g(c, a) = g((c(1), 0)f(c(2), 0)) + g((0, 1)f(c, a)) + g((c, a)f(0, 1))− g((0, a)f(0, 1))

= g′((c(1))f′(c(2))) + g((0, 1)(f ′(c) + aaf )) + g((c, a)af )− g((0, a)af )

= f ′ ∗ g′(c) + g(0, f ′(c)) + g(0, aaf ) + g′(caf ) + aafag − aafag

= f ′ ∗ g′(c) + f ′(c)ag + aafag + g′(caf )= f ′ ∗ g′(c) + (f ′ ∗ ag)(c) + (af ∗ g′)(c) + aafag

From this computation we can conclude that α(f ∗ g) = (f ′ ∗ g′ + f ′ ∗ ag + af ∗ g′, afag), which

is exactly the formula for multiplication in ∗C.

The following example [2] shows how we can construct a non-trivial coring out of a coalgebraby use of the Dorroh extension.

Example 3.13. Let k be a commutative ring and C a k-coalgebra, then ∗C = C∗ is a k-algebra with multiplication (f ∗ g)(c) = g(c(1))f(c(2)) and unit map i : k → C∗, i(r)(c) = rε(c).We obtain that C is an object in the monoidal category C∗MC∗ , with actions f c = f(c(1))c(2)

and c f = c(1)f(c(2)). Define the map ∆ as the following composition

C∆ //

∆ ((PPPPPPPPPPPPPP C ⊗k C

π

C ⊗C∗ C,

where the map π is canonical surjection. We claim that ∆ is C∗-bilinear

∆(f c) = ∆(f(c(1))c(2))= π ∆(f(c(1))c(2))= π(f(c(1))c(2) ⊗k c(3))= f(c(1))c(2) ⊗C∗ c(3)= f c(1) ⊗C∗ c(2) = f ∆(c)

and similar for the right-hand side. In this way, (C, ∆) becomes a non-counital C∗-coring.Remark that the composition ε = i ε : C → k → C∗ is not C∗-bilinear, and thus it can not

define a counit for the C∗-coring C. On one hand we find

ε(f c)(d) = ε(f(c(1))c(2))(d) = ε(f(c(1))c(2))ε(d) = f(c)ε(d)

and on the other hand,

(f ∗ ε(c))(d) = ε(c)(d(1))f(d(2)) = ε(c)ε(d(1)f(d(2)) = ε(c)f(d).

The right C∗-linearity follows in a similar way.By Theorem 3.11, we can equip C with a counit by considering C = C ⊕C∗ with comultipli-

cation

∆ : C → C ⊗ C ∼= (C ⊗ C)⊕ (C ⊗ C∗)⊕ (C∗ ⊗ C)⊕ (C∗ ⊗ C∗)

∆(c, f) = (c(1) ⊗ c(2))⊕ (c⊗ ε)⊕ (ε⊗ c)⊕ (f ⊗ ε)

and counit ε : C → C∗, ε(c, f) = f .

Suppose now that k is a field and G a semigroup. Denote by kG the set of all maps from Gto k and by

(kG) = f ∈ (kG)∗ | Ker (f) contains an ideal of finite codimension,the finite dual of kG. There exists a natural isomorphism of vector spaces

φ : kG → (kG)∗, φ(f)(∑

rigi) =∑

rif(gi).

We call Rk(G) = φ−1((kG)), this set can be described as (see e.g. [55, p. 41])

Rk(G) = f ∈ kG | ∃fi, gi ∈ kG, s.t. f(xy) =∑

fi(x)gi(y),∀x, y ∈ G


With this notation, we can define a comultiplication on Rk(G) given by ∆(f) =∑

i fi ⊗ gi. Ingeneral Rk(G) has no counit. If G is a monoid, i.e. G has a unit e, then Rk(G) has a counit,this counit is given by the evaluation in e. We call Rk(G) the representative coalgebra of thesemigroup G.

Proposition 3.14. Let G be a semigroup without unit and k a commutative field, then thereis a canonical coalgebra morphism between the Dorroh extension of the representative coalgebraRk(G) and the representative coalgebra Rk(G), where G is the Dorroh extension of the semigroupG.

Proof. Let us denote by e the unit of the Dorroh monoid G associated to the semigroup G.

Let us first prove that (kG) ∼= (kG) ⊕ kE, where E ∈ k bG is defined by E(e) = 1 and E(g) = 0for all g ∈ G. The direct sum decomposition kG = kG⊕ ke induces a direct sum decomposition(kG)∗ = (kG)∗ ⊕ kE. Take f ∈ kG, then there exists an ideal I ∈ ker f such that dim(kG/I)is finite and we can decompose f = f1 + f2 with f1 ∈ kG∗ and f2 ∈ kE. Furthermore, from theprevious direct sum decompositions we obtain ideals I1 ⊂ ker f1 ⊂ kG and I2 ⊂ ker f2 ⊂ ke suchthat I = I1⊕ I2. Therefore dim(kG/I) = dim(kG/I1)+dim(ke/I2). Since dim(ke/I2) can only

be 1 or 0, we conclude that dim(kG/I) must be finite. Consequently, we find an isomorphism of

vector spaces over k α : Rk(G) ∼= Rk(G).We must check that this isomorphism is an isomorphism of coalgebras. Consider f ∈ Rk(G).

Then f = (f ′, rE) where f : G→ k and r ∈ k. Applying the comultiplication in the representativecoalgebra, we find, ∆(f) =

∑i gi⊗hi = (g′i, siE)⊗(h′i, tiE), with gi = (g′i, siE) and hi = (h′i, tiE)

if and only if

f(xy) =∑i

gi(x′)hi(y′) + sitiδxδy,

where x = x′+ δxe, y′+ δye ∈ G with δx, δy ∈ 0, 1. From the left and right counit property, wededuce that (f ′, r) =

∑i(sihi, siti) = (tigi, siti) and consequently f ′ =

∑i sihi =

∑i tigi and

r =∑

i siti. Since tensor products preserve direct sums, we find the decomposition

(kG ⊕ kE)⊗ (kG ⊕ kE) ∼= (kG ⊗ kG)⊕ (kG ⊗ kE)⊕ (kE ⊗ kG)⊕ (kE ⊗ kE)

Then we can rewrite the comultiplication in Rk(G) as follows

∆(f) =∑i

(g′i ⊗ h′i)⊕ (g′i ⊗ tiE)⊕ (siE ⊗ h′i)⊕ (siE ⊗ tiE)(61)

=∑i

(g′i ⊗ h′i)⊕ (f ′ ⊗ E)⊕ (E ⊗ f ′)⊕ (rE ⊗ E)

Furthermore, α(f) = (f ′, r) ∈ Rk(G)⊕ k and

∆(f ′, r) =∑i

(gi, 0)⊗ (hi, 0) + (0, 1)⊗ (f ′, r) + (f ′, r)⊗ (0, 1)− (0, r)⊗ (0, 1)

Using again the fact that the tensor product preserves direct sums we find

(62) ∆(f ′, r) =∑i

(gi ⊗ hi)⊕ (f ′ ⊗ 1)⊕ (1⊗ f ′)⊕ (r ⊗ 1 + r ⊗ 1− r ⊗ 1)

If we apply α⊗ α on (61), then we obtain (62). This means that α is comultiplicative. Finally, itis easily verified that α preserves the counit, so α is a coalgebra morphism.

3.2. Corings and entwining Structures

3.2.1. Entwining structures and entwined modules. Entwining Structures (or mixed dis-tributative laws) were introduced in [34]. In this section we provide an alternative way to introducethese structures, motivated by categorical techniques. One should compare the results of this sec-tion with [105].

Let B and A be two k-algebras. Everything in this section can be repeated in any monoidalcategory. In particular, one can replace k by a non-commutative algebra R, the k-algebras by

3.2. CORINGS AND ENTWINING STRUCTURES 81

R-rings and Mk by RMR. We will denote in this section −⊗k− by the unadorned tensor product−⊗−. Consider P ∈Mk, then P induces a functor

(63) −⊗ P : MB →Mk.

Theorem 3.15. With notations as above, there exists a functor P making the followingdiagram commutative,

MB−⊗P //

∃P ''OOOOOOOOOOOOO Mk

MA

U

OO

where U denotes the forgetful functor, if and only if there exists a map

ψ : P ⊗A→ B ⊗ P,

such that that the following diagram is commutative

(64) P ⊗A⊗A

ψ⊗A

~~

P⊗µA

&&NNNNNNNNNNN

P ⊗A

ψ

B ⊗ P ⊗A

A⊗ψ

AAA

AAAA

AAAA

AAAA

AAAA

P

P⊗ηA

ggPPPPPPPPPPPPP

ηB⊗Pwwnnnnnnnnnnnnn

B ⊗ P

B ⊗B ⊗ P

µB⊗P

88ppppppppppp

Proof. Suppose that the functor P exists, then we find UP(B) = B ⊗ P . The forgetfulfunctor U maps an object X in MA to the same k-module, but forgets the A-action. This impliesthat P(B) is equal to B ⊗ P as k-module, endowed with a right A-module action. Therefore, wecan define the map ψ as the following composition.

ψ : P ⊗AηB⊗P⊗A // B ⊗ P ⊗A // B ⊗ P

where the second map in the above composition is given by the right A-action on B ⊗ P . Onecan easily check that this ψ makes the diagram (64) to commute.

Conversely, having a map ψ satisfying (64), we can define for any (M,ρM ) ∈ MB a rightA-action on M ⊗ P as follows

ψM : M ⊗ P ⊗AM⊗ψ // M ⊗B ⊗ P

ρM⊗P // M ⊗ P

Using (64) it follows from an easy computation that this defines indeed a right A-action. Con-sequently, we obtain a functor P : MB → MA, where P(M,ρM ) = (M ⊗ P,ψM ). Obviously,U P = −⊗ P .

Theorem 3.16. If the equivalent statements of Theorem 3.15 hold, then the functor P hasa right adjoint Q : MA →MB.

Proof. Let M be any right A-module. The functor Q is defined as follows. Q(M) =HomA(B ⊗ P,M), where the right B action is given by the following formula

(ϕ · b′)(b⊗ p) = ϕ(b′b⊗ p),

for all ϕ ∈ HomA(B ⊗ P,M), b, b′ ∈ B and p ∈ P .

We will consider a special case of Theorem 3.15 Let A be any k-algebra and C a k-coalgebra.Then we can consider the functor −⊗ C : MA →MC .


Lemma 3.17. Let A be any k-algebra and C a k-coalgebra. Assume that there exists a cotriplefunctor C : MA →MA such that the following diagram commutes

MA−⊗C //

C ''OOOOOOOOOOOOO Mk

MA

UA

OO

Then there exists a functor U ′ : MAC →MC such that the following diagram of forgetful functors

is commutative

MC UC//Mk

MAC

U ′

OO

FC//MA

UA

OO

where MAC is the category of C-comodules in MA (see Example 1.23).

Proof. Let (M,ρ) be an object in MAC with ρ : M → C(M). By similar arguments as

in the proof of Theorem 3.15, we know that C(M) = M ⊗ C as k-module and moreover thisk-module has a right A-module structure. The functor U ′ forgets this A-module structure.

Theorem 3.18. Let (A,µ, η) be a k-algebra and (C,∆, ε) a k-coalgebra. The following areequivalent.

(i) There exists a cotriple functor C : MA →MA such that the following diagram commutes

(65) MA−⊗C //

GC ''OOOOOOOOOOOOO MC UC//Mk

MAC

U ′

OO

FC//MA

UA

OO

where U ′ is the functor defined in Lemma 3.17, FC is the forgetful functor and GC is theinduction functor.

(ii) There exists a map ψ : C ⊗A→ A⊗ C, such that the following diagram commutes

(66) C ⊗A⊗A

ψ⊗A

~~

C⊗µ

&&NNNNNNNNNN C ⊗ C ⊗A

C⊗ψ

AAA

AAAA

AAAA

AAAA

AAAA

C ⊗Aε⊗A

&&NNNNNNNNNNNN

ψ

∆⊗A88ppppppppppp

A⊗ C ⊗A

A⊗ψ

AAA

AAAA

AAAA

AAAA

AAAA

C

C⊗η88pppppppppppp

η⊗C &&NNNNNNNNNNNN A C ⊗A⊗ C

ψ⊗C

~~

A⊗ C

A⊗ε

88pppppppppppp

A⊗∆

&&NNNNNNNNNNN

A⊗A⊗ C

µ⊗C

88ppppppppppA⊗ C ⊗ C

(iii) There exists an A-coring structure on C = A⊗C, where the left A-module structure of C isthe obvious one and the comultiplication and counit are induced by those of C.

Proof. (i) ⇒ (iii). We know from Theorem 3.15, that the functor C is given by −⊗C if weforget the A-module structure. We may suppose that C is a cotriple functor. By Theorem 3.16,we know that C has a right adjoint, thus C is right exact and preserves direct sums. Consequently,C(A) = A⊗ C is an A-coring by Theorem 3.5. Denote its comultiplication by ∆C and its counitby εC.


Take any M ∈ MA and compute the coaction of C(M) = M ⊗ C in MAC , then we find

ρM⊗C : M ⊗ C → (M ⊗ C) ⊗A (A ⊗ C) ∼= M ⊗ C ⊗ C ∼= C2(M). Having in mind thatU ′ GC = −⊗ C, we find that ρM⊗C = M ⊗∆C . In particular ∆C

∼= A⊗∆C .Define the map ε = A⊗εC : A⊗C → A. Obviously ε is left A-linear and by the commutativity

of (65) and the observations above about the comultiplications of C-comodules we know that εacts as a right counit on right C-comodules, in particular, ε is a right counit for C. By applicationof Lemma 3.1 we obtain that εC = ε.(iii) ⇒ (ii). Define ψ(c ⊗ a) = (1 ⊗ c) · a using the right A-module structure on A ⊗ C. The

commutativity of the left-hand side of diagram (66) follows then by the associativity and unitalityof the right A-module action on A⊗C. The commutativity of the right-hand side of diagram (66)follows by the right A-linearity of the comultiplication and counit of the coring A⊗ C.(ii) ⇒ (i). By the commutativity of the left-hand side of diagram (66), we find by Theorem 3.15

that there exists a functor C : MA →MA defined by C(M,ρM ) = (M ⊗ C,ψM = (ρM ⊗ C) ⊗(M ⊗ ψ)) such that the outer lines of diagram (65) commute. If we define δ : C → CC andε : C →MA as follows

δM = M ⊗∆ : CM = M ⊗ C → CCM = M ⊗ C ⊗ C

εM = M ⊗ ε : CM = M ⊗ C →M

for all M ∈ MA, then we find that (C, δ, ε) is a cotriple on MA. Consequently, we obtain byLemma 3.17 the functor U ′. The commutativity of the inner triangle of diagram (65) is thenexactly given by the commutativity of the right-hand side of diagram (66).

If the equivalent conditions of Theorem 3.18 hold then (A,C, ψ) is called an entwining struc-ture.

The commutativity of the diagram (66) translates to the following four conditions:

(ab)ψ ⊗ cψ = aψbΨ ⊗ cψΨ(67)

(1A)ψ ⊗ cψ = 1A ⊗ c(68)

aψ ⊗∆C(cψ) = aψΨ ⊗ cΨ(1) ⊗ cψ(2)(69)

εC(cψ)aψ = εC(c)a(70)

Here we used the sigma notation

ψ(c⊗ a) = aψ ⊗ cψ = aΨ ⊗ cΨ

We then call (A,C, ψ) a (right-right) entwining structure. An entwined module M is a k-moduletogether with a right A-action and a right C-coaction ρ, in such a way that

(71) ρ(ma) = m[0]aψ ⊗mψ[1]

for all m ∈M and a ∈ A.Recall from [30] that a B-coring D is named a coring extension of the A-coring C if and only

if C is a C-D bicomodule where we consider the regular left C-comodule structure on C. In [30,Theorem 2.6] it is proven that this condition is equivalent to the existence of a k-linear functorU : MC →MD such that the following diagram commutes

MCU //

FC ""EEE

EEEE

E MD

FD||xxxxxxxx

MA

UA ""EEE

EEEE

E MB

UB||yyyy

yyyy

Mk

Theorem 3.19. To an entwining structure (A,C, ψ), we can associate an A-coring C = A⊗Csuch that the coalgebra C is a right coring-extension of C. The structure maps are given by the


formulasa′(b⊗ c)a = a′baψ ⊗ cψ

∆C(a⊗ c) = (a⊗ c(1))⊗A (1⊗ c(2))εC(a⊗ c) = aεC(c)

The category M(ψ)CA of entwined modules and A-linear C-colinear maps is isomorphic to thecategory of right C-comodules.

Proof. The coring structure on A ⊗ C follows directly from Theorem 3.18. Consideringfunctor U ′ from statement (i) of the same Theorem, we obtain the following commutative diagramof functors

MCU ′ //

FC ""EEE

EEEE

E MC

UC

MA

UA ""EEE

EEEE

E

Mk

This implies that C is a right coring extension of C. Moreover, this implies that every C-comoduleM is completely determined by a k-module UAFC(M) = UCU ′(M), endowed with the rightA-module structure of FC(M) and the right C-comodule structure of U ′(M). The compatibilitycondition (71) is the translation of the right A-linearity of the C-coaction.

Remark 3.20. Let B be a ring with unit and B the Dorroh extension of B. Then MB = M bBis the category of all (possibly non-unital) right B-modules. Consider the functor

L : MB →MB

where we define L(M) = M1B = m1B | m ∈ M and send B-linear maps to B-linear maps.

Then the functor L is well-defined and L(M) is a direct summand of M in MB. Suppose that

there exists a functor C : MB → MB such that C ∼= CJL where J : MB → MB is the inclusionfunctor. If there exists a natural transformation δ : C → C2, such that (δ ∗ C) δ = (C ∗ δ) δ,then we can define a functor C = LCJ : MB →MB together with a natural transformation

δ = LδJ : C → C2 = LCJLCJ ∼= LC2J

such that (δ ∗ C) δ = (C ∗ δ) δ. Finally, suppose that there exists a natural transformation

ε : C → MB, then we say that (C, δ, ε) is a left lax cotriple on MB if (C, δ, LεJ) is a cotriple onMB.

Let C be a left unital B-bimodule then the functor C = −⊗BC satisfies the condition C = CJL.We say that C is a left lax cotriple on MB if and only if C is a left unital lax A-coring .

Let C be a k-algebra and consider a left lax cotriple (C, δ, ε) rendering the following diagramcommutative

MA−⊗C //

GCJ ''OOOOOOOOOOOOO MC UC//Mk

MC

U ′

OO

LFC//MA

UB

OO

Applying the machinery of Theorem 3.18 we find a correspondence between lax cotriples, laxcorings and a new type of entwining structures (A,C, ψ) that are named lax entwining structures.

For the exact definitions of lax corings and lax entwining structures, we refer to [44], wherethese notions were introduced.

The fundamental example of entwining structures in Hopf algebra theory is given by the fol-lowing structures.

Example 3.21. Let k be commutative ring. A k-bialgebra H, consists of a k-coalgebra(H,∆, ε), a k-algebra (H,µ, η) (remark that the underlyink k-module of coalgebra and algebraare the same), such that ∆ and ε are algebra morphisms or equivalently, µ and η are coalgebra


morphisms. It is well-known that for a bialgebra H the categories MH and MH are monoidal,hence we can define algebras and coalgebras in these categories. This gives rise to the notions ofmodule algebras, module coalgebras, comodule algebras and comodule coalgebras. The categoryof right Hopf-modules over a bialgebra H consists of object (M,µM , ρM ), where (M,µM ) ∈MH ,(M,ρM ) ∈MH such that the following compatibility condition holds

(m · h)[0] ⊗ (m · h)[1] = m[0]h(1) ⊗m[1]h(2)

for all h ∈ H and m ∈ M . We denote the category of all H-Hopf modules together with rightH-linear, right H-colinear maps by MH

H .Let H be a bialgebra, A a right H-comodule algebra, and C a right H-module coalgebra.

We call (H,A,C) a right-right Doi-Hopf structure or Doi-Koppinen structure. We associate anentwining structure (A,C, ψ) to (H,A,C) as follows, with ψ defined by ψ(c ⊗ a) = a[0] ⊗ ca[1].The corresponding entwined modules are called Doi-Hopf modules. They have to satisfy thecompatibility relation

ρ(ma) = m[0]a[0] ⊗m[1]a[1]

for all m ∈M and a ∈ A.In light of this Doi-Hopf structure (H,A,C) we can associate the A-coring C = A ⊗ C with

structure maps

b(a⊗ c) = ba⊗ c (a⊗ c)b = ab[0] ⊗ cb[1]

∆C(a⊗ c) = (a⊗ c(1))⊗A (1⊗ c(2)) εC(a⊗ c) = aεC(c)

for all a, b ∈ A and c ∈ C. In particular, if H is a bialgebra, then we find that H = H ⊗H is anH-coring with structure given by

h(f ⊗ g) = hf ⊗ g (f ⊗ g)h = fh(1) ⊗ gh(1)

∆H(f ⊗ g) = (f ⊗ g(1))⊗A (1⊗ g(2)) εC(f ⊗ g) = fεH(g)

for all f, g, h ∈ H. The category of H-modules are exactly the Hopf-modules of H.

Let A and S be k-algebras, and R : S ⊗A→ A⊗ S a k-linear map. We will write

R(s⊗ a) = aR ⊗ sR = ar ⊗ sr

(summation understood). The k-module A⊗ S, with multiplication

(72) (a#s)(b#t) = abR#sRt

will be denoted by A#RS. It is straightforward to verify that this multiplication is associative withunit 1A#1S if and only if

R(s⊗ 1A) = 1A ⊗ s(73)

R(1S ⊗ a) = a⊗ 1S(74)

R(st⊗ a) = aRr ⊗ srtR(75)

R(s⊗ ab) = aRbr ⊗ sRr(76)

for all a, b ∈ A and s, t ∈ S. We then call (A,S,R) a factorization structure, and A#RS thesmash product of A and S. There exists a bijective correspondence between smash products ofthe form (A,S,R) and algebra structures on A ⊗ S for which iA : A → A ⊗ S, iA(a) = a ⊗ 1Sand iS : S → A⊗ S, iS(s) = 1A ⊗ s are algebra maps.

One can easily see that factorization sturctures and entwining structures are closely related, asthey are semi-dual notions. In particular if C is finitely generated and projective as k-algebra thenthere exists a bijective correspondence between (right,right) entwining structures (A,C, ψ) andfactorization structures (A, (C∗)op, R). In this case the associated smash product is isomorphic tothe dual coring of the coring associated to the entwining structure.

If (A,C, ψ) is the entwining structure associated to a Doi-Hopf structure (A,C,H), then thisfactorization structure exists, even if C is not finite dimensional. Indeed, if we define

R : A⊗ (C∗)op → (C∗)op ⊗A, R(a⊗ f) = a[1]f ⊗ a[0]


then (A, (C∗)op, R) is a factorization structure and the smash product is a subalgebra of the dualring of the associated coring. Since this property does not hold in general for entwining structures,this leads to the introduction of factorizable entwining structures.

3.2.2. Factorizable enwining structures. Let k be a commutative ring, and consider aright-right entwining structure (A,C, ψ). We call (A,C, ψ) factorizable if there exists a mapα : A→ A⊗ End(C) such that ψ factorizes as follows:

ψ = (A⊗ θ) (α⊗ C) τ :

C ⊗Aτ // A⊗ C

α⊗C // A⊗ End(C)⊗ CA⊗θ // A⊗ C

where τ : C ⊗A→ A⊗ C is the switch map, and θ : End(C)⊗ C → C is the evaluation map.Using the notation α(a) = aα ⊗ λα (summation implicitely understood), this means that

ψ(c⊗ a) = aα ⊗ λα(c)

We will say that (A,C, ψ) is completely factorizable if ψ has an inverse ϕ, and (A,C, ψ) and(A,C, ϕ) are both factorizable.

Examples 3.22. (1) Let H be a bialgebra, (H,A,C) a right-right Doi-Hopf structure.The corresponding entwining structure (A,C, ψ) is factorizable: Take α(a) = a[0]⊗ma[1]

,

where mh : C → C, mh(c) = ch, for all c ∈ C.

(2) Assume that A is finitely generated projective. We know from [103] that (A,C, ψ) isinduced by a Doi-Hopf structure over a bialgebra, so (A,C, ψ) is factorizable. This canalso be seen directly: let (ei, e∗i ) | i = 1, · · · , n be a finite dual basis of A, and define

α(a) =∑i

ei ⊗ 〈e∗i , aψ〉(−)ψ

(3) Now assume that C is finitely generated projective, and let (ci, c∗i ) | i = 1, · · · , n be afinite dual basis of C. Then (A,C, ψ) is factorizable: take

α(a) = aψ ⊗ cψi c∗i

Indeed, we easily compute that

aα ⊗ λα(c) = aψ ⊗ cψi 〈c∗i , c〉 = aψ ⊗ cψ

Factorizable entwining structures are close to Doi-Hopf structures: the philosophy is that thebialgebra H is replaced by the algebra End(C). Of course this is just philosophy, since in general,there are no bialgebra structures on End(C) with the composition as multiplication. We will seethat factorizable entwining structures are more general then Doi-Hopf structures, and also thatthere exists non-factorizable entwining structures. Our examples will be inspired by the examplesgiven in [103]. Let (A,C, ψ) be an entwining structure over a field k. Recall from [103] that wecan construct the following endomorphisms of A, for every c ∈ C and c∗ ∈ C∗:

Tc,c∗ : A→ A, Tc,c∗(a) = 〈c∗, cψ〉aψIf (A,C, ψ) originates from a Doi-Koppinen structure (A,C,H), then Tc,c∗(a) = c∗(a[1]c)a[0], andwe see that every H-subcomodule of A is Tc,c∗-invariant. Since we are working over a field, everya ∈ A is contained in a finite dimensional H-subcomodule of A, so every a ∈ A is contained in afinite dimensional Tc,c∗-invariant subspace of A. This property will be used in the examples in thesequel.In a similar way, if (A,C, ψ) originates from an alternative Doi-Hopf datum, then every c ∈ C liesin a finite dimensional Ta,a∗-invariant subspace of C. Here Ta,a∗ is defined as follows:

Ta,a∗(c) = a∗(aψ)cψ

for every a ∈ A, a∗ ∈ A∗ and c ∈ C. Remark that α(a) =∑

i ei ⊗ Ta,e∗i in Example 3.22 2).We will now give an example of a factorizable entwining structure that does not originate froma Doi-Hopf datum. It follows from the observations above that A and C have to be infinitedimensional in such an example.


Example 3.23. Let A = k < (Xi)i∈I1 > be the free algebra with a family of generatorsindexed by I1 = N or I1 = Z.Let C be the k-module with free basis 1, t ∪ ti | i ∈ I2, where I2 = N or I2 = Z. We put acoalgebra structure on k by making 1 grouplike and t and ti primitive.We now define the entwining map ψ. For every a ∈ A and c ∈ C we define ψ(a⊗ 1) = a⊗ 1 andψ(1⊗ c) = 1⊗ c. Furthermore we define

ψ(Xi1 · · ·Xin ⊗ t) = Xi1+1 · · ·Xin+1 ⊗ t

ψ(Xi1 · · ·Xin ⊗ tj) = Xi1+1 · · ·Xin+1 ⊗ tj+n

and extend ψ linearly. A straightforward computation shows that (A,C, ψ) is an entwining struc-ture. Let us show that (A,C, ψ) is factorizable. We only need to define α on elements of the forma = Xi1 · · ·Xin ⊗ c, since we can again extend linearly. Write c = c+ ct+

∑i citi with c, c, ci ∈ k.

Then we have

ψ(a⊗ c) = a+ c+Xi1+1 · · ·Xin+1 ⊗ (ct+∑j

cjtj+n)

= a⊗ λa1(c) +Xi1+1 · · ·Xin+1 ⊗ λa2(c)

where λa1, λa2 : C → C are defined by

λa1(1) = 1, λa1(t) = 0, λa1(ti) = 0

λa2(1) = 0, λa1(t) = t, λa1(ti) = ti+1

so we find that ψ0(a) = a⊗ λa1 +Xi1+1 · · ·Xin+1 ⊗ λa2 and (A,C, ψ) is factorizable.Let us show that there is no Doi-Hopf structure inducing (A,C, ψ). Take c∗ ∈ C∗ such that〈c∗, t〉 = 1 (this is possible since we work over a field). Then Tc∗,t(Xi) = Xi+1, so every Tc∗,t-invariant subspace that contains X0 is infinite dimensional.In a similar way, we find that (A,C, ψ) does not originate from an alternative Doi-Hopf datum:take a∗ ∈ A∗ such that a∗(X1) = 1; then we find that Ta∗,X0(ti) = ti+1, so every Ta∗,X0-invariantsubspace containing t0 is infinite dimensional.If I1 = I2 = Z, then ψ is bijective, and ϕ = ψ−1 has the same properties as ψ, so we find acompletely factorizable entwining structure that cannot be derived from an (alternative) Doi-Hopfstructure.Remark that we could also have taken

ψ(Xi1 · · ·Xin ⊗ tj) = Xi1+1 · · ·Xin+1 ⊗ tj+P

k ik

Adapting Example 3.23, we can give an example of an entwining structure that is not factor-izable.

Example 3.24. Let A and C be as in Example 3.23, and let ψ : A⊗C → A⊗C be definedby

ψ(Xi1 · · ·Xin ⊗ tj) = Xi1+j · · ·Xin+j ⊗ tj+n

Consider the k-linear map p : C → k, given by p(1) = p(t) = p(ti) = 1. If (A,C, ψ) isfactorizable, then for all a ∈ A, the set

Aa = (A⊗ p)ψ(a⊗ c) | c ∈ Cis contained in a finite dimensional subspace of A. This is not the case, since AX1 containsX2, X3, · · · . Hence (A,C, ψ) is not factorizable.

Let (A,C, ψ) be a right-right entwining structure, and C = A ⊗ C the associated A-coring.The left dual ring is ∗C = AHom(A⊗ C,A) ∼= #(C,A), with multiplication

(f#g)(c) = f(c(2))ψg(cψ(1))

A and (C∗)op are then subalgebras of #(C,A), via the algebra monomorphisms i : A→ #(C,A)and j : (C∗)op → #(C,A) given by

i(a)(c) = ε(c)a and j(c∗)(c) = 〈c∗, c〉1A


and we easily compute that

(j(c∗)#i(a))(c) = 〈c∗, c〉afor all a ∈ A, c ∈ C and c∗ ∈ C∗.

Proposition 3.25. If (A,C, ψ) is a factorizable entwining structure, then

i(A)#j((C∗)op) ⊂ j((C∗)op)#i(A)

and consequently i(A)#j((C∗)op) is a subalgebra of #(C,A).

Proof. For all a ∈ A, c ∈ C and c∗ ∈ C∗, we have

(i(a)#j(c∗))(c) = i(a)(c(2))ψj(c∗)(cψ(1))

= ε(c(2))aψ〈c∗, cψ(1)〉1A

= 〈c∗, cψ〉aψ= 〈c∗ λα, c〉aα=

(j(c∗ λα)#j(aα)

)(c)

hence

i(a)#j(c∗) = j(c∗ λα)#j(aα) ∈ j((C∗)op)#i(A)as needed.

If C∗ is C-locally projective as a k-module (e.g. C∗ is projective and C is dense in C∗∗, thisis for example the case when k is a field), then

j ⊗ i : (C∗)op ⊗A→ #(C,A)

is injective, and we obtain the following corollary of Proposition 3.25.

Corollary 3.26. Let (A,C, ψ) be a factorizable entwining structure. Then we have afactorization structure ((C∗)op, A,R), with R : A⊗ (C∗)op → (C∗)op ⊗A given by

R(a⊗ c∗) = c∗ λα ⊗ aα

if moreover C∗ is C-locally projective as a k-module, then j ⊗ i : (C∗)op#RA → #(C,A) is analgebra monomorphism.

If (A,C, ψ) is a completely factorizable entwining structure, then R is bijective.

3.3. Rational modules

Let C be an A-coring. Applying (60), the adjunction between the forgetful functor CF andthe induction functor CG induces an isomorphism CEnd(C) ∼= AHom(C, A) = ∗C. Since CEnd(C)becomes in a natural way a k-algebra under composition, we find that ∗C is a k-algebra as well.Furthermore, as explained in Example 1.46, ∗C is even an A-ring and comodules over C becomemodules over ∗C. We denote the unit map of ∗C by i : A→ ∗C, i(a)(c) = ε(c)a.

In this section we study a technique to reverse this construction. We want to turn modulesover ∗C into comodules over C. Usualy this is done in cases where the coring C is weakly locallyprojective as a left module over its base ring A. We discuss rationality properties of modules overa suitable subring R of ∗C, in case the coring is weakly R-locally projective. The subring R ⊂ ∗Cis not even an A-module. Entwining structures form an inspiring example for this situation.

3.3.1. R-rational modules. Let R be a subring (not nessecary with a unit) of ∗C. Suppose Cis weakly R-locally projective as a left A-module. Recall from Theorem 2.59 that this is equivalentto the fact that the map

αM,R : M ⊗A C → HomZ(R,M), αM,R(m⊗A c)(r) = m · r(c)

is injective, for all M ∈MA (i.e. C satiesfies the α condition for R).For every M ∈MR, we define

δM,R : M → HomZ(R,M), δM,R(m)(f) = m · f.

3.3. RATIONAL MODULES 89

For the remaining part of this section, we will denote by T the subring of ∗C generated by Aand R. This means that T is the Z-module generated by elements of the form a1 ∗ r1 ∗ a2 ∗ r2 ∗· · · ∗ rn ∗ an+1 with ai ∈ A and ri ∈ R. Note that T has a unit 1A, and that S = RA ⊂ T .

The first result is a straightforward observation.

Lemma 3.27. If R is a right A-module, then for all M ∈MT , the morphisms αM,R and δM,Rfactorize in the following way

αM,R : M ⊗A C // HomT (R,M) // HomZ(R,M);

δM,R : M // HomT (R,M) // HomZ(R,M).

Definition 3.28. A right T -module M is called R-rational if Im (δM,R) ⊂ Im (αM,R),or, equivalently, if for every m ∈ M , there exist finitely many mi ∈ M and ci ∈ C such thatm · f =

∑imif(ci), for all f ∈ R. RMR will be the full subcategory of MT consisting of

R-faithful R-rational T -modules.

Proposition 3.29. Let R be any subring of ∗C. Then we have the following properties:

(i) every cyclic R-submodule of an R-rational T -module M is contained in a finitely generatedright A-module;

(ii) any T -submodule of an R-rational T -module is R-rational;(iii) any quotient of an R-rational T -module is R-rational;(iv) the direct sum of a family of R-rational T -modules is again R-rational.

Proof. (i). Take any m ∈M , and consider the cyclic R-module mR. Since M is R-rational,

there exists (not necessary unique) finite sets of elements mj ∈ M and cj ∈ C such that for allr ∈ R we find m · r =

∑jmjr(cj). This means mR is contained in the right A-module generated

by the elements mj .(ii). Let M be an R-rational T -module and N a T -submodule of M . Take n ∈ N and f ∈ R.

Then we know that n · f =∑

i nif(ci), with ni ∈ M and ci ∈ C. We have to show that wecan find n′j ∈ N and c′j ∈ C with the same property. By Theorem 2.59 the ci have a dual basis

(ej , rj) | j ∈ J ⊂ C×R. We easily compute that

n · f =∑i

nif(ci) =∑i,j

nif(rj(ci)ej)

=∑i,j

nirj(ci)f(ej) =∑j

n · rjf(ej)

using the fact that M is R-rational. Since N is an R-module, we know that n · rj ∈ N . Now justtake n′j = n · rj and c′j = ej . Parts (iii) and (iv) can be proven in the same way.

Let M ∈ MA, and recall that an A-module is subgenerated by M if it is isomorphic to asubobject of a quotient of a direct sum of copies of M . The full subcategory of MA consisting ofmodules subgenerated by M is the smallest full Grothendieck subcategory of MA containing M(see [114]). This category is denoted by σ[M ].

Corollary 3.30. Let C be an A-coring, andR any subring of ∗C. Then we have the followinginclusions of categories:

MC ⊆ σ[CS ] ⊆ RMR

Moreover, σ[CS ] is a full subcategory of RMR.

Proof. Take M ∈ MC. Since A is a generator for MA, there exists a surjective right A-module morphism π : A(I) → M . Therefore we can construct the following diagram with exact


rows and collumn.

0

A(I)

π // M

ρM

// 0

C(I) ∼= A(I) ⊗A Cπ⊗AC

// M ⊗A C // 0

This implies that M is isomorphic as a C-comodule (and a fortiori as an S-module) to a subobjectof a quotient of a direct sum copies of C, and therefore M ∈ σ[CS ].It is clear that C is an R-rational T -module, and it follows from Proposition 3.29 that everyM ∈ σ[CS ] is an R-rational T -module.

We will now look for conditions under which the inclusions of categories in Corollary 3.30become isomorphisms.

Let R be a ring and S and T two subrings of R. Observe that ST ⊂ TS, if and only if for alls ∈ S and t ∈ T , we can find si ∈ S and ti ∈ T such that st =

∑i tisi in R. Let us present some

examples of this situation.

Examples 3.31. (1) Let (H,A,C) be a right-right Doi-Hopf structure over a commu-tative ring k. As we have seen, C = A ⊗ C is an A-coring. (C∗)op is a subring of∗C ∼= Hom(C,A), and A(C∗)op ⊂ (C∗)opA, since

(i(a) ∗ f)(b⊗ c) = f((b⊗ c)a) = ba[0]f(ca[1])

= bf(ca[1])a[0] = ((a[1]f) ∗ i(a[0]))(b⊗ c)

Recall that the left H-action on C∗ is given by (hf)(b⊗ c) = bf(ch). It follows thati(a) ∗ f = (a[1]f) ∗ i(a[0]), and A(C∗)op ⊂ (C∗)opA, as needed.

(2) If (A,C, ψ) is a factorizable entwining structure, then the same property holds as in theprevious example, by Proposition 3.25

(3) Let S and T be two k-algebras, k is a commutative ring, and (T, S,R) a factorizationstructure.

iT : T → T#S, iT (t) = t#1S and iS : S → T#S, iS(s) = 1T#s

are algebra maps, and ST ⊂ TS since

s · t = (1#s)(t#1) = tR#sR= (tR#1)(1#sR) = tR · sR ∈ TS

for all s ∈ S and t ∈ T .

Remark 3.32. In the setting of Example 3.31 (1), we can consider #(C,A)-rational modules,as well as C∗-modules, and C∗ is not a right A-module. This situation has been exhaustively studiedin [57]. This has been our motivation, in order to recover these results, to allow that R is not aright A-module such that AR ⊂ RA is satisfied.

Lemma 3.33. Let R be a subring (without unit) of ∗C, T the subring of ∗C generated by Rand A, and S = RA. Then the following assertions are equivalent:

(i) AR ⊂ RA;(ii) S is a ring and a left A-module, and T = A+ S;(iii) S is T -bimodule;(iv) S is a left A-module.

In this case:

(1) S is a T -ring and an A-ring;(2) RMS = RMR;(3) HomB(S,M) ∈MT , for all rings B ⊂ T and M ∈MB;(4) αM,S ∈MT for all M ∈MT .


Proof. (i) ⇒ (ii). Take any generator g = a1 ∗ r2 ∗ · · · ∗ rn ∗ an+1 of T as a Z-module. Ifn = 0, then g = a ∈ A. If n > 1, then it follows from AR ⊂ RA that g ∈ S = RA, henceT = A+ S. S is closed under multiplication since S2 = RARA ⊂ R2A2 ⊂ RA = S. S is a leftA-module since AS = ARA ⊂ RA2 = S.(ii) ⇒ (iii). S is a ring, so it is an S-bimodule. It is a right A-module, and, by (ii), a leftA-module. Since T = A+ S, S is a T -bimodule.(iii) ⇒ (iv) is trivial.

(iv) ⇒ (i). If S is a left A-module, then AR ⊂ AS ⊂ S = RA.

(1) Trivial.

(2). R ⊂ S, so S-rationality implies R-rationality. Conversely, if M is R-rational, then for all

m ∈M and∑

i fi ∗ ai ∈ S, we have

m ·∑i

fi ∗ ai =∑i

(m · fi)ai =∑i

(m[0]fi(m[1]))ai = m[0](∑i

fi ∗ ai)(m[1])

and it follows that M is S-rational. It is easy to see that M is R-faithful if and only if M isS-faithful.(3). Take ϕ ∈ HomZ(S,M), f ∈ S and g ∈ T . Then g ∗ f ∈ S, and we define ϕ · g by

(ϕ · g)(f) = ϕ(g ∗ f).(4). By part (3) we know that HomZ(S,M) is a right T -module. Let us check that αM,S :M ⊗A C → HomZ(S,M) is right T -linear. Indeed,

αM,S((m⊗A c) · g)(s) = αM,S(m⊗A c(1)g(c(2)))(s)= m · s(c(1)g(c(2))) = m · (g ∗ s)(c)

= αM,S(m⊗A c)(g ∗ s) =((αM,S(m⊗A c)

)· g)(s)

for all g ∈ T , s ∈ S and m⊗ c ∈M ⊗A C.

Proposition 3.34. Let C be an A-coring, R a subring of ∗C, and M ∈ RMR. If C is weaklyR-locally projective as a left A-module and AR ⊂ RA, then δM,R defines a right C-comodulestructure on M .

Proof. From the R-rationality of M and the fact that C is weakly R-locally projective as aleft A-module (or equivalently, C satisfies the α condition for R), it follows that, for any m ∈M ,there exists a unique

∑imi ⊗A ci ∈M ⊗A C such that m · f =

∑imif(ci), for every f ∈ R. So

we have a well-defined map δM : M →M ⊗A C, δM (m) =∑

imi ⊗A ci, which is equal to δM,Rif we regard the injection αM,R as an inclusion. Let us use the notation

δM (m) = m[0] ⊗A m[1]

We will show that δM defines a C-comodule structure on M . First, δM is right A-linear. SinceAR ⊂ RA, there exist ak and fk such that i(a) ∗ f =

∑k fk ∗ i(ak), hence

(ma) · f = m · (i(a) ∗ f) = m · (∑k

fk ∗ i(ak))

=∑k

(m · fk)ak =∑k

(m[0] · fk(m[1]))ak

=∑k

m[0] · (fk(m[1])ak) =∑k

m[0] · (fk ∗ i(ak))(m[1])

= m[0] · (i(a) ∗ f)(m[1]) = m[0] · f(m[1]a)

for all f ∈ R, m ∈M and a ∈ A, so δM (ma) = m[0] ⊗A m[1]a = δM (m)a.Let us next show the mixed coassociativity. For all m ∈M , we have to show that

ym = (M ⊗A ∆C)δM (m)− (δM ⊗A C)δM (m) = 0


For all f, g ∈ R, we have

m · (f ∗ g) = m[0](f ∗ g)(m[1])

= m[0]g((m[1])(1)f((m[1])(2)))

= (M ⊗A g) (M ⊗A C⊗A f)(m[0] ⊗A (m[1])(1) ⊗A (m[1])(2))

= (M ⊗A g) (M ⊗A C⊗A f)(M ⊗A ∆C)δM (m)= (m · f) · g = (m[0] · f(m[1])) · g

= (m[0])[0]g((m[0])[1]f(m[1]))

= (M ⊗A g) (M ⊗A C⊗A f)((m[0])[0] ⊗A (m[0])[1] ⊗A m[1])

= (M ⊗A g) (M ⊗A C⊗A f)(δM ⊗A C)δM (m)

where we used the right A-linearity of δM,R. It follows that

(M ⊗A g)((M ⊗A C⊗A f)(ym)) = 0

for all f, g ∈ R. This means exactly that

αM,R((M ⊗A C⊗A f)(ym)) = 0, ∀f ∈ RSince C satisfies the α-condition for R, we find

(M ⊗A C⊗A f)(ym) = 0, ∀f ∈ Ror αM⊗AC,R(ym) = 0. Applying again the α-condition, we find that ym = 0, as needed.Finally, for every f ∈ R, we have, using the mixed coassociativity,

(m−m[0]εC(m[1])) · f = m · f −m[0]f(m[1]εC(m[2]))

= m · f −m[0]f(m[1]) = m · f −m · f = 0

From the fact that M is faithful as a right R-module, we then deduce that m = m[0]εC(m[1]).

Theorem 3.35. Let C be an A-coring and R a subring of ∗C. Suppose that AR ⊂ RA. ThenC is weakly R-locally projective as a left A-module if and only if the categories

RMR, RMS , RM∗C, σ[CS ], σ[C∗C] and MC

are isomorphic full subcategories of MS .

Proof. Suppose first that C is weakly R-locally projective as a left A-module. We definea functor F : RMR → MC as follows: F (M) = M as a right A-module, with C-comodulestructure as in Proposition 3.34; for f : M → N in RMR, we put F (f) = f .Let us first prove that f is right C-colinear. Take m ∈M , then δM (f(m)) = f(m)[0] ⊗A f(m)[1],if and only if f(m) · g = f(m)[0]g(f(m)[1]) for every g ∈ R. But f(m) · g = f(m · g) =f(m[0]g(m[1])) = f(m[0])g(m[1]), since f is right R-linear and right A-linear. Using the α-condition, we find f(m)[0] ⊗A f(m)[1] = f(m[0]) ⊗A m[1]. It is easy to see that F is the inverse

of the composition of the inclusions MC ⊂ σ[CS ] ⊂ RMR.Since C is weaklyR-locally projective, C is also ∗C-locally projective. Furthermore, A∗C ⊂ ∗CA = ∗Csince ∗C is an A-bimodule. This means that we can apply our machinery for R = ∗C, and we findRM∗C

∼= σ[C∗C] ∼= MC.Conversely, it is proven in [3, Theorem 2.8] that σ[CS ] ∼= σ[C∗C] ∼= MC, implies that C is

weakly locally projective and S is dense in the finite topology on ∗C. By Theorem 2.59, this isequivalent to C being weakly R-locally projective as a left A-module.

Theorem 3.36. If ∗C ∈ RMR, then C is finitely generated and projective as a left A-module.In particular, the faithful functor MC → M∗C is an isomorphism of categories if an only if C isfinitely generated and projective as a left A-module.

Proof. If ∗C is a rational module, then we know that ∗C is a right C-comodule. Let ρ : ∗C →∗C⊗A C denote the right C-coaction of ∗C. Put ρ(ε) =

∑i fi ⊗A ci ∈ ∗C⊗A C. This means that

ε ∗ f = f =∑i

fi ∗ f(ci)


for all f ∈ R. Every right ∗C-module M is R-rational: for all m ∈M , we have

m · f =∑i

(m · fi)f(ci)

and this means thatρ(m) =

∑i

m · fi ⊗A ci.

In particular, for M = C, we find, for all c ∈ C:

ρ(c) = c(1) ⊗A c(2) =∑i

c · fi ⊗A ci =∑i

c(1)fi(c(2))⊗A ci.

Applying ε to the first factor, we find

c = ε(c(1))c(2) =∑i

ε(c(1)fi(c(2)))ci

=∑i

ε(c(1))fi(c(2))ci =∑i

fi(ε(c(1))c(2))ci

=∑i

fi(c)ci

and it follows that (ci, fi) | i = 1, · · · , n is a finite dual basis of C.

3.3.2. The rational functor.

Definition 3.37. Assume that C is weakly R-locally projective as a left A-module, and letM be a right T -module. We define the R-rational part of M as

RatR(M) = δ−1M,R(αM,R(M ⊗A C))

The rational part of M is meant to be the ∗C-rational part:

Rat(M) = Rat∗C(M)

Observe that m ∈ RatR(M) if and only if there exist mi ∈ M and ci ∈ C such that m · f =∑imi · f(ci) for all f ∈ R. We find that M is R-rational if and only if RatR(M) = M .

Proposition 3.38. Let R ⊂ ∗C be a subring, assume that C is weakly R-locally projectiveas a left A-module, and take M ∈M∗C.

(i) Let R′ be another subring of ∗C. If R ⊂ R′, then C is also R′-locally projective as a leftA-module, and

RatR′(M) ⊂ RatR(M) and RatR′(∗C) = RatR(∗C)

If M is R′-rational then M is also R-rational, ∗C is R′-rational if and only if ∗C is R-rational.(ii) RatR(M) is a right R-submodule of M . If AR ⊂ RA, then it is a right T -module, and

consequently it is the biggest T -submodule of M that is R-rational. RatR(M) is then alsoa C-comodule.

(iii) RatR(∗C) is a two-sided ideal of ∗C;(iv) RatR∗C is dense in ∗C with respect to the finite topology if and only if RatR(∗C) has a right

local units and it has local units on C, hence it has local units on all M ∈MC;(v) Let N be a right C-comodule and consider f ∈ HomR(N,M), then Im f ⊂ RatR(M).

Proof. (i) The first statement is obvious. Take g ∈ RatR(∗C). Then there exist gj ∈ ∗C andcj ∈ C such that

g ∗ h =∑j

gj ∗ i(h(cj))

for all h ∈ R. Now(g ∗ h)(c) = h(c(1)g(c(2)))

and ∑j

(gj ∗ i(h(cj)))(c) =∑j

gj(c)h(cj) =∑j

h(gj(c)cj),


so it follows that

h(c(1)g(c(2))−

∑j

gj(c)cj)

= 0

for all h ∈ R. C satisfies the α-condition for R, so we have a dual basis (ei, ri) | i ∈ I ⊂ C×R,and we find that

c(1)g(c(2))−∑j

gj(c)cj =∑i∈I

ri

(c(1)g(c(2))−

∑j

gj(c)cj)ei = 0

Applying an arbitrary f ∈ R′ to this equation, we find

g ∗ f =∑j

gj ∗ f(cj)

and it follows that g ∈ RatR′(∗C).(ii) For m ∈ RatR(M), there exists a unique m[0] ⊗A m[1] ∈M ⊗A C such that

m · f = m[0]f(m[1])

for all f ∈ R. For all f, g ∈ R, we then have

(m · f) · g = m · (f ∗ g)= m[0](f ∗ g)(m[1]) = m[0]g(m[1](1)f((m[1](2)))

and this means that m · f ∈ RatR(M), as needed. If A ∗R ⊂ R∗A, then the same argument asin the first part of the proof of Proposition 3.34 shows that

(ma) · f = m[0]f(m[1]a)

for all f ∈ R, hence ma ∈ RatR(M) if m ∈ RatR(M) and a ∈ A. Consequently RatR(M) ∈MT . It follows from Theorem 3.35 that RatR(M) is a right C-comodule.(iii). It follows from part (ii) that RatR(∗C) is a right C-comodule, hence a right ∗C-module. Let

us show that RatR(∗C) is also a left ideal in ∗C. For f ∈ RatR(∗C) and g, h ∈ ∗C, we have

(g ∗ f) ∗ h = g ∗ (f ∗ h)= g ∗ (f[0] ∗ h(f[1])) = (g ∗ f[0]) ∗ h(f[1])

It follows that (g ∗ f)[0] ⊗A (g ∗ f)[1] = g ∗ f[0] ⊗A f[1], and g ∗ f ∈ RatR(∗C).(iv). This is a consequence of part (iii), Corollary 2.48 and Proposition 2.47. (v). Since N is aright C-comodule, N is R-rational. By Proposition 3.29 the image of f is also R-rational, henceIm f ⊂ RatR(M).

By the previous observations, we can define a functor

RatR : MT → RMR,

If we consider moreover the inclusion functor J : RMR → MT . Then (J,RatR) is an adjointpair of functors. The unit ν and counit ζ are given by

νM : M → RatR(M)ζN : RatR(N) → N

for all M ∈ RMR and N ∈ MT , where νM is an isomorphism since RatR(M) = M and ζN isthe canonical injection. Since the unit ν of this adjunction is an isomorphism, we recover that thefunctor J is fully faithful. Furthermore, if we consider the isomorphism of categories MC ∼= RMRthen we obtain the pair of adjoint functors (J ′,RatR)

MT

RatR //MC

J ′oo

where the functor J ′ is again fully faithful.

3.4. COMATRIX CORINGS 95

3.4. Comatrix corings

3.4.1. Comatrix corings over firm rings. In Theorem 1.30 we stated that in a generalbicategory, we can associate to any adjoint pair a monad and a comonad. In Section 2.3.1 wespecified this theorem for the bicategory Frm(k), of firm k-algebras, firm bimodules and bilinearmaps, stating that a comatrix coring context (A,R,M,M †, η, ε) between the firm rings A and Rgives rise to a comatrix coring M ⊗R M †. In this Section we will give the explicit description ofthis construction.

Let (Σ,Σ′, µ) be a dual pair over the rings B and A. Consider a firm B-ring R and a non-unital ring morphism ι : R→ Σ⊗A Σ′. Then Σ becomes a left R-module by restriction of scalars.With these notations, we have the following theorem that generalizes the construction of comatrixcorings given in [63].

Theorem 3.39. If RΣ is firm as a left R-module. Then Σ′ ⊗R Σ is an A-coring with comul-tiplication

∆ : Σ′ ⊗R ΣΣ′⊗RdR,Σ // Σ′ ⊗R R⊗R Σ

Σ′⊗Rι⊗RΣ // Σ′ ⊗R Σ⊗A Σ′ ⊗R Σ

and counit

µ : Σ′ ⊗R Σ // A .

Moreover, Σ is an R-Σ′ ⊗R Σ bicomodule, and Σ† = Σ′ ⊗R R is a Σ′ ⊗R Σ-R bicomodule withthe coactions

ρΣ : ΣdR,Σ // R⊗R Σ

ι⊗RΣ // Σ⊗A Σ′ ⊗R Σ

and

λΣ† : Σ′ ⊗R RΣ′⊗RdR // Σ′ ⊗R R⊗R R

Σ′⊗Rι⊗RR // Σ′ ⊗R Σ⊗A Σ′ ⊗R R .

Proof. For φ⊗R u ∈ Σ′ ⊗R Σ we have

(77) ∆(φ⊗R u) = φ⊗R er ⊗A ϕr ⊗R ur

where we denote ι(r) = er ⊗A ϕr and dR,Σ(u) = r ⊗R ur. Therefore,

(78)

(Σ′ ⊗R Σ⊗A ∆)∆(φ⊗R u) = (Σ′ ⊗R Σ⊗A ∆)(φ⊗R er ⊗A ϕr ⊗R ur)= φ⊗R er ⊗A ϕr ⊗A et ⊗A ϕt ⊗R urt= (Σ′ ⊗R ι⊗R ι⊗R Σ)(φ⊗R r ⊗R t⊗R urt)= (Σ′ ⊗R ι⊗R ι⊗R Σ)(φ⊗R srs ⊗R t⊗R urt)= (Σ′ ⊗R ι⊗R ι⊗R Σ)(φ⊗R s⊗R rst⊗R urt)= (Σ′ ⊗R ι⊗R ι⊗R Σ)(φ⊗R s⊗R rs ⊗R turt)= (Σ′ ⊗R ι⊗R ι⊗R Σ)(φ⊗R s⊗R rs ⊗R ur).

On the other hand,

(∆⊗A Σ′ ⊗R Σ)∆(φ⊗R u) = (∆⊗A Σ′ ⊗R Σ)(φ⊗R er ⊗A ϕr ⊗R ur)= φ⊗R es ⊗A ϕs ⊗R ers ⊗A ϕr ⊗R ur.

So it follows that ∆ is coassociatative if we can show that

(79) (ι⊗R ι)(s⊗R rs) = es ⊗A ϕs ⊗R ers ⊗A ϕr.To prove (79) let µR,Σ : R ⊗R Σ → Σ be the left action map, and µR : R ⊗R R → R themultiplication map. It follows from (51) that

µZ′ (ι⊗R Σ′ ⊗A Σ) = µR,Σ ⊗A Σ.

Since ι is an algebra map, we deduce

(80) ι µR = (µR,Σ ⊗A Σ′) (R⊗R ι).Now, both R and Σ are firm, which allows us to rewrite (80) as

(dR,Σ ⊗A Σ′) ι = (R⊗R ι) dR,

from which (79) follows after composition with ι ⊗R Σ ⊗A Σ′ on the left. The counit property

follows easily from (51) and (52). The arguments to prove that ρΣ and λΣ† are comodule structure


maps are similar. To check this, it is useful to use the following expressions for the right and leftcoactions:

(81) ρΣ(u) = er ⊗A ϕr ⊗R ur

andλΣ†(φ⊗R r) = φ⊗R es ⊗A ϕs ⊗R rs

Obviously, they are R-linear.

Corollary 3.40. Let (Σ,Σ′, µ) be a dual pair over the rings B and A and Z ′ = Σ ⊗A Σ′

be the elementary B-ring. Consider a firm B-ring R and a non-unital ring morphism ι : R→ Z ′.Then Σ′ ⊗R R⊗R Σ is an A-coring with comultiplication ∆ and counit ε given by

Σ′ ⊗R R⊗R Σ

Σ′⊗Rd2R⊗RΣ **UUUUUUUUUUUUUUUU

∆ // Σ′ ⊗R R⊗R Σ⊗A Σ′ ⊗R R⊗R Σ

Σ′ ⊗R R⊗R R⊗R R⊗R ΣΣ′⊗RR⊗Rι⊗RR⊗RΣ

33ggggggggggggggggggggg

µ : Σ′ ⊗R R⊗R Σ → A, µ(ϕ⊗R r ⊗R u) = µ(ϕ⊗R ru)Moreover Σ⊗RR is a R-Σ′⊗RR⊗RΣ bicomodule and R⊗RΣ′ is a Σ′⊗RR⊗RΣ-R bicomodule.

Proof. We can consider the firm left R-module Σ = R⊗RΣ. Then (Σ,Σ, µ) is again a dualpair. Furthermore, consider the map

ι : RdR // R⊗R R

R⊗Rι // R⊗R Σ⊗R Σ′ = Σ⊗R Σ′

Then the R-action on Σ is identical with the action on Σ induced by ι. This means that allconditions of Theorem 3.39 are satisfied, and the statement follows.

Corollary 3.41. Let A and B be two (unital) rings and R a firm B-ring.

(i) Let Σ be an R-firmly projective right A-module, then Σ∗ ⊗R Σ is an A-coring with comulti-plication and counit given by

∆ : Σ∗ ⊗R ΣΣ∗⊗RdR,Σ // Σ∗ ⊗R R⊗R Σ

Σ∗⊗Rι⊗RΣ // Σ∗ ⊗R Σ⊗A Σ∗ ⊗R Σ

ε : Σ∗ ⊗R Σ ev // A

where we denoted ι : R→ Σ⊗A Σ∗ for the associated ring morphism.(ii) Let (Σ,Σ†, µ) be an R-firm dual pair, then Σ†⊗RΣ is an A-coring with comultiplication and

counit given by

∆ : Σ† ⊗R ΣΣ†⊗RdR,Σ

∼=dΣ†,R

⊗RΣ// Σ† ⊗R R⊗R Σ

Σ†⊗Rι†⊗RΣ // Σ† ⊗R Σ⊗A Σ† ⊗R Σ

ε : Σ† ⊗R Σµ // A

where we denoted ι† : R→ Σ⊗A Σ† for the associated ring morphism.(iii) The bijective correspondence of Proposition 2.50 between R-firmly projective modules and

R-firm dual pairs induces an isomorphism of A-corings

Σ∗ ⊗R Σ ∼= Σ† ⊗R Σ.

Proof. (i). Consider the canonical dual pair (Σ,Σ∗, ev). By definition of the R-firmlyprojectivity of Σ, there exists a non-unital ring morphism ι : R → Σ ⊗A Σ∗ such that Σ isfirm as a left R-module under the action induced by ι. The construction follows now immediatelyby Theorem 3.39.(ii). By definition an R-firm dual pair, (Σ,Σ†, µ) satisfies the conditions of Theorem 3.39 and we

find that Σ† ⊗R Σ is an A-coring with structure maps as given in the statement of the theorem.(iii) Recall from Proposition 2.50 that Σ† ∼= Σ∗⊗RR as A-R bimodules. We define α = Σ∗⊗RµR,Σand β(ψ ⊗R u) = ζ(ψr)⊗R r ⊗R u for ψ ⊗R u ∈ Σ† ⊗R Σ

f : Σ† ⊗R Σβ // Σ∗ ⊗R R⊗R Σ α // Σ∗ ⊗R Σ


then f(ψ ⊗R u) = ζ(ψ)⊗R u. Conversely,

f−1 : Σ∗ ⊗R Σα−1// Σ∗ ⊗R R⊗R Σ

β−1

// Σ† ⊗R Σ

where α−1(ϕ⊗R u) = ϕ⊗R r ⊗R ur and β−1(ϕ⊗R r ⊗R u) = ϕ(er)ϕr ⊗R u = ϕr ⊗R u for allϕ ⊗R u ∈ Σ∗ ⊗R u and ϕ ⊗R r ⊗R u ∈ Σ∗ ⊗R R ⊗R Σ. To prove that f is a homomorphism ofA-corings we need to check that

(82) ζ(ϕ)⊗R es ⊗A fs ⊗R us = ζ(ϕ)⊗R es ⊗A ζ(fsrs)⊗R ur

From the right R-linearity of ζ, we immediately obtain that

ζ(ϕ)⊗R es ⊗A ζ(fsrs)⊗R ur = ζ(ϕ)⊗R es ⊗A fs ⊗R rsur

A similar computation as in (78) shows that s⊗R rs ⊗R ur = s⊗R t⊗R urt and consequently

ζ(ϕ)⊗R es ⊗A fs ⊗R rsur = ζ(ϕ)⊗R es ⊗A fs ⊗R tust = ζ(ϕ)⊗R es ⊗A fs ⊗R us

and f is an isomorphism of corings.

Definition 3.42. We refer to the A-coring Σ† ⊗R Σ as the comatrix coring associated tothe R-firm dual pair (Σ,Σ†, µ).

If (Σ,Σ′, µ) is a dual pair, R a firm B-ring, ι : R → Σ ⊗A Σ′ a non-unital ring morphismsuch that RΣ is a firm module with action induced by ι. Then Σ is R-firmly projective as aright A-module (see Proposition 2.53) and (Σ,Σ†, µ†) is a firm dual pair with Σ† = Σ′ ⊗R R andµ† = µ (Σ′ ⊗R µR,Σ).

Proposition 3.43. With notations and under the conditions described above, the A-coringsconstructed in Theorem 3.39 and Corollary 3.41 are isomorphic.

Σ′ ⊗R Σ ∼= Σ∗ ⊗R Σ ∼= Σ† ⊗R Σ.

Proof. Let us define f : Σ′ ⊗R Σ → Σ∗ ⊗R Σ, f(ϕ ⊗R u) = ζ(ϕ) ⊗R u with inversef−1 : Σ∗ ⊗R Σ → Σ′ ⊗R Σ, f−1(ϕ ⊗R u) = ϕ(er)ϕr ⊗R ur where we make use of the mapι′ : R → Σ ⊗A Σ′, ι′(r) = er ⊗A ϕr. Then this can be proven to be an isomorphism of coringsin the same way as in Corollary 3.41, part (iii). The second isomorphism is a direct application ofCorollary 3.41, part (iii).

Remark 3.44. Starting from µ : Σ′ ⊗B Σ → A and ι : R→ Σ⊗A Σ′ and assuming Σ′ to befirm as a right R-module, an A-coring structure is defined on Σ′ ⊗R Σ whose comultiplication isgiven by

∆′(φ⊗R u) = φr ⊗R er ⊗A ϕr ⊗R u,and with counit µ : Σ′ ⊗R Σ → A. A straightforward version of Theorem 3.39 shows that Σ′ is aleft comodule, with coaction

(83) λΣ′(φ) = φr ⊗R er ⊗A ϕrand †Σ = R⊗R Σ is a right comodule, with coaction

ρ†Σ(r ⊗R u) = s⊗R ers ⊗A ϕrs ⊗R u,

where r = srs. Here, RΣ is not assumed to be firm.

When both modules RΣ and Σ′R are firm, it is equivalent to construct the comatrix coringusing the “firmness” of Σ or Σ′, as the following proposition establishes.

Proposition 3.45. If RΣ and Σ′R are assumed to be firm, then ∆ = ∆′, and the isomorphismdΣ′,R : Σ′ ∼= Σ′ ⊗R R = Σ† (resp. dR,Σ : Σ ∼= R ⊗R Σ = †Σ) is an isomorphism of Σ′ ⊗R Σ-Rbicomodules (resp. of R-Σ′ ⊗R Σ bicomodules).

Proof. To prove that ∆ = ∆′ it is enough to check that for every φ ⊗R u ∈ Σ′ ⊗R Σ onehas

φr ⊗R r ⊗R u = φ⊗R r ⊗R ur.


This equality follows by applying the isomorphism Σ′ ⊗R µR,Σ = µΣ′,R ⊗R Σ to both sides. Toshow that dΣ′,R is an isomorphism of left Σ′ ⊗R Σ-comodules we have to check the equality

(84) φr ⊗R es ⊗A ϕs ⊗R rs = φr ⊗R er ⊗A (ϕr)s ⊗R s.

Applying the isomorphism Σ′ ⊗R Σ⊗A µΣ′,R to both sides of (84), we obtain

φr ⊗R es ⊗A ϕsrs = φr ⊗R er ⊗A (ϕr)ss,

which holds by (51). An analogous argument shows that dR,Σ is an isomorphism of R-Σ′ ⊗R Σbicomodules.

Remark 3.46. (1) The isomorphism of A-R bimodules αA : Σ∗ ⊗R R ∼= A ⊗A Σ′ ⊗RR ∼= Σ′ ⊗R R deduced from Proposition Proposition 2.53 induces a natural isomorphismbetween the functors −⊗AΣ∗⊗RR and −⊗AΣ′⊗RR : MA →MR. This, together withProposition Proposition 3.43 allows to replace any given dual pair (Σ,Σ′, µ) satisfyingthe equivalent conditions of Proposition 2.53 by its canonical R-firm dual pair (Σ,Σ∗, ev)in our theory.

(2) If Σ is R-firmly projective as a right A-module, then Σ† = Σ∗⊗RR is R-firmly projectiveas a left A-module and (Σ,Σ†, µ) is an R-firm dual pair. We can then construct thecomatrix coring Σ⊗RΣ† associated to Σ and the comatrix coring †(Σ†)⊗R Σ† associatedto Σ†. Since Σ ∼= †(Σ†) by Theorem 2.51, Proposition 3.45 implies that both corings areisomorphic.

Example 3.47. Let B → A be a ring morphism. Then A becomes a B-ring and we canconsider the dual pair (A,A, µ), with µ : A⊗B A→ A. Furthermore we can consider a morphismof B-rings ι : B → A⊗B A, ι(b) = b⊗B 1 = 1⊗B b. Then by Theorem 3.39 we find an A-coringstructure on A⊗B A. One can easily verify that this is precisely the Sweedler coring presented inExample 3.3.

Example 3.48. If Σ is a B-A bimodule such that ΣA is finitely generated and projective,and (ei, e∗i ) ⊆ Σ × Σ∗ is a finite dual basis for Σ, then the canonical ring homomorphismι : B → Σ ⊗A Σ∗ given by ι(b) = bei ⊗A e∗i = ei ⊗A e∗i b (sumation is understood implicitly)induces on Σ ⊗B Σ∗ the structure of an A-coring, in virtue of Theorem 3.39. This coring isprecisely the coring constructed in Theorem 2.80. These corings were introduced in [63], and arecalled finite comatrix corings.

Example 3.49. Let P be a set of finitely generated and projective right modules over a ringA, and consider Σ =

⊕P∈P P . For each P ∈ P let ιP : P → Σ and πP : Σ → P be the canonical

injection and projection, respectively. Then the elements uP = ιPπP with P ∈ P form a set oforthogonal idempotents in EndA(Σ). Consider any homomorphism of rings T → EndA(Σ), andlet R =

∑P,Q∈P uPTuQ, which is a (non unital) subring of the image of T in EndA(Σ). Consider

the unique homomorphism of rings ι : R → Σ ⊗A Σ∗ defined on each uPTuQ by the compositemap

uPTuQ // HomA(Q,P ) ' // P ⊗A Q∗ ⊆ Σ⊗A Σ∗,

where the first map assigns πP fιQ to every f ∈ uPTuQ. Clearly the ring R has enough idempo-tents, which implies that R is a firm ring. Moreover, Σ is easily shown to be a firm left R-module,with the left R-action given by the restriction of its canonical structure of left T -module. There-fore, the comatrix A-coring Σ∗ ⊗R Σ stated in Theorem 3.39 makes sense. By this constructionwe recover the infinite comatrix corings from [64].

An explicit expression for the comultiplication of Σ∗ ⊗R Σ in this case is obtained as follows.For each P ∈ P, let eαP , e

∗αP be a finite dual basis for PA. Then we obtain the identity

ι(uP ) =∑αP

ιP (eαP )⊗R e∗αPπP ,

from which, in conjunction with (77), we easily deduce, for φ ∈ Σ∗ and x ∈ Σ, that

∆(φ⊗R x) =∑P∈F

φ⊗R ιP (eαP )⊗A e∗αPπP ⊗R x,


where F is a finite subset of P such that x =∑

P∈F ePx. The Σ∗ ⊗R Σ-comodule structures of

Σ and Σ† = Σ⊗R R are then given by

ρΣ(x) =∑P∈F

ιP (eαP )⊗A e∗αPπP ⊗R x,

andλΣ†(φ⊗R r) =

∑Q∈G

φ⊗R ιQ(eβQ)⊗A e∗αQ

πQ ⊗R r,

respectively. In the second case, G is a finite subset of P such that r =∑

Q∈P eQr.

Example 3.50. Let Σ be a B-A bimodule that is weakly M∗-locally projective as B-Abimodule. Take now a set of generators U for ΣA, and let E be a set consisting of local dual basesfor each of the elements of U (i.e. E ⊂ Z = Σ⊗A Σ∗ is a complete set of local units for Σ). LetR be the sub B-ring of Σ ⊗A Σ∗ generated by the elements of E. By construction, R acts withlocal units on Σ and thus Σ is a firm left R-module. Clearly, R is a ring with left local units and,therefore, it is a firm ring. It follows now from our general theory (Theorem 3.39) that Σ∗ ⊗R Σis an A-coring. The counit and comultiplication are given explicitly by the following formulas

ε(ϕ⊗R u) = ϕ(u)

and∆(ϕ⊗R u) = ϕ⊗R ei ⊗A fi ⊗R u = ϕ⊗R e⊗R u,

where ϕ ∈ Σ∗ and e = ei⊗A fi is a dual basis for u ∈ Σ. That this comultiplication is well-definedcan be also checked directly as follows. If e′ is another dual basis for u, let then e′′ be a local unitfor both e and e′, i.e. e′′e = e and e′′e′ = e′. We compute

ϕ⊗R e⊗R u = ϕ⊗R e′′e⊗R u = ϕ⊗R e′′ ⊗R eu= ϕ⊗R e′′ ⊗R u = ϕ⊗R e′′ ⊗R e′u= ϕ⊗R e′′e′ ⊗R u = ϕ⊗R e′ ⊗R u.

We can recover the same coring as well from the following construction. Let ΣA be again aB-A bimodule that is weakly M∗-locally projective as B-A bimodule. As in Remark 2.79 we canconstruct a coring with local comultiplications Σ∗⊗BΣ. To every element e ∈ E, we can associatea comultiplication

∆e : Σ∗ ⊗B Σ → Σ∗ ⊗B Σ⊗A Σ∗ ⊗B Σ, ∆e(ϕ⊗B u) = ϕ⊗B e⊗B u.Furthermore, since E was a complete set of local units in Z, the set of local comultiplicationsassociated to E is also complete. Then Theorem 2.81 allows us to construct a usual coring C, asthe multiple coequalizer of this complete set of comultiplications. It is not hard to see that

C ∼= Σ∗ ⊗R Σ.

Example 3.51. We will construct examples over rings with idempotent local units in Sec-tion 3.5. These can be interpreted as the intermediate case of Examples 3.49 and 3.50.

Example 3.52. Let ι : B → A be a separable ring extension, i.e. there exists a ring morphismE : A→ B such that E ι = B. Then we know from Example 2.12 that S := A⊗B A is a firmring with multiplication and comultiplication

π(a⊗B a′ ⊗A a′′ ⊗B a′′′) = a⊗B E(a′a′′)a′′′, ∆(a⊗B a′) = a⊗B 1⊗A 1⊗B a′.Consider now an A-B bimodule Σ, a B-A bimodule Σ′ and a B-bilinear map µ : Σ′ ⊗A Σ → B,i.e. (Σ,Σ′, µ) is a dual pair over A and B. Consider the associated elementary A-ring Σ ⊗B Σ′.Since S is generated as an A-bimodule by the element 1⊗B 1, any morphism of A-rings j : S =A⊗B A→ Σ⊗B Σ′ is completely determined by a single element j(1⊗B 1) = ei⊗B fi ∈ Σ⊗B Σ′

such that aei ⊗B fi = ei ⊗B fia for all a ∈ A. Assuming that such a morphism j exists, wecan apply Corollary 3.40, and we find that Σ′ ⊗S S ⊗S Σ = Σ′ ⊗S A ⊗B A ⊗S Σ is a B-coring.Comultiplication and counit are given explicitly by

∆(ϕ⊗S a⊗B a′ ⊗S u) = ϕ⊗S a⊗B 1⊗S ei ⊗B fi ⊗S 1⊗B a′ ⊗S uε(ϕ⊗S a⊗B a′ ⊗S u) = µ(ϕ⊗B aei)µ(fi ⊗B a′u)


If we consider the trivial situation where ι = A : A→ A is an isomorphism, then we recover againthe finite comatrix corings from Example 3.48.

3.4.2. Locally finite duality. Let C be an A-coring and R a firm ring. We denote by MCRfp

the category of all right C-comodules that are R-firmly projective as a right A-module, such thatΣ ∈ RMC. If Σ ∈ MC

Rfp, than there exists a ring morphism ι : R → Σ ⊗A Σ∗, such that Σ is afirm R-module under the R-action induced by ι and Σ becomes an R-C bicomodule. In the sameway, we introduce the category C

RfpM. With this notation, we have the following proposition.

Proposition 3.53. Let C be an A-coring and R a firm ring. there exists a pair of inverseequivalences between the categories MC

Rfp and CRfpM.

Proof. This is a reformulation of Proposition 1.35 in the bicategory Mk taking a D fixed asthe firm ring R. Explicitly, take Σ ∈MC

Rfp. We know from Proposition 2.50 that Σ† := Σ∗ ⊗R Ra R-firmly projective is as a left A-module. The left C-coaction λΣ† on Σ† is defined as follows

Σ∗ ⊗R RλΣ†

//

∼=

C⊗R Σ∗ ⊗R R

Σ∗ ⊗R R⊗R R

Σ∗⊗Rι⊗RR

A⊗A C⊗R Σ∗ ⊗R R

∼=

OO

Σ∗ ⊗R Σ⊗A Σ∗ ⊗R RΣ∗⊗Rρ

Σ⊗AΣ∗⊗RR

// Σ∗ ⊗R Σ⊗A C⊗R Σ∗ ⊗R R

ev⊗AC⊗RΣ∗⊗RR

OO

Taking Λ ∈ CRfpM, we find in the same way †Λ ∈ MC

Rfp. Finally, it follows from Theorem 2.51

(iii) that †(Σ†) ∼= Σ, hence both constructions are mutual inverses.

Corollary 3.54. Let C be an A-coring and R a firm ring. To any Σ ∈ MCRfp we can

associate two pairs of adjoint functors (GΣ,HΣ) and (ΣG,ΣH) as follows,

MR

GΣ=−⊗RΣ //MC

HΣ=HomC(Σ,−)⊗RR

oo RMΣG=Σ†⊗R− // CM

ΣH=R⊗RCHom(Σ†,−)

oo

where Σ† = Σ∗ ⊗R R ∈ CRfgM.

Proof. This follows from Corollary 3.10 and its left-handed version, in combination withProposition 3.53.

Let C be again an A-coring. Then MCfgp will be the notation for the category of right C-

comodules that are finitely generated and projective as a right A-module. In the same way, CfgpM

denotes the category of left C-comodules that are finitely generated and projective as a left A-module

Corollary 3.55. Let C be an A-coring. We have a pair of inverse equivalences between thecategory MC

fgp and CfgpM. Furthermore let B be any unital ring, then to any Σ ∈ BMC

fgp we canassociate two pairs of adjoint functors

MB

GΣ=−⊗RΣ //MC

HΣ=HomC(Σ,−)

oo BMΣG=Σ∗⊗B− // CM

ΣH=CHom(Σ∗,−)

oo

Proof. This follows from Proposition 3.53 and Corollary 3.54 in combination with the factthat R-firmly projective modules are finitely generated and projective if R is a ring with unit (seeTheorem 2.64).

For sake of completeness, let us give the explicit formula of the left C-comodule action of Σ∗

if Σ ∈MCfgp. Let

∑i ei⊗A fi ∈ Σ⊗AΣ∗ be a finite dual basis for Σ, then we have for all f ∈ Σ∗

(85) λΣ∗(f) =∑i

f(ei[0])ei[1] ⊗A fi.


Suppose C is finitely generated and projective as a left A-module. Then ∗C is a right C-comodule. If fi ⊗A ci ∈ ∗C⊗A C is a dual basis for C, then we find that the left right C-coactionof ∗C is given by

ρ(f) =∑i

fi ⊗A ei · f =∑i

fi ⊗A ei(1)f(ei(2))

=∑ij

fi ⊗A fj(ei(1)f(ei(2)))ej =∑ij

fi ⊗A (f ∗ fj)(ei)ej

=∑ij

fi(f ∗ fj)(ei)⊗A ej =∑j

f ∗ fj ⊗A ej

for all f ∈ ∗C. So we obtain two equivalent formulas for the right C-coaction of ∗C,

(86) ρ(f) =∑i

fi ⊗A ei · f =∑i

f ∗ fi ⊗A ei

and in particular,∑

i fi ⊗A ei commutes with all f ∈ ∗C, i.e. it is a so-called Casimir element.

3.4.3. Comatrix corings and cotriples coming from adjunctions. Recall that the con-structions of the previous section are a special case of Theorem 1.30 in the bicategory Frm(k) offirm rings. Theorem 1.30 can of course be applied as well in the bicategory CATof categories,functors and natural transformations.

Let A and B be two categories, F : B → A and G : A → B two covariant functors, such that(F,G) is an adjunction, i.e. there exist natural transformations η : B ⇒ GF and ε : FG ⇒ A,which are respectively called the unit and the counit of the adjuction and which satisfy

(ε ∗ F ) (F ∗ η) = F, (G ∗ ε) (η ∗G) = G.

By Theorem 1.30, we can construct a triple (GF,GεF, η) on B and a cotriple (FG,FηG, ε) onA.

If C = (C, δ, ε) is a cotriple on A, then by Proposition 3.8 there exists a pair of adjoint functors

GC : A → AC ; GC(X) = C(X);FC : AC → A; FC(X) = X.

The proof of the next result is an easy exercise.

Proposition 3.56. Let C be a cotriple on the category A. The cotriple on A associated tothe adjoint pair (FC ,GC) is exactly the given cotriple C.

Moreover, if one starts with an adjunction (F,G) where F : B → A and G : A → B, constructsa cotriple C = FG out of this adjunction, then we can compare the original adjunction with thenewly obtained adjunction between A and AC , as we can see from the following well-knowntheorem.

Proposition 3.57. Let (F,G) be a pair of adjoint functors between the categories B andA and denote by C = FG the cotriple associated to this adjunction. Then there exists a uniquefunctor K : B → AC such that F = FCK and KG = GC .

BF //

K &&NNNNNNNNNNNNN AG

oo

GC

AC

FC

OO

Proof. The functor K can be defined as K(B) = (G(B), ηG(B)) for all B ∈ B, where η isthe unit of the adjunction (F,G).

Consider now a pair of adjoint functors (F,G)

MR

F //MAG

oo

such that G preserves colimits. Then C = FG is a cotriple on MA. Moreover, since F has a rightadjoint, it preserves colimits so C = FG preserves colimits as well. Consequently, by Theorem 3.5


C = −⊗A C for an A-coring C. By Theorem 2.51 we know that there exists an R-A bimodule Σand an A-R bimodule Σ† such that F ∼= −⊗R Σ and G ∼= −⊗A Σ† and there exists a comatrixcoring context (R,A,Σ,Σ†, η, ε). Thus we can construct the comatrix coring Σ† ⊗R Σ.

Theorem 3.58. With notation introduced above, the A-coring C associated to the cotripleC is isomorphic to the comatrix coring Σ† ⊗R Σ associated to the comatrix coring context(R,A,Σ,Σ†, η, ε).

Proof. The isomorphism is induced by Theorem 2.13

C ∼= C(A) = FG(A) ∼= A⊗A Σ† ⊗R Σ.

We leave it to the reader to verify that this is an isomorphism of A-corings.

3.5. Corings from colimits

3.5.1. Corings from colimits. Let Z be a (small) category and let (M,⊗, A) be a monoidalcategory. Then (Func(Z,M),⊗, A) is also a monoidal category. The tensor ⊗ and the unit Aare given by the following formulas:

(F ⊗G)(Z) = F (Z)⊗G(Z) and (F ⊗G)(f) = F (f)⊗G(f);

A(Z) = A and A(f) = A,

for all F,G : Z →M, Z,Z ′ ∈ Z and f : Z → Z ′ in Z. A coalgebra in (Func(Z,M),⊗, A) willbe called a Z-coalgebra in M. The result of this Section is the following.

Proposition 3.59. Let (G,∆, ε) be a Z-coalgebra in M, and assume that colimG = (C, c)exists. Then C is a coalgebra in M.

Proof. We give a proof of the statement in the case where M is a strict monoidal category.Recall that this is no restriction since every monoidal category is equivalent to a strict monoidalcategory, see Theorem 1.5 and Corollary 1.6.

For every Z ∈ Z, consider the morphism

dZ = (cZ ⊗ cZ) ∆Z : G(Z) → C ⊗ C.

Let f : Z → Z ′ be a morphism in Z, and look at the diagram

G(Z)

G(f)

∆Z // G(Z)⊗G(Z)cZ⊗cZ //

G(f)⊗G(f)

C ⊗ C

G(Z ′)∆Z′// G(Z ′)⊗G(Z ′)

cZ′⊗cZ′// C ⊗ C

The left hand square commutes since ∆ : G→ G⊗G is a natural transformation, and the righthand square commutes because (C, c) is a cocone on G. It follows that (C ⊗C, d) is a cocone onG, and we conclude that there exists a morphism ∆C : C → C ⊗ C in M such that

∆C cZ = dZ = (cZ ⊗ cZ) ∆Z ,

for all Z ∈ Z. We then have

(∆C ⊗ C) ∆C cZ = (∆C ⊗ C) (cZ ⊗ cZ) ∆Z

= (cZ ⊗ cZ ⊗ C) (∆Z ⊗ cZ) ∆Z

= (cZ ⊗ cZ ⊗ cZ) (∆Z ⊗G(Z)) ∆Z

= (cZ ⊗ cZ ⊗ cZ) (G(Z)⊗∆Z) ∆Z

= (C ⊗∆C) ∆C cZ ,for all Z ∈ Z. It follows (see [25, Prop. 2.6.4]) that (∆C ⊗ C) ∆C = (C ⊗∆C) ∆C , so ∆C

is a coassociative comultiplication on C.The counit is defined in a similar way: (A, ε) is a cocone on G, so there exists a morphismεC : C → A in M such that εC cZ = εZ , for all Z ∈ Z. The counit property is verified asfollows: for all Z ∈ Z, we have

(εC ⊗ C) ∆C cZ = (εC ⊗ C) (cZ ⊗ cZ) ∆Z = (A⊗ cZ) (εZ ⊗G(Z)) ∆Z = cZ .

3.5. CORINGS FROM COLIMITS 103

Proposition 3.60. Let (G,∆, ε) be as in Proposition 3.59. If (H, ρ) is a right G-comodule,and colimH = (M,m) exists, then M is a right C-comodule.

Proof. For every Z ∈ Z, consider the composition

rZ = (mZ ⊗ cZ) ρZ : H(Z) → H(Z)⊗G(Z) →M ⊗ C.

Arguments similar to the ones presented above show that (M ⊗C, r) is a cocone on H. It followsthat there exists a morphism ρM : M → M ⊗ C such that ρM mZ = rZ , for every Z ∈ Z.Standard computations show that ρM is coassociative and satisfies the counit property.

3.5.2. Colimit comatrix corings. Let R be a B-ring with idempotent local units and A aB-ring with unit. Let P be a firm R-A bimodule. Then P satisfies the conditions of Lemma 2.41.In particular, there exists a split direct system P s : Z → (BMA)s such that colimP = (P, σ)and Pi is finitely generated and projective as a right A-module. Recall from Proposition 2.5 andLemma 2.43 the construction of the dual of the split direct system P ∗ with colimP ∗ = (P †, τ∗)such that P † is a firm A-R bimodule.

Proposition 3.61. We have a directed system G : Z → AMA, G(i) = P ∗i ⊗Ri Pi, and

G(aji) : P ∗i ⊗Ri Pi → P ∗j ⊗Rj Pj , G(aji)(ϕi ⊗Ri pi) = ϕi τij ⊗Rj σji(pi).

Proof. We first show that G(aji) is well-defined. For all ϕi ∈ P ∗i , pi ∈ Pi and bi ∈ Ri, wehave

G(aji)(ϕi ⊗Ri bi · pi) = ϕi τij ⊗Rj σji(bipi)(38)= τ∗ij(ϕi)⊗Rj βji(bi)σji(pi) = τ∗ij(ϕi)βji(bi)⊗Rj σji(pi)

(42)= τ∗ij(ϕibi)⊗Rj σji(pi) = G(aji)(ϕibi ⊗Ri pi).

If i ≤ j ≤ k, then we have

(G(akj) G(aji))(ϕi ⊗Ri ti · pi) = ϕi τji τjk ⊗Rk(σkj σji)(pi)

= ϕi τik ⊗Rkσki(pi) = G(aki)(ϕi ⊗Ri pi).

Let Ei =∑

i zi ⊗A z∗i be a left B-linear finite dual basis of Pi ∈ MA. Then Ei is the uniqueelement of Pi ⊗A P ∗i satisfying the formulas

(87) pi =∑

ziz∗i (pi); ϕi =

∑z∗i ϕ(zi),

for all pi ∈ Pi and ϕi ∈ P ∗i . With this notation, we have the following lemma.

Lemma 3.62. (i) For all bi ∈ Ri,

(88)∑

bizi ⊗A z∗i =∑

zi ⊗A z∗i bi,

(ii) If i ≤ j, then

(89) Ei =∑

τij(zj)⊗A z∗j σji = eizj ⊗A zj|Pi.

Proof. (i). This follows from the fact that Pi is a (Ri, A)-bimodule.

(ii). We show that the right hand side of (89) satisfies (87). For all pi ∈ Pi, we have∑τij(zj)(z∗j σji)(pi) =

∑τij

(zjz

∗j

(σji(pi)

))= τij(σji(pi)) = pi.

For the remaining part of this Section, we will concentrate on modules that are strongly locallyprojective. We will need a more restrictive characterisation than Lemma 2.41.

Lemma 3.63. The following statements are equivalent

(i) P satisfies the equivalent condtions of Lemma 2.41 and in addition Pi is finitely generatedand projective as a right A-module for all i ∈ I and colimP ∗ = (P †, τ †).


(ii) S = P ⊗A P † is a ring with idempotent local units, P is a firm left S-module, P † is afirm right S-module and there exists a unique morphism of rings with idempotent local unitsη : R→ P ⊗A P † such that

(90) η(βi(bi)) =∑

σi(bizi)⊗A z∗i τi =∑

σi(zi)⊗A z∗i bi τi,

for all i ∈ I, bi ∈ Ri.(iii) P is strongly P †-locally projective as a right A-module and P † is strongly P -locally projective

as a left A-module. P is a firm left R-module and P † is a firm right R-module.

Proof. (i) ⇒ (ii) By Lemma 2.3 P ∗s is a split direct system and obviously P ∗i is finitely

generated and projective as a left A-module for every i ∈ I. The first part of statement (ii)follows now from Corollary 2.68. The second equality in (90) is an immediate consequence of (88).Let us show that η is well-defined. Take b ∈ R, and assume that b = βi(bi) = βj(bj), for somei, j ∈ I, bi ∈ Ri, bj ∈ Rj . Take k ≥ i, j, and let bk = βki(bi) = βkj(bj). We compute∑

σi(bizi)⊗A z∗i τi(89)=∑

(σk σki)(biτik(zk))⊗A z∗k σki τik τk(35,41)

=∑

σk(βki(bi)zk)⊗A (z∗kβki(1Ri)) τk(88)=

∑σk(βki(1Ri)βki(bi)zk)⊗A z∗k τk

=∑

σk(βki(1Ribi)zk)⊗A z∗k τk

=∑

σk(bkzk)⊗A z∗k τk.

In a similar way, we prove that∑σj(bjzj)⊗A z∗j τj =

∑σk(bkzk)⊗A z∗k τk,

and it follows that the right hand side of (90) is independent of the choice of i. Next we prove thatη is a ring morphism. Take two elements b, b′ ∈ R and chose i big enough such that b = βi(bi)and b′ = βi(b′i). Let us denote Ei =

∑zi ⊗A z∗i =

∑zi ⊗A z∗i

η(b)η(b′) =∑

σi(bizi)z∗i τi σi(b′izi)⊗A z∗i τi

=∑

σi(bib′izi)⊗A z∗i τ = η(bb′)

Finally, the idempotent local units of P ⊗AP † are of the form∑σi(zi)⊗A z∗i τi, these are exactly

given by η(1Ri), so η is a morphism of rings with idempotent local units.(ii) ⇒ (iii). By Corollary 2.68 we only have to prove that P and P † are firm R-modules under

the action induced by the morphism η. This is a consequence of the fact that η is a morphism ofrings with enough idempotents. Take p ∈ P , then we know that there exists an idempotent e ∈ Rsuch that η(e) ∈ P ⊗A P † is a local unit for p. Thus e · p = η(e)p = p and P is a firm R-module.Analogously one proves that P † is a firm right R-module.

(iii) ⇒ (i). Follows from Corollary 2.68 and Lemma 2.41.

For every i ∈ I, consider bimodule maps

coevPi : Ri → Pi ⊗A P ∗i , coevPi(bi) = biEi = EibievPi : P ∗i ⊗Ri Pi, evPi(ϕi ⊗Ri pi) = ϕi(pi)

Then (Ri, A, Pi, P ∗i , coevPi , evPi) is a comatrix coring context, so we have a comatrix coring(G(i),∆i, εi) with

∆i(ϕi ⊗Ri pi) = ϕi ⊗Ri Ei ⊗Ri pi and εi(ϕi ⊗Ri pi) = ϕi(pi).

G(i) is a finite comatrix coring, as constructed in Example 3.48.

Proposition 3.64. Suppose the equivalent conditions of Lemma 3.63 hold, and consider thedirected system G from Proposition 3.61. Then (G,∆, ε) is a coalgebra in Func(Z,AMA).


Proof. It suffices to show that ∆ and ε are natural transformations, or, equivalently, thatG(aji) is a morphism of corings, for every i ≤ j, or

(G(aji)⊗A G(aji)) ∆i = ∆j G(aji) ; εi = εj G(aji).

For all ϕi ∈ P ∗i and pi ∈ Pi, we compute

(∆j G(aji))(ϕi ⊗Ri pi) = ∆j

(ϕi τij ⊗Rj σji(pi)

)= ϕi τij ⊗Rj Ej ⊗Rj σji(pi)

=∑

ϕi τij σji τij ⊗Rj zj ⊗A z∗j ⊗Rj (σji τij σji)(pi)(37,41)

=∑

ϕi τij ⊗Rj (σji τij)(zj)⊗A z∗j σji τij ⊗Rj σji(pi)(89)=

∑ϕi τij ⊗Rj σji(zi)⊗A z∗i τij ⊗Rj σji(pi)

=∑

G(aji)(ϕi ⊗Ri zi)⊗A G(aji)(z∗i ⊗Ri pi)

=((G(aji)⊗A G(aji)) ∆i

)(ϕi ⊗Ri pi)

and

εj(G(aji)(ϕi ⊗Ri pi)

)= (ϕi τij σji)(pi) = ϕi(pi) = εi(ϕi ⊗Ri pi).

Proposition 3.65. Under the same conditions as in Proposition 3.64, colimG = (P † ⊗RP, g), with

gi : G(i) = P ∗i ⊗Ri Pi → P † ⊗R P, gi(ϕi ⊗Ri pi) = ϕi τi ⊗R σi(pi).

Proof. We first show that gi is well-defined. For all bi ∈ Ri, we have

gi(ϕibi ⊗ pi) = ϕibi τi ⊗R σi(pi) = (ϕi τi σi)bi τi ⊗R σi(pi)(47)= (ϕi τi)βi(bi)⊗R σi(pi) = (ϕi τi)⊗R βi(bi)σi(pi)

(46)= (ϕi τi)⊗R σi(biτi(σipi)) = (ϕi τi)⊗R σi(bpi) = gi(ϕi ⊗ bipi).

Let us now prove that (P † ⊗R P, g) is a cocone on G. Indeed, if i ≤ j, then

(gj G(aji))(ϕi ⊗Ri pi) = gj(ϕi τij ⊗Rj σji(pi))= ϕi τij τj ⊗R σj(σji(pi)) = ϕi τi ⊗R σi(pi) = gi(ϕi ⊗Ri pi).

Let (M,m) be another cocone on G. Then mi : P ∗i ⊗Ri Pi →M and mj G(aji) = mi if j ≥ i.We define f : P † ⊗ P → M as follows. For ϕ ∈ P † and p ∈ P , we can find i ∈ I, ϕi ∈ P ∗i andpi ∈ Pi such that p = σi(pi) and ϕ = ϕi τi; we then define

f(ϕ⊗ p) = mi(ϕi ⊗Ti pi).

We have to show that f is well-defined. If k ≥ i, then we have that ϕ = ϕk τk and p = σk(pk)with ϕk = ϕi τik and pk = σki(pi). We then find that

mk(ϕk ⊗Rkpk) = mk(ϕi τik ⊗Rk

σki(pi))= (mk G(aki))(ϕi ⊗Ri pi) = mi(ϕi ⊗Ri pi).

We will now show that f induces a map f : P † ⊗R P →M . To this end, we need to prove that

f(ϕb⊗ p) = f(ϕ⊗ bp),

for all ϕ ∈ P †, p ∈ P and b ∈ R. We can find i ∈ I, bi ∈ Ri, ϕ ∈ P ∗i and pi ∈ Pi such thatb = βi(bi), p = σi(pi) and ϕ = ϕi τi. Then we compute that

f(ϕb⊗ p) = f((ϕi τi)βi(bi)⊗ σi(pi))(47)= f(((ϕi τi σi)bi) τi ⊗ σi(pi)) = f(ϕibi τi ⊗ σi(pi))= mi(ϕibi ⊗Ri pi) = mi(ϕi ⊗Ri bipi)= f(ϕi τi ⊗ σi(bipi)) = f(ϕi τi ⊗ σi(biτi(σi(pi))))

(46)= f(ϕi τi ⊗ βi(bi)σi(τi(σi(pi)))) = f(ϕ⊗ bp).


Finally,

f(gi(ϕi ⊗Ri pi)) = f(ϕi τi ⊗R σi(pi)) = mi(ϕi ⊗Ri pi).

The following result now follows immediately from Propositions 3.59, 3.64 and 3.65.

Corollary 3.66. If the equivalent conditions of Lemma 3.63 hold, then G = P †⊗R P is anA-coring, with comultiplication and counit given by the following formulas, for all i ∈ I, ϕi ∈ P ∗iand pi ∈ Pi:

∆(ϕi τi ⊗R σi(pi)) =∑

∆(ϕi τi ⊗R σi(zi)⊗A z∗i τi ⊗R σi(pi)),

ε(ϕi τi ⊗R σi(pi)) = ϕi(pi).

As before, Ei =∑zi ⊗A z∗i is the finite dual basis of Pi ∈MA.

We will now show that P † ⊗R P can be constructed starting from a comatrix coring context,as described in Section 3.4.1. We already know that P and P † are firm bimodules.

Proposition 3.67. If the equivalent conditions of Lemma 3.63 hold, then (R,A, P, P †, η, ε)is a comatrix coring context, where ε : P † ⊗R P → A is the restriction of the evaluation mapP ∗ ⊗R P → A.

Proof. We have to show that a diagram like (53) commutes. This means that bp =b−ε(b+ ⊗A p) and ϕb = ε(ϕ ⊗A b−)b+, where we denoted η(b) = b− ⊗A b+ ∈ P ⊗A P †. Takeb = βi(bi) ∈ R, p = σi(pi) ∈ P and ϕ = ϕi τi ∈ P †. Then

b−ε(b+ ⊗A p) =∑

σi(bizi)(z∗i τi)(σi(pi))

=∑

σi(biziz∗i (pi)) = σi(bipi) = βi(bi)σi(pi) = bp,

and (ε(ϕ⊗A b−)b+

)(p) =

(∑ϕ(σi(zi))(z∗i bi) τi

)(p) =

∑ϕ(σi(zi))z∗i (biτi(p))

=∑

ϕ(σi(ziz∗i (bipi))) = ϕ(σi(bipi)) = ϕ(bp) = (ϕb)(p),

hence ε(ϕ⊗A b−)b+ = ϕb.

Example 3.68. Let B be a k-algebra with orthogonal idempotent local units and let ei | i ∈I be a complete set of idempotents. For all i, j ∈ I, let Bij = eiBej . Then B =

⊕i,j∈I Bij ,

and a firm left B-module P can then be written as P =⊕

i∈I Pi, with Pi = eiP a left Bi = Bii-module. For each i ∈ I, we take a (Bi, A)-bimodule Pi which is finitely generated and projectiveas a right A-module, and we put P =

⊕i∈I Pi. It is not hard to see that P † =

⊕i∈I P

∗i , and we

have a comatrix coring P † ⊗B P . In this way we recover the infinite comatrix corings of Example3.49

Example 3.69. As a special case of the previous example, consider now the case where theorthogonal idempotents are central in B, then the situation simplifies to B = ⊕i∈IBi, whereBi = Bei.

We remark that in the situation of Example 3.69, the ei are central idempotents. This conditionis also needed in the proof of our next result. We have seen that the comatrix coring is the colimitof the directed system G discussed in Proposition 3.61. If we work over an algebra with centralidempotent local units, then this system is split.

Proposition 3.70. Let B be a k-algebra with central idempotent local units and supposethe equivalent conditions of Lemma 3.63 hold, then the direct system G of Proposition 3.61 splits.Gs : Z → AMs

A, with Gs(aji) = (gji, hij), where

hij(ϕj ⊗Bj pj) = ϕj σji ⊗Bi τij(pj) = ϕj|Pi⊗Bi eipj ,

for all ϕj ∈ P ∗j and pj ∈ Pj .


Proof. Let us show that hij is well-defined; all the rest is obvious. First we compute forϕj ∈ P ∗j , bj ∈ Bj and pi ∈ Pi that

(ϕjbj)(σjipi) = ϕj(bjpi) = ϕj(bjeipi) = ϕj(eibjeipi) = (ϕj σji)(eibjei)(pi),where we used the fact that ei is central. Then we compute

hij(ϕjbj ⊗ pj) = ϕjbj σji ⊗Bi eipj = (ϕj σji)(eibjei)⊗Bi eipj

= ϕj σji ⊗Bi eibjeieipj = ϕj σji ⊗Bi eibjpj = hij(ϕj ⊗ bjpj).

3.5.3. Factorizing split direct systems. In this Section, we consider split direct systemsP s : Z → Ms

A,fgp that factorize through a k-linear category A: we assume that there exists asplit direct system

M s : Z → As, M s(i) = Mi, Ms(aji) = (µji, νij)

and a functor ω : A →MA such that P s = ω M s, or

Pi = ω(Mi), σji = ω(µji), τij = ω(νij).

For every i ∈ I, Ti = EndA(Mi) is a k-algebra with unit. For i ≤ j, we have a multiplicative map

ρji : Ti → Tj , ρji(ti) = µji ti νij .This defines a direct system T : Z → Fk, T i = Ti, T (aji) = ρji. If ti ∈ Ti = EndA(Mi), thenω(ti) ∈ EndA(Pi). Hence Pi is a (Ti, A)-bimodule, with left Ti-action given by

ti · pi = ω(ti)(pi).

We claim that (35) holds. Indeed, for all i ≤ j, ti ∈ Ti and pj ∈ Pj , we have

ρji(ti) · pj = (µji ti νij) · pj = ω(µji ti νij)(pj)= (σji ω(ti) τij)(pj) = σji(ti · τij(pj)).

Applying the results of Section 3.5.2, we obtain a comatrix coring. We will now assume thatcolimM = (M,µ) exists, and that ω preserves colimits. We will give an explicit description ofcolimT , and provide some alternative descriptions of the comatrix coring. Using Proposition 2.1, we obtain morphisms νi : M → Mi. Let σi = ω(µi), τi = ω(νi). We consider the k-algebraT = EndA(M). For every i ∈ I, ei = µi νi is an idempotent in T . We also have

(91) ei µi = µi and νi ei = νi,

and, for i ≤ j:

(92) ej ei = ei ej = ei.

(91) is immediate; (92) can be seen as follows:

ej ei = ej µi νi = ej µj µji νi(91)= µj µji νi = µi νi = ei;

ei ej = µi νi ej = µi νij νj ej(91)= µi νij νj = µi νi = ei.

Lemma 3.71. Let i ∈ I and t ∈ T . There exists ti ∈ Ti such that

t = µi ti νiif and only if

t = ei t ei.In this situation, ti is unique, and is given by the formula ti = νi t µi; furthermore, for everyj ≥ i, t = µj tj νj , with

(93) tj = µji ti νij .

Proof. We leave the first part as an easy exercise to the reader. For j ≥ i, we compute

µj µji ti νij νj = µi ti νi = t.


Proposition 3.72. T † = t ∈ T | ∃i ∈ I : t = ei t ei is a subalgebra of T withidempotent local units. In particular, T † is a firm k-algebra. colimT = (T †, ρ), with

ρi : Ti → T †, ρi(ti) = µi ti νi.

Proof. It is clear that the ei form a set of idempotent local units. It follows from Lemma 3.71that Ti = eiTei. (T †, ρ) is a cocone on T since, for all j ≥ i and ti ∈ Ti, we have

(ρj ρji)(ti) = µj µji ti νij νj = µi ti νi = ρi(ti).

Assume that (M,m) is another cocone on T †. This means that mi : Ti →M and mj ρji = mi ifi ≤ j. The map f : T † →M , f(µitiνi) = mi(ti), is well-defined: assume that t = µitiνi =µj tj νj . Take k ≥ i, j. By (91), t = µk tk νk, with tk = µki ti νik = ρki(ti), and itfollows that mk(tk) = mk(ρki(ti)) = mi(ti). In a similar way, we have that mk(tk) = mj(tj).Finally, for every i ∈ I and ti ∈ Ti, we have that (f ρi)(ti) = f(µi ti νi) = mi(ti).

It is easy to show that Pi ∼= ei · P , so that the comatrix coring P † ⊗T † P is a special case ofthe comatrix coring studied in Section 3.5.2. In general, P † is a proper submodule of P ∗ and T † isa proper subalgebra of T . But we have the following remarkable result, which can be understoodas a special case of Corollary 3.41 (iii).

Proposition 3.73. The map

κ : P † ⊗T † P → P ∗ ⊗T P, κ(ϕ⊗T † p) = ϕ⊗T pis an isomorphism of A-bimodules.

Proof. We first define a map λ : P ∗⊗T P → P †⊗T † P as follows: take ϕ ∈ P ∗ and p ∈ P .There exists i ∈ I such that p = σi(pi), and we define

λ(ϕ⊗T p) = ϕ σi τi ⊗T † p.The right hand side does not depend on the choice of i: assume that j ∈ I is such that p = σj(pj)for some pj ∈ Pj , and take k ≥ i, j. Then we have that p = σk(pk) with pk = σki(pi). Wecompute that

(94) σk τk σi τi = σk τk σk σki τi = σk σki τi = σi τi,hence

(ϕ σi τi)⊗T † p = (ϕ σk τk σi τi)⊗T † p= (ϕ σk τk)⊗T † (σi τi)(p) = (ϕ σk τk)⊗T † p,

and, in a similar way,(ϕ σj τj)⊗T † p = (ϕ σk τk)⊗T † p.

Our next aim is to show that

(95) λ(ϕ⊗ t · p) = λ(ϕ · t⊗ p),

for all ϕ ∈ P ∗, t ∈ T , and p ∈ P . There exists i ∈ I such that

p = σi(τi(p)) = σi(pi) and t · p = (σi τi)(t · p) = (σi τi ω(t) σi)(pi).For all k ≥ i, we then also have that

(96) p = σk(pk) and t · p = (σk τk ω(t) σk)(pk).For all p ∈ P , we have

τi(p) =∑

ziz∗i (τi(p)),

hence(ω(t) σi τi)(p) =

∑(ω(t) σi)(zi)z∗i (τi(p)).

There exists k ∈ I such that all (ω(t)σi)(zi) ∈ σk(Pk), or (ω(t)σi)(zi) = (σkτkω(t)σi)(zi).Then we find for all p ∈ P that

(ω(t) σi τi)(p) =∑

(ω(t) σi)(zi)z∗i (τi(p))

=∑

(σk τk ω(t) σi)(zi)z∗i (τi(p)) = (σk τk ω(t) σi τi)(p).


We can take k ≥ i. Using (94), we then find

(97) ω(t) σi τi = σk τk ω(t) σi τi = σk τk ω(t) σk τk σi τi.We now compute

λ(ϕt⊗ p) = ϕ ω(t) σi τi ⊗T † σi(pi)(97)= ϕ σk τk ω(t) σk τk σi τi ⊗T † σi(pi)= ϕ σk τk ω(t) σk τk ⊗T † (σi τi σi)(pi)= ϕ σk τk ω(t) σk τk ⊗T † σi(pi)= ϕ σk τk σk τk ω(t) σk τk ⊗T † σk(pk)= ϕ σk τk ⊗T † (σk τk ω(t) σk τk σk)(pk)

(96)= ϕ σk τk ⊗T † tp = λ(ϕ⊗ tp),

proving (95). We conclude that λ induces a well-defined map λ : P ∗ ⊗T P → P † ⊗T † P . Let usfinally show that λ is the inverse of κ. Take ϕ ∈ P † and p ∈ P . Then there exists i ∈ I such thatϕ = ϕi τi and p = σi(pi) for some pi ∈ Pi, ϕi ∈ P ∗i . Then

λ(κ(ϕ⊗T † p)) = λ(ϕ⊗T p) = ϕ σi τi ⊗T † p = ϕ⊗T † (σi τi)(p) = ϕ⊗T † p.Take ϕ ∈ P ∗, and p = σi(pi) ∈ P . Then

κ(λ(ϕ⊗T p)) = κ(ϕ σi τi ⊗T † p) = ϕ σi τi ⊗T p = ϕ⊗T (σi τi)(p) = ϕ⊗T p.

We will now describe the infinite comatrix coring P † ⊗T † P as the colimit of a richer system.On I × I, we define a preorder as follows.

• (i, i) ≤ (j, j) if i ≤ j in I;• (i, j) ≤ (i, i), for all i, j ∈ I;• (i, j) ≤ (j, j), for all i, j ∈ I.

This preorder induces a partial order ≤ on I × I. We have a corresponding category Y. If i ≤ jin I, then the corresponding morphism (i, i) → (j, j) in Y is denoted by aji. The morphism(i, j) → (i, i) is denoted by lij , and the morphism (i, j) → (j, j) by rij . Note that we have afunctor ξ : Z → Y, ξ(i) = (i, i), ξ(aji) = aji.

Proposition 3.74. We have a functor F : Y → AMA such that F ξ = G.

Proof. For i, j ∈ I, Tji = HomA(Mi,Mj) is a (Tj , Ti)-bimodule, and we have

F (i, j) = P ∗j ⊗Tj Tji ⊗Ti Pi.

We now define F on the morphisms. Let F (aji) = G(aji); F (lij) and F (rij) are given by

F (lij) : P ∗j ⊗Tj Tji ⊗Ti Pi → P ∗i ⊗Ti Pi, F (lij)(ϕj ⊗Tj tji ⊗Ti pi) = ϕj tji ⊗Ti pi;

F (rij) : P ∗j ⊗Tj Tji ⊗Ti Pi → P ∗j ⊗Tj Pj , F (rij)(ϕj ⊗Tj tji ⊗Ti pi) = ϕj ⊗Tj tji(pi).We have to prove that F (aji) F (lij) = F (rij) if i ≤ j. We compute easily that

F (aji) F (lij)(ϕj ⊗Tj tji ⊗Ti pi) = ϕj tji β(aji)⊗Tj α(aji)(pi)= ϕj ⊗Tj (tji β(aji) α(aji))(pi)= ϕj ⊗Tj tji(pi) = F (rij)(ϕj ⊗Tj tji ⊗Ti pi).

In a similar way, we prove that F (aji)F (rij) = F (lij) if i ≤ j. All other verifications are easy.

Proposition 3.75. colimF = (P † ⊗T † P, f) with

fij = gi F (lij) = gj G(rij),

for all i, j ∈ I.

Proof. It is easy to show that (P † ⊗T † P, f) is a cocone on F . If (M,m) is another coconeon F , then we have a cocone (M,n) on G, with ni = m(i,i). We then have an A-bimodule map

f : P † ⊗T † P , and it is straightforward to show that it satisfies the necessary requirements.


References

Most results from Section 3.1.1 and Section 3.1.2 are well-known and are included for sake ofcompleteness. We refer to [28], [36] or [46] and to [73] and [112] for the adjuction propertiesrelated to firm rings. The construction of the Dorroh extension of a coring was done in [113]. Thefactorizable entwining structures from Section 3.2.2 were introduced in the author’s joint paperwith S. Caenepeel and Shuanhong Wang [48]. From the same paper originate most results ofSection 3.3, we refer also to [3], [65] and [115] for related results. Section 3.4 is inspired on thejoint paper with J. Gomez-Torrecillas [73]. Section 3.5 is part of a work with S. Caenepeel and E.De Groot [41].

IIPart II :

Galois Theory

The Tupolevs are without any doubtthe most underestimated scientists of modern History.

– Elma Gretarsson

Chapter 4Galois Comodules

Galois theory for corings and comodules studies functors, or more precisely equivalences, be-tween categories of modules and categories of comodules. If a functor from a module category to acomodule category has a right adjoint, than it can be represented by a (bi)comodule and the prop-erties of the functor are completely encoded in the properties of this (bi)comodule. In particular,the functor establishes an equivalence of categories if the comodule is a so-called Galois comodulethat satisfies some additional flatness conditions. This theory generalizes some well known resultsfrom various fields, such as the classical Galois theory of field extensions, Hopf-Galois theories anddescent theory. There is as well an interesting link with the theory on cotripleability of functorsand Beck’s theorem. To unify our Galois theory for corings with this categorical theory, we showhow our techniques can be transfered to the framework of bicategories.

In the first Section, we describe in more detail the descent problem and the example of HopfGalois theory. In Section 4.2 we describe extensively the Galois theory for comodules. We show howdifferent notions of ‘Galois’ comodules that appear in the literature are related and prove structuretheorems for general functors between categories of modules and comodules. In Section 4.3 wemove our viewpoint to bicategories and develop a Galois theory in this general setting. This unifiesthe Galois theory of comodules with several other theories, such as Galois theory in the bicategoryof corings [31], the theory on coendomorphism corings [118], [36, Section 23] and comonadicityof functors [26, Section 4.7], [70]. Dualizing our results we obtain the Galois theory of C-rings[35].

4.1. Introduction : Motivating problems

4.1.1. The descent problem. The descent problem can be described as follows. Let A andB be two rings. Consider the categories MA and MB and a functor

F : MB →MA.

It is a natural question to ask which right A-modules are of the form F (N) for some N ∈ MB,i.e. for which M ∈ MA can we find N ∈ MB such that M = F (N). Furthermore, does Festablish an equivalence of categories between MB and a particular subcategory of MA ?

This problem has a solution that can be formulated in the language of corings and comonads insome particular and interesting situations. A first restriction can be made by considering functorsF that have a right adjoint. By the Eilenberg-Watts Theorem (see Theorem 2.13), F has a rightadjoint if and only if F preserves colimits if and only if F ∼= −⊗B Σ for some Σ ∈ BMA.

A complete solution to the problem can be formulated if Σ is finitely generated and projectiveas a right A-module; then we can construct the comatrix A-coring D = Σ∗ ⊗B Σ and consider its

113

114 CHAPTER 4. GALOIS COMODULES

category of comodules MD. The functor F factorizes in the following way

F : MB−⊗BΣ//MD

FD//MA

where FD denotes as usual the forgetful functor. The needed subcategory of MA is exactly thecategory of right D-comodules. Moreover, − ⊗B Σ establishes an equivalence between MB andMD if Σ is faithfully flat as a left B-module. Conversely, if Σ is flat as a left B-module and −⊗BΣinduces an equivalence between MB and MD, then Σ is faithfully flat as a left B-module. Thisis known as the structure theorem for Galois comodules (see Corollary 4.35).

More generally, if Σ is R-firmly projective for a firm B-ring R, then we can construct thecomatrix A-coring D = Σ∗ ⊗R Σ ∼= Σ† ⊗R Σ where Σ† = Σ∗ ⊗R R. We now obtain the followingdiagram of functors

MBF //MA

MR −⊗RΣ//

UR

OO

MD

FD

OO

where UR and FD are the forgetful functors. One can formulate a generalization of the structuretheorem for Galois comodules in the following way. If Σ is faithfully flat as a left R-modulethen we obtain an equivalence between the categories MR and MD through the functor −⊗R Σ.Conversely, if Σ is flat as a left R-module and −⊗RΣ induces an equivalence of categories betweenMR and MD, then Σ is faithfully flat as a left R-module (see Theorem 4.27).

Although this result encaptures the finitely generated case and many other interesting examples(such as direct sums of finitely generated and projective comodules, comodules coming fromcolimits), remark that the result is less satisfactory than the original finite case since we don’tknow very much about the functor UR. For this reason, it might be interesting to obtain (at leastin some cases) a result that omits the functor UR. This problem is still under investigation.

4.1.2. Hopf-Galois theory. The classical Galois theory of finite field extensions has a formu-lation in terms of Hopf algebras. Let k be a field, and consider a field L ⊃ k, with a group ofk-automorphisms G. Put F = LG, the field fixed by G. Then it is known that L/F is a Galoisextension with Galois group G if and only if |G| = [L : F ] (see [79, Artin’s Lemma, p. 229]). Letus denote |G| = n, G = g1, . . . , gn, then the elements gi form a basis for kG as k-vectorspace.Define pi ∈ kG∗ by pi(gj) = δij and take a basis u1, . . . , un for L as a F -vectorspace. It iswell-known that kG is a Hopf algebra that acts on L. Therefore, L is a comodule algebra for thedual Hopf algebra H = kG∗. The coaction of L is given explicitly by

(98) ρ : L→ L⊗H, ρ(a) =∑i

gi · a⊗ pi

for all a ∈ L. There exists as well another map

can : L⊗F L→ L⊗H, can(a⊗ b) =∑i

a(gi · b)⊗ pi,

for all a ⊗ b ∈ L ⊗F L. Let us show that can is always injective. We can represent any elementin the kernel of the morphism can in the form

∑j aj ⊗ uj ∈ L ⊗F L. Then can(

∑j aj ⊗ bj) =∑

ij aj(gi · uj)⊗ pi = 0. Since the elements pi are lineary independent, we find∑j

aj(gi · uj) = 0, for all i.

As the elements uj constitute a basis for L, the above system of equations can only have a the trivalsolution, i.e. all aj are zero (cf. the proof of Artin’s Lemma), so we conclude that

∑j aj ⊗uj = 0

and can is injective.Since can is injective, it is bijective if and only if the dimentions of the source and target are

equal. We find dimF (L⊗F L) = [L : F ]2 and dimF (L⊗H) = [L : F ]|G|, and therefore, can isbijective if and only if [L : F ] = |G|, i.e. L is a Galois extension of F .

This example has led to a range of generalizations, all under the name of Galois theory (werefer to [38] and [116] for a profound overview). A comodule algebra A over the Hopf algebra H,

4.2. GALOIS COMODULES 115

is called an H-Galois extension of AcoH = a ∈ A | ρ(a) = a ⊗ 1H if and only if the canonicalmap

can : A⊗AcoH A→ A⊗k H, can(a⊗ a′) = aa′[0] ⊗ a′[1],

is an isomorphism, where ρA(a) = a[0] ⊗ a[1] denotes the H-coaction on A.We know from Example 3.21 that bialgebras (and hence Hopf algebras) give rise to examples

of corings. Therefore, Hopf-Galois theory can be generalized in the framework of corings. In thisway Hopf-Galois and descent theory are unified by one general theory.

4.2. Galois comodules

4.2.1. Comonadic-Galois comodules. Let R be a firm ring, C an A-coring and Σ an R-Cbicomodule. This implies the existence of a ring morphism

(99) : R→ T = EndC(Σ), (r)(u) = ru.

Recall from Corollary 3.10 the existence of the following diagram of adjoint functors

(100) MR

−⊗RΣ //

GΣ=−⊗RΣ


oo

GC=−⊗AC

MC

FC

OO

HΣ=HomC(Σ,−)⊗RR


such that there exist natural isomorphisms

(101)−⊗R Σ ' FCGΣ : MR →MA

HomA(Σ,−)⊗R R ' HΣGC : MA →MR

That is, the adjunction in the upper row of diagram (100) factorizes (up to natural isomorphism)trough MC.

Also the adjuctions (FC,GC) and (GΣ,HΣ) satisfy a factorization property, altough not up toisomorphism, as one can see from the following theorem, of which the proof is left to the reader.

Theorem 4.1. Let R be a firm ring, C an A-coring and Σ an R-C bicomodule. Then thereexist natural transformations as follows

can′M : HomA(Σ,M)⊗R R⊗R Σ → GC(M) = M ⊗A C,(102)

can′M (ϕ⊗R r ⊗R u) = ϕ(ru[0])⊗A u[1];

ev′N : HomC(Σ, N)⊗R R⊗R Σ → FC(N) = N,(103)

ev′N (ψ ⊗R r ⊗R u) = ψ(ru);

δP : GΣ(P ) = P ⊗R Σ → P ⊗R Σ⊗A C,(104)

δP (p⊗R u) = p⊗R u[0] ⊗A u[1];

τN : HΣ(N) = HomC(Σ, N)⊗R R→ HomA(Σ, N)⊗R R,(105)

τN (ψ ⊗R r) = FC(ψ)⊗R r;

for M ∈MA, N ∈MC and P ∈MR.

The natural transfomations defined in Theorem 4.1 are not choosen arbitrary. They are inducedby the units and counits of the adjunctions in diagram (100). With notation as in Corollaries 3.9and 3.10 we obtain,

ev′N = FC(ζN );

δP = %P⊗RΣ = ρP⊗RΣ = P ⊗R ρΣ.

Also can′ is obtained in such a canonical way, as we will see in Proposition 4.2. First remark thatsince we assumed that Σ is a firm left R-module, the natural transformations can′ and ev′ are up


to natural isomorphism identical to the following ones,

canM : HomA(Σ,M)⊗R Σ → GC(M) = M ⊗A C,(106)

canM (ϕ⊗R u) = ϕ(u[0])⊗A u[1];

evN : HomC(Σ, N)⊗R Σ → FC(N) = N,(107)

evN (ψ ⊗R u) = ψ(u);

Proposition 4.2. With notation as above, the following assertions hold

(i) δP is injective for all P ∈MR;(ii) if R is flat as a left R-module, then τN is injective for all N ∈MC;(iii) for all M ∈MA, the following identity holds up to isomorphism

canM ∼= evM⊗AC

Proof. (i). Check that P ⊗R Σ⊗A εC is a left inverse for δP .

(ii). Clearly the forgetful map HomC(Σ,M) → HomA(Σ,M) is injective. If R is flat as a left Rmodule, then the functor −⊗R R is exact, so τN is injective.(iii). From (60) we obtain the isomorphism θΣ,M : HomC(Σ,M⊗AC) → HomA(Σ,M). Therefore,the following diagram commutes

HomC(Σ,M ⊗A C)⊗R ΣθΣ,M⊗RΣ

//

evM⊗AC ++WWWWWWWWWWWWWWWWWWWWWWHomA(Σ,M)⊗R Σ

canM

M ⊗A C

Definition 4.3. We call an R-C bicomodule an R-C comonadic-Galois comodule if thenatural transformation (106) is an natural isomorphism.

Under the extra assumption that the morphism from (99) is an isomorphism, this coincideswith the notion of Galois comodule introduced by Wisbauer in [117].

Recall that a right C-comodule N is called (C, A)-injective if and only if the comultiplicationρN : N → N ⊗AC has a left inverse γN : N ⊗AC → N in MC. If N = M ⊗AC, where M ∈MA

and ρN = M⊗A∆C then M is (C, A)-injective, since M⊗A ε⊗AC is a right C-colinear left inversefor ρN .

Theorem 4.4. Let R be a firm ring, C an A-coring and Σ ∈ RMC. Then the followingassertions are equivalent.

(i) Σ is an R-C comonadic-Galois comodule;(ii) For every (C, A)-injective N ∈MC, the evaluation map

evN : HomC(Σ, N)⊗R Σ → N, evN (f ⊗R u) = f(u);

is an isomorphism.

Proof. (i) ⇒ (ii). Let N be an (A,C)-injective right C-comodule, where we denote γ :N ⊗A C → N for the left inverse of the coaction ρN of N . For all L ∈MC, consider the followingdiagram

(108) HomC(L,N)i // HomA(L,N)

j1 //j2// HomA(L,N ⊗A C).

where the maps j1 and j2 are defined as follows

j1(f)(l) = f(l)[0] ⊗A f(l)[1], j2(f)(l) = f(l[0])⊗A l[1].

Define maps α : HomA(L,N) → HomC(L,N) and β : HomA(L,N ⊗A C) → HomA(L,N) bythe formulas

α(f)(l) = γ(f(l[0])⊗A l[1]), β(g) = γ g.


Now it is easy to check that j1 i = j2 i, α i = HomA(L,N), β j1 = HomC(L,N) andβj2 = iα, this means that (108) is a contractable equalizer . Any functor preserves contractableequalisers (see e.g. [13, Proposition 3.3.2] or [88, VI.6]). Consequently, if we apply the functorGΣ = −⊗RΣ to (108), we obtain a contractable equalizer inMC. By the coassociativity condition,we can associate to L a second equalizer in MC,

LρL

// L⊗A CρL⊗AC //

L⊗A∆// L⊗A C⊗A C

Now take L = Σ, than we can compare both equalizers by the canonical map,

HomC(Σ, N)⊗A Σ

evN

i // HomA(Σ, N)⊗A Σ

canN

j1 //j2// HomA(Σ, N ⊗A C)⊗A Σ

canN⊗AC

Σ

ρΣ // Σ⊗A CρΣ⊗AC //

Σ⊗A∆// Σ⊗A C⊗A C

Since canN and canN⊗AC are isomorphisms, we find that evN is an isomorphism by the universalproperty of the equalizer.

(ii) ⇒ (i). This follows from Proposition 4.2 and the fact that M ⊗A C is (A,C)-injective forall M ∈MA.

4.2.2. The canonical cotriple morphism. In the previous section we have seen that everyR-C bicomodule induces a commutative diagram of adjoint functors (100). We will now provean Eilenberg-Watts type theorem that states that any adjunction between a category MR ofR-modules and a category MC of C-comodules can be obtained from a diagram like (100) for aproper choice of Σ.

Theorem 4.5. Let R be a firm ring and C an A-coring. Let F : MR → MC be a functorand Σ = F (R). Then the following assertions are equivalent:

(i) F has a right adjoint G : MC →MR;(ii) F is right exact and preserves direct sums;(iii) F ' −⊗R Σ for some Σ ∈ RMC.

In this situation the right adjoint G of F is unique up to isomorphism and given by

(109) G = HomC(Σ,−)⊗R R;

and the adjunction (F,G) can be extended to a commutative diagram of adjoint functors (100).

Proof. One can repeat the proof of Theorem 2.13.

Recall from Proposition 3.8 that any cotriple C = (C, δC , εC) on a category A induces a pairof adjoint functors FC : AC → A : GC .

Lemma 4.6. If D = (D, δD, εD) and C = (C, δC , εC) are two cotriples on a category A, andϕ : D → C is a morphism of cotriples, i.e.

ϕX : DX → CX

is a natural transformation such that (ϕ ∗ ϕ) δD = δC ϕ and εC ϕ = εD, then there exists afunctor

Φ : AD → AC ,such that there exists a natural isomorphism α and a natural transformation β as follows

α : FD → FCΦ;β : ΦGD → GC .

A AGD

||||

|||| GC

BBB

BBBB

B

AD

FD==||||||||Φ // AC

FC``BBBBBBBB

ADΦ // AC


Proof. We define Φ(X, ρX) = (X,ϕX ρX) for all (X, ρX) ∈ AD. Then

FCΦ(X, ρX) = FD(X, ρX) = X,

so α is just the identity. Furthermore, we define

βX = (ϕX , CϕX −) : ΦGD(X) = (DX,ϕDX δDX) → GC(X) = (CX, δCX)

then one can check that this defines a natural transformation β : ΦGD → GC .

In our next approach, we start from an adjunction (F,G) between two categories of modules.

MR

F //MAG

oo

From Theorem 2.13 we know that F ' −⊗RΣ and G ' HomA(Σ,−)⊗RR for some Σ ∈ RMA.Furthermore, D = FG defines a cotriple on MA (see Section 3.4.3). Denote by MD the categoryof D-comodules. Applying Proposition 3.57, we obtain a unique functor K : MR → MD suchthat the following diagram commutes in the sense that F = FDK and KG = GD.

MR

F //

K ''OOOOOOOOOOOOO MAG

oo

GD

MD

FD

OO

Let C be an A-coring and denote by C = − ⊗A C the associated cotriple on MA (seeTheorem 3.5). Suppose that there exists a cotriple morphism

φ : D → C, φM : DM = HomA(Σ,M)⊗R Σ → CM = M ⊗A C,

for all M ∈ MA. Combining the properties of the Kleisli functor K with Lemma 4.6 we obtainthe following diagram of functors

MR

F //

K ''OOOOOOOOOOOOO MAG

oo

GD

MA

GC

MD

FD

OO

Φ//MC

FC

OO

We have natural isomorphisms

F ' FDK ' FCΦK

and there exists a natural transformation

ΦKG ' ΦGD → GC.

Theorem 4.7. Let (F,G) be a pair of adjoint functors with F : MR → MA and C anA-coring. With notation as introduced above, the following assertions are equivalent.

(i) There exists a morphism of cotriples ϕ : D → C;(ii) there exists a functor K ′ : MR →MC such that F = FC K ′;(iii) there exists an R-C bicomodule Σ such that K ′ = − ⊗R Σ and the adjoint pair (F,G) can

be extended to a diagram of the form (100).

Proof. (i) ⇒ (ii). This follows from the above reasoning. One has to take K ′ = ΦK.

(ii) ⇒ (iii). By Theorem 2.13, we know that there exists a Σ ∈ RMA such that F ' −⊗RΣ.

For any M ∈MR, we can compute that FCK(M) = F (M) = M ⊗A Σ. This means that the C-comodule K(M) has M⊗RΣ as underlying A-module structure. In particular, K(R) = R⊗RΣ ∼=Σ is a right C-comodule. Let us check that the right C-coaction of Σ is left R-linear. To this end,take any r ∈ R and consider the morphism fr : R → R, fr(s) = rs in MR. Then K(fr) is aright colinear map and moreover FCK(fr) = F (fr) ∼= fr ⊗R Σ : R ⊗R Σ ∼= Σ → R ⊗R Σ ∼= Σ.


We find that as a right A-linear map K(fr) can be computed as K(fr)(u) = ru for all u ∈ Σ.Therefore the diagram

K(R) ∼= ΣρΣ //

K(fr)

Σ⊗A C

K(fr)⊗AC

K(R) ∼= Σ

ρΣ// Σ⊗A C

which expresses that K(fr) is right C-colinear, implies that ρΣ is left R-linear, so Σ ∈ RMC.Finally, let us prove that K(M) = (M ⊗R Σ,M ⊗R ρΣ) for all M ∈ MR. Indeed, for allm ∈ M we define the right R-linear map fm : R → M, fm(r) = mr for all r ∈ R. ThenK(fm) : R ⊗ Σ ∼= Σ → M ⊗R Σ is again right C-colinear and by a similar argument as beforewe obtain that as a right A-linear map we can compute K(fm)(u) = m⊗R u for all u ∈ Σ. Thecommutativity of the diagram

ΣρΣ //

K(fm)

Σ⊗A C

K(fm)⊗AC

M ⊗R ΣρK(M)

// M ⊗R Σ⊗A C

implies that ρK(M)(m⊗R u) = m⊗R u[0] ⊗A u[1].(iii) ⇒ (i). The conditions of Theorem 4.1 are satisfied and so there exists a natural trans-

formation can : D → C. Let us check that can′ (and therefore also can) is a morphism ofcotriples. To this end, we have to verify the commutativity of the following diagram. We denoteH = HomA(Σ,M)⊗RR⊗R Σ and N = HomA(Σ,M)⊗RR for M ∈MA. Furthermore, we usethe notation of Corollary 3.10 for the unit of the adjunction (F,G).

HomA(Σ,M)⊗R R⊗R Σ

can′M

δDM=ηN⊗RΣ

// HomA(Σ,HomA(Σ,M)⊗R R⊗R Σ)⊗R R⊗R Σ

can′H

HomA(Σ,M)⊗R R⊗R Σ⊗A C

can′M⊗AC

M ⊗A C

δCM=M⊗A∆C

// M ⊗A C⊗A C

Take any ϕ⊗R r ⊗R u ∈ HomA(Σ,M)⊗R R⊗R Σ, then we can compute

(M ⊗A ∆C) can′M (ϕ⊗R r ⊗R u) = ϕ(ru[0])⊗A u[1] ⊗A u[2].

On the other hand, ηN ⊗RΣ(ϕ⊗R r⊗Ru) = ψϕ⊗Rrs⊗R s⊗Ru, where ψϕ⊗Rrs(v) = ϕ⊗R rs⊗R vfor all v ∈ Σ. Therefore we find

can′H (ηN ⊗R Σ)(ϕ⊗R r ⊗R u) = ϕ⊗R rs ⊗R su[0] ⊗A u[1]

= ϕ⊗R rss⊗R u[0] ⊗A u[1]

= ϕ⊗R r ⊗R u[0] ⊗A u[1]

and we conclude that (can′M ⊗A C) can′H (ηN ⊗RΣ)(ϕ⊗R r⊗R u) = (M ⊗A∆C) can′M (ϕ⊗Rr ⊗R u) as needed. The counit is preserved if and only if the following diagram commutes for allM ∈MA Remark that εDM = εM in the notation of Corollary 3.10 for the counit of the adjunction(F,G)

HomA(Σ,M)⊗R R⊗R ΣεDM //

can′M

M

∼=

M ⊗A CεCM=M⊗AεC

// M ⊗A A


The commutativity of this diagram is assured by

(M ⊗A εC) can′M (ϕ⊗R r ⊗R u) = ϕ(ru[0])εC(u[1]) = ϕ(ru[0]εC(u[1]))

= ϕ(ru) = εDM (ϕ⊗R r ⊗R u)

for any ϕ⊗R r⊗R u ∈ HomA(Σ,M)⊗R R⊗R Σ, where we used the right A-linearity of ϕ in thesecond equality and the counit property of Σ in the third equality.

Remark 4.8. It follows from the proof of the previous theorem that for any morphism ofcotriples ϕ : D → C, there exists a second morphism of cotriples can : D → C. However, thereis no reason why ϕ and can should be same.

We will call can the canonical morphism of cotriples associated to the R-C bicomodule Σ. Wethus conclude that Σ is an R-C comonadic Galois comodule if and only if can is an isomorphismof cotriples.

4.2.3. Firm Galois comodules. Let Σ be a B-A bimodule that is R-firmly projective right A-module. This means that there exists a non-unital ring morphism ι : R→ Σ⊗AΣ∗, ι(r) = er⊗Afr,such that u = erfr(ur) (summation understood). Then we know from Section 3.4 that we canconstruct two isomorphic comatrix corings Σ∗ ⊗R Σ and Σ† ⊗R Σ, where Σ† = Σ∗ ⊗R R.

Theorem 4.9. Consider an A-coring C and a B-A bimodule that is an R-firmly projectiveright A-module. Then the following statements are equivalent.

(i) There exists a coring morphism can : Σ∗ ⊗R Σ → C;(ii) there exists a coring morphism can′ : Σ† ⊗R Σ → C;(iii) Σ is an R-C bicomodule;(iv) Σ† is a C-R bicomodule;

Proof. (i) ⇔ (ii). This follows from the isomorphism of corings Σ∗ ⊗R dR,Σ : Σ∗ ⊗R Σ →Σ† ⊗R Σ (see Corollary 3.41) from which we can deduce the following commutative diagram ofA-coring morphisms

(110) Σ∗ ⊗R Σ can //

Σ∗⊗RdR,Σ

C

Σ† ⊗R Σcan′

77nnnnnnnnnnnnnn

(i) ⇒ (iii)&(iv). By Theorem 3.39 we know that Σ is an R-Σ∗ ⊗R Σ bicomodule and Σ† is aΣ∗ ⊗R Σ-R bicomodule. Appying can we obtain the statement immediately.(iii) ⇒ (i). It follows from the proof of Theorem 4.7 that the natural transformation can− is acotriple morphism. The correspondence of Theorem 3.5 between corings and cotriples implies thatcan = canA : HomA(Σ, A)⊗R Σ = Σ∗ ⊗R Σ → A⊗A C ∼= C is a coring morphism.(iii) ⇔ (iv) follows from Proposition 3.53.

Definition 4.10. An R-C bicomodule Σ is called a firm R-C Galois comodule if Σ is R-firmlyprojective as a right A-module and can (or equivalently can′) is an isomorphism of A-corings. Wecall can the canonical map.

Remark 4.11. It follows from Remark 3.46 that Σ is a firm R-C Galois comodule if and onlyif Σ† is a firm C-R Galois comodule.

In the remaining part of this Chapter, we will use the notation can both for the morphism canand can′. This is justified by diagram (110).

Theorem 4.12. Let C be an A-coring and Σ an R-C bicomodule that is R-firmly projectiveas a right A-module. Then the ring morphism i : R → Σ ⊗A Σ†, i(r) = er ⊗R fr ⊗R sr canbe corestricted to a morphism η : R → Σ ⊗C Σ†. Furthermore, for any N ∈ MR, we obtain amorphism

νN : N → (N ⊗R Σ)⊗C Σ†, νN (n) = nr ⊗R er ⊗A fr.


Proof. Consider the following commutative diagram

Σ∼= //

ρΣ

R⊗R Σi⊗RΣ //

R⊗RρΣ

Σ⊗A Σ∗ ⊗R Σ

Σ⊗Acan

Σ⊗A C ∼=// R⊗R Σ⊗A C

∼= //

i⊗RΣ⊗AC

Σ⊗A C

ρΣ⊗AC

Σ⊗A Σ∗ ⊗R Σ⊗A CΣ⊗Rcan⊗AC // Σ⊗A C⊗A C

The left upper square in this diagram commutes since Σ is an R-C bicomodule. We can see fromthe following calculation that the right upper square commutes as well

er ⊗A fr(u[0])u[1] = erfr(u[0])⊗A u[1] = ru[0] ⊗A u[1]

for all r ⊗R u ∈ R ⊗R Σ. The lower right square is commutative by the same argument as theupper right square, one just has to tensor all morphisms with C over A. The commutativity of thisdiagram implies the following equality for all u ∈ Σ,

(111) er[0] ⊗A er[1] ⊗A fr(ur[0])ur[1] = er ⊗A fr(ur[0])u

r[1] ⊗A u[2].

Take now any r ∈ R and consider i(r) = ur ⊗A fr ⊗R sr. If we want to check whether i(r) ∈Σ⊗C Σ†, we have to verify if

(ρΣ ⊗A Σ†) i(r) = (Σ⊗A λΣ†) i(r).

The right hand side of this equation is

ur ⊗A fr(et[0])et[1] ⊗A ft ⊗R srt = ur ⊗A fr(ert[0])ert[1] ⊗A ft ⊗R s

t

where we made use of a classical argument in calculation over firm structures. On the other hand,the left hand side can be computed as follows.

ur[0] ⊗A ur[1] ⊗A fr ⊗R sr = ur[0] ⊗A ur[1] ⊗A fr ⊗R trst

= ur[0] ⊗A ur[1] ⊗A frtr ⊗R st

= ur[0] ⊗A ur[1] ⊗A fr(ert )ft ⊗R st

= ur[0] ⊗A ur[1]fr(ert )⊗A ft ⊗R st

= ur[0] ⊗A ur[1]fr(ert[0])ε(ert[1])⊗A ft ⊗R s

t

= ur ⊗A fr(ert[0])ert[1]ε(e

rt[2])⊗A ft ⊗R s

t

= ur ⊗A fr(ert[0])ert[1] ⊗A ft ⊗R s

t

where we used (111) to calculate the sixth equality. This proves that η is well defined. A similarcomputation shows that νN is well defined for all N ∈MR.

Theorem 4.13. Let Σ be an R-C bicomodule that is R-firmly projective as right A-module andsuppose that R is flat as a left R-module. Then the natural isomorphism α : HomA(Σ,−)⊗RR '− ⊗A Σ† from Theorem 2.51 induces a natural isomorphism HomC(Σ,−) ⊗R R ' − ⊗C Σ†. Inparticular, we have an adjoint pair (−⊗R Σ,−⊗C Σ†)

(112) MR

−⊗RΣ //MC

−⊗CΣ†oo .

The unit and counit of the adjunction (−⊗R Σ,−⊗C Σ†) are given by the following formula

νN : N → (N ⊗R Σ)⊗C Σ†, ν(n) = nr ⊗R er ⊗A frζM : (M ⊗C Σ†)⊗R Σ →M, ζM (m⊗A ϕ⊗R u) = mϕ(u)

(= mµ(ϕ⊗R u)

)where N ∈MR and M ∈MC.


Proof. Consider the diagram of equalizers

HomC(Σ,M)⊗R R

M ⊗C Σ†

HomA(Σ,M)⊗R R

β′′⊗RR

δ′′⊗RR

αM // M ⊗A Σ†

ρM⊗AΣ†

M⊗Aλ

Σ†

HomA(Σ,M ⊗A C)⊗R R

αM⊗AC // M ⊗A C⊗A Σ†,

where λΣ† denotes the left C-coaction of Σ†, β′′ = HomA(Σ, ρM ) and δ′′(−) = (−⊗A C) ρΣ. Inorder to deduce that αM induces an isomorphism αM : HomC(Σ,M)⊗R R 'M ⊗C Σ† we needto check the following two identities:(a) αM⊗AC (β′′ ⊗R R) = (ρΣ ⊗A Σ†) αM(b) αM⊗AC (δ′′ ⊗R R) = (M ⊗A λΣ†) αM .To check (a), pick h⊗R r ∈ HomA(Σ,M)⊗R R and compute

αM⊗AC(β′′ ⊗R R)(h⊗R r) = αM⊗AC(ρΣh⊗R r)= ρΣh(es)⊗A ϕs ⊗R rs= h(es)[0] ⊗A h(es)[1] ⊗A ϕs ⊗R rs= (ρΣ ⊗A Σ†)(h(es)⊗A ϕs ⊗R rs)= (ρΣ ⊗A Σ†)αM (h⊗R r).

We shall deduce (b) by proving that αM⊗AC (δ′′ ⊗R R) α−1M = M ⊗A λΣ† as follows: given

m⊗A φ⊗R r ∈M ⊗A Σ†, write h⊗R r = α−1M (m⊗A φ⊗R r), which means that h(u) = mφ(u)

for every u ∈ Σ. Therefore,

(αM⊗AC (δ′′ ⊗R R) α−1M )(m⊗A φ⊗R r) = αM⊗AC(δ′′ ⊗R R)(h⊗R r)

= h(es[0])⊗A es[1] ⊗A ϕs ⊗R rs= mφ(es[0])⊗A es[1] ⊗A ϕs ⊗R rs= m⊗A φ(es[0])es[1] ⊗A ϕs ⊗R rs

= (M ⊗A λΣ†)(m⊗A φ⊗R r).

The expressions of the unit and counit are deduced from the expressions of the unit and counit ofthe adjunction (−⊗R Σ,HomC(Σ,−)⊗R R) (see Corollary 3.10) in combination with the aboveisomorphism HomC(Σ,−)⊗R R ' −⊗C Σ†. Remark that ν is well-defined by Theorem 4.12.

4.2.4. Comonadic-Galois versus firm Galois comodules.

Proposition 4.14. Let C be an A-coring, R a firm ring and Σ ∈ RMC. If Σ is a firm R-CGalois comodule, then S ⊗S Σ ∈ SMC is an S-C comonadic-Galois comodule, for any firm ring Ssuch that R ⊂ S ⊂ T = EndC(Σ).

Proof. Remark that Σ is not always a firm left S-module, even if it is firm as a left R-module, for this reason we have to consider S ⊗S Σ. Suppose Σ is a firm R-C Galois comodule.We construct an inverse map for the natural morphism (106). Define

νM : M ⊗A C → HomA(Σ,M)⊗S S ⊗S Σ,

as the composite of the following morphisms

M ⊗A C

M⊗Acan−1

νM // HomA(Σ,M)⊗S S ⊗S Σ

M ⊗A Σ∗ ⊗R Σ // HomA(Σ,M)⊗R R⊗R Σ

OO

We use the following notation can−1 : C → Σ† ⊗R Σ, can−1(c) = fc ⊗A uc, then we check

(113) νM canM (φ⊗S s⊗S u) = νM (φs(u[0])⊗A u[1]) = φs(u[0])fu[1]⊗S r ⊗S (uu[1]

)r


Since Σ ∈ RMC, the following diagram commutes

ΣρΣ,C

//

ρΣ,D

Σ⊗A C

Σ⊗A Σ∗ ⊗R ΣΣ⊗Acan

55jjjjjjjjjjjjjjj

i.e. ρΣ,C(u) = u[0]⊗A u[1] = erfr(ur[0])⊗A ur[1]. Applying Σ⊗A can−1 on this equation, we obtain

u[0] ⊗A fu[1]⊗R uu[1]

= er ⊗A fr ⊗R ur. If we apply now the firmness property of Σ to the last

factor in the tensor product, we find u[0] ⊗A fu[1]⊗R r ⊗R (uu[1]

)r = er ⊗A fr ⊗R r′ ⊗ urr′. With

this equality we can rewrite the last formula of (113) in the following way

φs(u[0])fu[1]⊗S r ⊗S (uu[1]

)r = φs(er)fr ⊗S r′ ⊗S urr′

= φs · r ⊗S r′ ⊗S urr′= φs⊗S r ⊗ r′urr

′

= φ⊗S s⊗S uFor the converse, first remark that εC = ev can−1

A , where ev is the evaluation map

ev : Σ∗ ⊗S Σ → A, ev(ϕ⊗A u) = ϕ(u).

This follows from the straightforward computation

εC canA(ϕ⊗S u) = ε(ϕ(u[0])u[1]) = ϕ(u) = ev(ϕ⊗S u),

and from can can−1 = C. Apply this new identity in the penultimate equality of the nextcomputation

canM νM (m⊗A c) = canM (mfc ⊗S r ⊗S (uc)r) = mfcr((uc)r[0])⊗A (uc)r[1]= mfc((r(uc)r)[0])⊗A (r(uc)r)[1] = mfc((uc)[0])⊗A (uc)[1]= m⊗A fc((uc)[0])(uc)[1] = m⊗A fc(1)(uc(1))c(2)

= m⊗A εC(c(1))c(2) = m⊗A c,where we used in the third equality that r is colinear and in the sixth equality that can is amorphism of right comodules.

Theorem 4.15. Let C be an A-coring, R a firm ring and Σ ∈ RMC. If Σ is R-firmlyprojective, then Σ is a firm R-C Galois comodule, if and only if Σ is an R-C comonadic-Galoiscomodule.

Proof. If Σ is an R-C Galois comodule, then it is an R-C comonadic-Galois comodule byProposition 4.14. To prove the converse, remember that since we know that Σ is R-firmly pro-jective, we can construct the comatrix coring Σ∗ ⊗R Σ. Moreover, Σ is comonadic-Galois, so inparticular canA : Σ∗ ⊗R Σ → C is an isomorphism (of right C-comodules). This map is exactlythe canonical coring morphism can (110), so Σ is also an R-C Galois comodule.

Theorem 4.16. Let C be an A-coring, R a firm ring and Σ ∈ RMC. We denote T = EndC(Σ)(i) If Σ is an R-C comonadic Galois module, then Σ is also a T -C comonadic-Galois comodule

(i.e. Σ is a Galois comodule in the sense of Wisbauer).(ii) If R is a left ideal in T and Σ is an T -C comonadic Galois module then Σ is also an R-C

comonadic-Galois comodule (and thus both properties are equivalent).

Proof. (i) Follows immediately from the commutativity of the following diagram and thesurjectivity of π,

HomA(Σ,M)⊗T ΣcanM,T // M ⊗A C

HomA(Σ,M)⊗R Σ

canM,R

44jjjjjjjjjjjjjjjjjπM

OO

(ii) If R is a left ideal in T , then we know by Lemma 2.15 that

πM : HomA(Σ,M)⊗R Σ → HomA(Σ,M)⊗T Σ


is an isomorphism for all M ∈MA. Consequently canM,T is an isomorphism if and only if canM,R

is an isomorphism.

Remark 4.17. That the converse of statement (i) of Theorem 4.16 does not hold in general,follows from the following example. Let C = A be the trivial A-coring, and take Σ = A. ThenEndA(A) = A and for all M ∈ MA, the canonical canM,A is the trivial isomorphism canM,A :HomA(A,M) ⊗A A → M ⊗A A. Take any nontrival ring morphism R → A, such that R is noideal in A. Then we can compute the maps πM and canM,R as

HomA(A,M)⊗R A ∼= M ⊗R A→M ⊗A A ∼= M

Which are clearly not isomorphisms in general.A sufficient condition to obtain that R is an ideal in T , and thus that the Galois-property in

the sense of Wisbauer [117] is equivalent to R-C comonadic Galois is, that the functor F = −⊗Σis a full and faithful functor (see Section 4.2.5).

Take any Σ ∈ SMC, for any firm ring S ⊂ T = EndC(Σ). As before, let Z = Σ⊗A Σ∗ be theelementary (possibly non-unital) S-ring associated to the dual pair (Σ,Σ∗, ev). If moreover Σ isan S-C comonadic-Galois comodule, then we can also construct comultiplications on Z and Σ asfollows.

dZ,Σ = (Σ⊗A can−1A ) ρΣ,C

dZ = dZ,Σ ⊗A Σ∗

Remark that dZ,Σ is equal to the map that is obtained by composing the obvious map Σ →EndA(Σ)⊗S Σ, u 7→ EndA(Σ)⊗S u, with the isomorphism

(Σ⊗A can−1A ) canΣ : EndA(Σ)⊗S Σ → Σ⊗A Σ∗ ⊗S Σ.

Consider the following diagram

ΣρΣ //

ρΣ

Σ⊗A CΣ⊗Acan−1

//

ρΣ⊗AC

Σ⊗A Σ∗ ⊗S Σ

ρΣ⊗AΣ∗⊗SΣ

(1) (2)

Σ⊗A CΣ⊗A∆ //

Σ⊗Acan−1

Σ⊗A C⊗A CΣ⊗AC⊗Acan−1

//

Σ⊗Acan−1⊗AC

Σ⊗A C⊗A Σ∗ ⊗S Σ

Σ⊗Acan−1⊗AΣ∗⊗SΣ

(3) (4)

Σ⊗A Σ∗ ⊗S ΣΣ⊗AΣ∗⊗Sρ

Σ

// Σ⊗A Σ∗ ⊗S Σ⊗A CΣ⊗AΣ∗⊗SΣ⊗Acan−1

// Σ⊗A Σ∗ ⊗S Σ⊗A Σ∗ ⊗S Σ

The commutativity of this diagram can be checked as follows. The square (1) commutes bycoassociatityty of Σ, the commutativity of (2) and (3) follow by direct computation (use can instead of can−1) and the commutativity of (4) is trivial. We obtain that

(dZ ⊗S Σ) dZ,Σ = (Z ⊗S dZ,Σ) dZ,Σ,

after tensoring the above equality with Σ∗ we find

(dZ ⊗S Z) dZ = (Z ⊗S dZ) dZ .

So Z is a (non-counital) S-coring and Σ is a left Z-comodule. Moreover since εC = ev can−1,we have

µZ,Σ dZ,Σ = Σ;(114)

µZ dZ = Z.

But we have more.


Proposition 4.18. Let C be an A-coring and S any firm ring. Take Σ ∈ SMC. The followingstatements are equivalent

(i) The following canonical map is an isomorphism of A-A bimodules

can : Σ∗ ⊗S Σ → C, can(f ⊗S u) = f(u[0])u[1];

(e.g. Σ is a S-C comonadic-Galois comodule)(ii) we have comatrix coring context (Z,A,Σ,Σ†, η, ε) with Σ† = Σ∗ ⊗S Z (i.e. Σ is firmly

projective) and the following map is an isomorphism of A-corings

can : Σ† ⊗Z Σ → C, can(ϕ⊗S z ⊗S x) = can(ϕ⊗S µZ,Σ(z ⊗S x)).

If there exists a firm ring R and a ring morphism ι : R→ Z, such that Σ ∈ RMC and R is a leftideal in S, then Σ is an R-C Galois comodule.

Proof. Suppose first that can is an isomorphism. Let us prove that Σ is firmly projective.Consider the maps µZ , µZ,Σ, dZ and dZ,Σ introduced above, where Z is considered as a (non-unital)S-ring and a (non-counital) S-coring. Denote by µZ : Z ⊗Z Z → Z and dZ : Z → Z ⊗S Z →Z ⊗Z Z the induced multiplication and comultiplication map, and introduce µZ,Σ and dZ,Σ in asimilar way.

By (114), dZ,Σ is a right inverse of µZ,Σ and, consequently, µZ,Σ dZ,Σ = Σ. Let us show thatdZ,Σ is also a left inverse for µZ,Σ. Denote dZ,Σ(u) = ez ⊗A fz ⊗Z uz. Since Σ ∈ ZMA, we find

dZ,Σ µZ,Σ(u⊗A ϕ⊗Z v) = ez ⊗A fz ⊗R uzϕ(v) = ez ⊗A fz ⊗Z (uz ⊗A ϕ) · v= ez ⊗A fz · (uz ⊗A ϕ)⊗Z v = ez ⊗A fz(uz)ϕ⊗Z v= ezfz(uz)⊗A ϕ⊗Z v = u⊗A ϕ⊗Z v

So we find that dZ,Σ µZ,Σ = Σ. Consequently dZ µZ = Z, so Z is a firm ring and Σ is a firmleft Z-module, i.e. Σ is firmly projective. By Theorem 2.51 this implies that we have a comatrixcoring context (Z,A,Σ,Σ†, η, ε) if we define

ε : Σ† ⊗Z Σ ∼= Σ∗ ⊗S Σ → A, ε(ϕ⊗S z ⊗Z u) = ϕ(zu);

η = dZ : Z → Σ⊗Z Σ† = Σ⊗A Σ∗ ⊗S Z.Let us check that can is an A-coring isomorphism. Denote the comatrix coring by D, then we findby µZ,Σ dZ,Σ = Σ that

(can⊗A can) ∆D(ϕ⊗S x⊗A ψ ⊗Z y) = (can⊗A can)(φ⊗S x[0] ⊗A can−1(x[1])ψ(y))

= ϕ(x[0])x[1] ⊗A x[1]ψ(y)

= ∆C can(ϕ⊗S x⊗A ψ ⊗Z y)The implication (ii) ⇒ (i) is trivial.Finally, it follows from Lemma 2.15 that C ∼= Σ∗ ⊗S Σ ∼= Σ∗ ⊗R Σ, and Σ is an R-C Galois

comodule.

4.2.5. Structure theorems. In Section 4.2.2 we have seen how a diagram of the type (100)can be obtained in different (but equivalent) ways. In particular we know that all adjoint pairs(F,G),

(115) MR

F //MC

Goo

can be extended to a diagram of this type, that is F ' −⊗R Σ and G ' HomC(Σ,−)⊗R R suchthat

MR

−⊗RΣ //

−⊗RΣ


oo

GC=−⊗AC

MC

FC

OO

HomC(Σ,−)⊗RR



In the present section, we want to characterize all adjoint pairs (F,G), in different steps, until(F,G) establishes an equivalence of categories between MR and MC.

Theorem 4.19. Let R be a firm ring and C an A-coring. Let (F,G) be an adjoint pair as in(115). Denote Σ = F (R) and Σ† = G(C). Then the following assertions are equivalent.

(i) GGC preserves colimits;(ii) Σ is R-firmly projective as a right A-module;(iii) There is an R-firm dual pair (Σ,Σ†, µ).

Proof. (i) ⇒ (iii) By (101) we know that HomA(Σ,−)⊗R R ' GGC : MA →MR. Since

this functor preserves colimits, it has a right adjoint H, and thus (FCF,GGC,H) is a ‘triple’ ofadjoint functors in the sense of Theorem 2.51. Since FCF (R) ∼= Σ and GGC(A) = Σ∗ ⊗R R ∼=HomC(Σ,C)⊗R R = Σ† we find by the same Theorem that (Σ,Σ†, µ) is an R-firm dual pair.(iii) ⇒ (i) Again by Theorem 2.51 we know that (−⊗RΣ,−⊗AΣ†) is an adjoint pair of functors.

Furthermore, −⊗AΣ† preserves colimits. Now by Theorem 2.51(vi) we find a natural isomorphismα : GGC = HomA(Σ,−)⊗R R ' −⊗A Σ†.(ii) ⇔ (iii). Follows by Proposition 2.50.

Corollary 4.20. Let R be a firm ring and C an A-coring. Let (F,G) be an adjoint pairas in (115). Denote Σ = F (R) and Σ† = G(C). If G has a right adjoint, then Σ is R-firmlyprojective as a right A-module.

Proof. We know from Corollary 3.9(i) that GC has a right adjoint HC. If we denote H forthe right adjoint of G, then HCH will be a right adjoint for GGC and the statement follows fromTheorem 4.19.

Our next aim is to characterize adjoint pairs (F,G) for which G is a fully faithful functor.

Recall that we denote by MR the category of all (possibly non-firm) right R-modules and R-linearmaps. If we consider as before an adjoint pair (F,G) as in (115), then we know from Corollary 3.10

that F and G factorize in the following way trough MR

MR

J // MR

F ′=−⊗RΣ //

−⊗RRoo MC

G′=HomC(Σ,−)

oo

We will denote the counit of (F ′, G′) by ζ ′ and the counit of (F,G) by ζ.Let us recall some basic facts about generators. We include the proof of our next Lemma for

the sake of completeness.

Lemma 4.21. Let C be an A-coring, and Σ ∈MC.

(i) Σ generates MC: if 0 6= g : M → N in MC, then there exists f ∈ HomC(Σ,M) such thatg f 6= 0.

(ii) for all M ∈MC, ζ ′M : HomC(Σ,M)⊗T Σ →M is surjective.

(iii) for all M ∈MC, ζ ′M : HomC(Σ,M)⊗T Σ →M is bijective.

The first two statements are equivalent. If C is flat as a left A-module, then all three statementsare equivalent.

Proof. (i) ⇒ (ii). The image of ζ ′M is a right C-comodule, and we can consider the canonical

projection g : M → M/Im (ζ ′M ) in MC. For all f ∈ HomC(Σ,M) and u ∈ Σ, (g f)(u) =g(ζ ′M (f ⊗ u)) = 0, hence g = 0, and ζ ′M is surjective.

(ii) ⇒ (i). Take m ∈M such that g(m) 6= 0. We have fi ∈ HomC(Σ,M) and ui ∈ Σ such that

m =∑

i fi(ui). If g(ϕi(ui)) = 0 for all i, then g(m) = g(∑

i fi(ui)) = 0, which is impossible.Hence there exists i such that g fi 6= 0.(ii) ⇒ (iii). (along the lines of [36, 43.12]). Assume that C is flat as a left A-module. We have

to show that every ζ ′M is injective. Take∑k

i=1 fi ⊗mi ∈ Ker ζ ′M , i.e.∑k

i=1 fi(mi) = 0.Consider the projection πi : Σk → Σ onto the i-th component, and

f =k∑i=1

fi πi ∈ HomC(Σk,M).


Ker f ∈ MC, since C is flat (see [36]). Also (m1, · · · ,mk) ∈ Ker f , since f(x1, · · · , xk) =∑ki=1 fi(xi). By assumption, the map

ζ ′Ker f : HomC(Σ,Ker f)⊗T Σ → Ker f

is surjective, hence we can find aj ∈ Σ and gj ∈ HomC(Σ,Ker f) such that

l∑j=1

gj(aj) = (m1, · · · ,mk)

and

k∑i=1

fi ⊗T mi =k∑i=1

fi ⊗Tl∑

j=1

(π gj)(aj)

=l∑

j=1

(k∑i=1

fi πi

) gj ⊗T aj

=l∑

j=1

f gj ⊗T aj = 0

since Im gj ⊂ ker f .

Take any M ∈ MC. Then it is easy to see that the canonical injection jM : HomC(Σ,M) →HomA(Σ,M) is the equalizer of (HomA(Σ, ρM ), (−⊗A C) ρΣ) in the categories MR and MT .

(116) HomC(Σ,M)jM // HomA(Σ,M)

HomA(Σ,ρM ) //

(−⊗AC)ρΣ// HomA(Σ,M ⊗A C)

Recall from the proof of Lemma 2.16 that the functor − ⊗R R : MR → MR is always exact.Tensoring (116) with R over R, we obtain the following equalizer in MR.(117)

HomC(Σ,M)⊗R Rj′M=jM⊗RR // HomA(Σ,M)⊗R R

HomA(Σ,ρM )⊗RR//

(−⊗AC)ρΣ⊗RR

// HomA(Σ,M ⊗A C)⊗R R

Theorem 4.22. Let R be a firm ring and C an A-coring. Let (F,G) be an adjoint pair as in(115). Denote Σ = F (R), then the following assertions are equivalent.

(i) G : MC →MR is fully faithful;

(ii) G′ = HomC(Σ,−) : MC → MR is fully faithful;

(iii) Σ is R-C comononadic-Galois and − ⊗R Σ : MR → MC preserves the equalizer jM for allM ∈MC;

(iv) Σ is R-C comononadic-Galois and − ⊗R Σ : MR → MC preserves the equalizer j′M for allM ∈MC;

If moreover C is flat as a left A-module then any of the previous conditions (i)-(iv) are furthermoreequivalent to

(v) Σ is R-C comononadic-Galois and Σ is flat as a left R-module;

If R is a left ideal in T , then the conditions (i)-(iv) are furthermore equivalent to

(vi) G′′ = HomC(Σ,−) : MC →MT is fully faithful;(vii) Σ is T -C comononadic-Galois and −⊗T Σ preserves the equalizers jM for all M ∈MC.(viii) (if AC is flat as left A-module) Σ is a generator in MC;(ix) (if AC is flat as a left A-module) Σ is T -C comononadic-Galois and Σ is flat as a left T -module.

Proof. (i) ⇔ (ii). Recall that G (or G′) is fully faithful if and only if the counit of the

adjunction is an isomorphism. If we compare the counit of (F,G) to the counit of (F ′, G′) (see


Corollary 3.10) we obtain for all N ∈MC

ζN : HomC(Σ, N)⊗R R⊗R Σ → N, ζN (ϕ⊗R r ⊗R u) = ϕ(ru)

ζ ′N : HomC(Σ, N)⊗R Σ → N, ζ ′N (ϕ⊗R u) = ϕ(u)

The isomorphism R⊗RΣ ∼= Σ coming from the firmness of Σ as a left R-module, implies that ζNand ζ ′N are isomorphic up to isomorphism.(ii) ⇔ (iii). Recall that FC(ζ ′N ) = evN (see (107)). If G′ is fully faithful then ζ ′N is an iso-

morphism for all N ∈ MC, and Σ is R-C comonadic-Galois by Theorem 4.4. Consider now thefollowing diagram.

HomC(Σ,M)⊗R Σj⊗RΣ //

ζM

HomA(Σ,M)⊗R ΣHomA(Σ,ρM )⊗RΣ//

((−⊗AC)ρΣ)⊗RΣ

//

canM

HomA(Σ,M ⊗A C)⊗R Σ

canM⊗AC

M

ρM// M ⊗A C

M⊗A∆C //

ρM⊗AC

// M ⊗A C⊗A C

One can easily check that this diagram commutes (following equally aligned arrows). The lowerrow is an equalizer. If G′ is fully faithful, then all vertical arrows are isomorphism, so the upperrow is also an equalizer, which implies that the equalizer jM is preserved by −⊗R Σ. Conversely,if Σ is R-C comonadic-Galois and −⊗R Σ preserves the equalizer jM , then both upper and lowerrow in the diagram are equalizers and since canM and canM⊗AC are isomorphisms we obtain fromthe universal property of the equalizer that ζM must be an isomorphism as well.(iii) ⇔ (iv). Consider the commutative diagram of functors

MR

−⊗RΣ //

−⊗RR ""EEE

EEEE

EAb

MR

−⊗RΣ

==

Recall that the functor − ⊗R R : MR → MR is exact (see proof of Lemma 2.16) and everyequalizer of the form (117) is obtained from an equalizer of the form (116) after appying this

exact functor. Then obviously − ⊗R Σ : MR → Ab preserves the equalizers (116) if and only if−⊗R Σ : MR → Ab preserves the equalizers (117).

Suppose now that C is flat as a left A-module.(ii) ⇔ (v). Since clearly (v) implies (iv), we only have to prove the implication (ii) ⇒ (v). As in

the proof of (ii) ⇔ (iii) we obtain that Σ is R-C comonadic-Galois. Hence, we have to show thatΣ is flat as a left R-module. By Proposition 2.20 it suffices to show that for any right ideal offinite type J = f1R+ · · ·+ fkR of R, the map µJ : J ⊗R Σ → JΣ, µJ(g⊗u) = g(u) is injective.Let us consider the surjective map φ : Σn → JΣ, f(u1, . . . , un) =

∑i fi(ui). We put K = Kerφ,

then K ∈MC, since AC is flat. Moreover, we have an exact sequence

0 // HomC(Σ,K)α // HomC(Σ,Σn)

β // J // 0.

Tensoring by Σ over R, we obtain the following commutative diagram with exact rows:

HomC(Σ,K)⊗R Σα⊗Σ //

ζ′K

HomC(Σ,Σn)⊗R Σβ⊗Σ //

ζ′Σn

J ⊗R Σ //

µJ

0

0 // K // Σnφ // JΣ // 0

Since G′ is fully faithful, the counit ζ ′K and ζ ′Σn are isomorphisms. After this, it follows from theproperties of the diagram that µJ is injective.

Suppose from now on that R is an ideal in T .(i) ⇔ (vi). We obtain from Lemma 2.15 that the counit of associated to G′′ is up to isomorphismidentical to ζ.(vi) ⇔ (vii). This can be proven as in the first part.


(vii) ⇔ (viii). Follows from Lemma 4.21.

(ix) ⇒ (vii) Clear.

(v) ⇒ (ix). By Theorem 4.16 the Σ is T -C comonadic Galois, so we only have to prove that theflatness of Σ as a left R-module implies the flatness of Σ as a left T -module. Take any sequenceN → M → P in MT . By Lemma 2.15, we know that N ⊗T Σ ∼= N ⊗R Σ as Z-module, andthe same counts for M and P . Therefore Σ is flat as a left T -module whenever Σ is flat as a leftR-module.

Suppose now that Σ is R-firmly projective as a right A-module. Then we know from Theo-rem 4.19 that Σ† = G(C) ∼= Σ∗ ⊗R R is a C-R bicomodule. If R is flat as left R-module, we findfor all M ∈MC the commutative diagram of equalizers in MR.

(118) HomC(Σ,M)⊗R Rj′M //

α′M

HomA(Σ,M)⊗R R

αM

HomA(Σ,ρM )⊗RR//

(−⊗AC)ρΣ⊗RR

// HomA(Σ,M ⊗A C)⊗R RαM⊗AC

M ⊗C Σ†

eqM,Σ† // M ⊗A Σ†

ρM⊗AΣ† //

M⊗AλΣ†

// M ⊗A C⊗A Σ†

where we used the isomorphism α′ : G ' −⊗CΣ† from Theorem 4.13 and the natural isomorphismα : HomA(Σ,−)⊗R R→ −⊗A Σ† form Theorem 2.51.

Theorem 4.23. Let R be a firm ring and C an A-coring. Let (F,G) be an adjoint pair asin (115). Suppose that Σ = F (R) is R-firmly projective as a right A-module.Then the followingassertions are equivalent.

(i) G : MC →MR is fully faithful;

(ii) G′ = HomC(Σ,−) : MC → MR is fully faithful;(iii) Σ is R-C comononadic-Galois, −⊗R Σ preserves the equalizers jM for all M ∈MC;(iv) Σ is a firm R-C Galois comodule, −⊗R Σ preserves the equalizers jM for all M ∈MC;If R is flat as left R-module, then the statements (i)-(iv) (and possible equivalent statements) arefurthermore equivalent to(v) −⊗C Σ† : MC →MR is fully faithful;(vi) Σ is R-C comononadic-Galois, −⊗R Σ preserves the equalizers eqM,Σ† for all M ∈MC;

(vii) Σ is a firm R-C Galois comodule, −⊗R Σ preserves the equalizers eqM,Σ† for all M ∈MC;

(viii) Σ is a firm R-C Galois comodule and for all M ∈ MC, the following isomorphism holds(M ⊗C Σ†)⊗R Σ ∼= M ⊗C (Σ† ⊗R Σ).

If R is a left ideal in T , then the assertions (i)-(iv) (and possible equivalent statements) arefurthermore equivalent to(ix) G′′ = HomC(Σ,−) : MC →MT is fully faithful;(x) Σ is T -C comononadic-Galois and −⊗T Σ preserves the equalizers eqM,Σ† for all M ∈MC;

(xi) Σ is T -C comononadic-Galois and −⊗T Σ preserves the equalizers jM for all M ∈MC;(xii) If moreover C is flat as a left A-module, then (ix)-(xi) are furthermore equivalent to

Σ is T -C comononadic-Galois and and Σ is flat as a left T -module;If C is flat as left A-module, then the statements (i)-(iv) (and possible equivalent statements) arefurthermore equivalent to(xiii) Σ is R-C comononadic-Galois, and Σ is flat as a left R-module;(xiv) Σ is a firm R-C Galois comodule, and Σ is flat as a left R-module.(xv) Σ is a generator in MC.If R is flat as a left R-module, then the flatness of Σ as a left R-module (in particular statements(xiii) and (xiv)) implies the flatness of C as a left A-module.

Proof. The equivalence between (i), (ii) and (iii) follows immediately from Theorem 4.22andthe equivalence with (iv) follows from Theorem 4.15.Suppose now that R is flat as a left R-module. The equivalence with (v) follows from the naturalisomorphism α′ : G ' − ⊗C Σ† in Theorem 4.13. Consider diagram (118). All vertical arrowsin this diagram are isomorphisms. Consequently this diagram implies that − ⊗R Σ preserves anequalizer of the form jM if and only if −⊗R Σ preserves the equalizer eqM,Σ† . Therefore, (vi) is


equivalent to (iii) and (vii) is equivalent to (iv). Obviously, (viii) is equivalent to (vii).The equivalence to the statements (ix)-(xii) as well to the statements (xiii)-(xv) follows fromTheorem 4.22 and the above observations.For the last statement, we have to prove that C is flat as a left A-module if R is flat as a leftR-module. Since Σ is R-firmly projective as a right A-module, and R is flat as a left R-module,we know from Theorem 2.51 that Σ† is flat as a left A-module. Therefore, Σ† ⊗R Σ is flat as aleft A-module because we know that Σ is flat as a left R-module. Finally, since G is fully faithful,Σ is an R-C Galois comodule and hence Σ† ⊗R Σ ∼= C. We conclude that C is flat as a leftA-module.

We now concentrate on the functor F : MR →MC. If F (R) = Σ is R-firmly projective, weare able to state sufficient and necessary conditions under which F is a full and faithful functor.Let us first prove the following lemma.

Lemma 4.24. Let R be a firm ring and Σ a left R-module. If Σ is R-firmly projective as aright A-module, then Σ is totally faithful as a left R-module if and only if Σ is faithful as a leftR-module an the morphism ` : R→ EndA(Σ) is pure as a left R-linear map.

Proof. We already know from Proposition 2.21 that if Σ is totally faitful, then Σ is faithfuland ` is pure. To prove the converse, take any N ∈ MR and suppose N ⊗R Σ = 0. Since Σ isR-firmly projective, the map ` : R→ EndA(Σ) factorizes as

Ri // Σ⊗A Σ∗ // EndA(Σ)

If we tensor ` with N ⊗R −, then we obtain an injective morphism

N ⊗R ` : N ∼= N ⊗R R→ N ⊗R EndA(Σ)

which factorizes as

N ∼= N ⊗R RN⊗Ri // N ⊗R Σ⊗A Σ∗ // N ⊗R EndA(Σ)

and since N ⊗R Σ is zero, we obtain that N ⊗R ` is the zero-morphism. Because we know it wasinjective, we conclude that N = 0 and Σ is totally faithful.

Recall from Lemma 2.9 that the regular right R-module is a generator in MR. Therefore, wecan find for any object N ∈ MR two free right R-modules F1 = R(I1) and F2 = R(I2) togetherwith following exact sequence in MR

(119) F1g // F2

// N // 0

As in Lemma 3.6, one can easily check that N is exacly the cokernel of g. Since −⊗R Σ : MR →MC is right exact, we obtain the following exact sequence in MC

(120) F1 ⊗R Σ // F2 ⊗R Σ // N ⊗R Σ // 0

Theorem 4.25. Let R be a firm ring that is flat as a left R-module and C an A-coring.Let (F,G) be an adjoint pair as in (115) and denote Σ = F (R). Then following statements areequivalent.

(i) F is fully faithful;(ii) R is a left ideal in T and for all N ∈ MR the natural map fN : N ⊗R EndC(Σ) ⊗R R →

HomC(Σ, N ⊗R Σ)⊗R R is an isomorphism;

Suppose that Σ = F (R) is a R-firmly projective as a right A-module. Then assertions (i)-(ii) arefurthermore equivalent to

(iii) (if R is flat as a left R-module) R is a left ideal in T and for all N ∈MR the following mapis an isomorphism N ⊗R (Σ⊗C Σ†) ∼= (N ⊗R Σ)⊗C Σ†;

(iv) (if R is flat as a right R-module) G preserves the exact sequences of the form (120);

In particular, F is fully faithful if R is flat as a right R-module and G preserves coequalizers, ifR is flat as a left and right R-module and Σ† is left C-coflat or if Σ is totally faithful as a leftR-module. Conversely, if F is fully faithful and the map t : Σ ⊗A Σ† → EndA(Σ) is a pure leftR-monomorphism, then Σ is totally faithful.


Proof. (i) ⇔ (ii). If F ' −⊗R Σ is fully faithful, we know that the unit of the adjunctionis an isomorphism. Evaluating this unit in R, we obtain an isomorphism of right R-modules

R ∼= HomC(Σ, R⊗R Σ)⊗R R ∼= HomC(Σ,Σ)⊗R R = T ⊗R R.It follows from Lemma 2.14 that R is a left ideal in T . Consider any N ∈MR, if R is a left idealin T , then we define an isomorphism vN : N → N⊗REndC(Σ)⊗RR as the following composition

N ∼= N ⊗R R ∼= N ⊗R (T ⊗R R) = N ⊗R (EndC(Σ)⊗R R)

where we used Lemma 2.14 in the second isomorphism. Consider the commutative diagram

NνN //

vN

HomC(Σ, N ⊗R Σ)⊗R R

N ⊗R EndC(Σ)⊗R RfN

44iiiiiiiiiiiiiiiii

where fN (n⊗R ψ⊗R r) = [u 7→ n⊗R ψ(u)]⊗R r and explicitly we find vN (n) = nr ⊗R 1Σ ⊗R r,hence the diagram commutes. Since vN is an isomorphism, then νN is an isomorphism if and onlyif fN is an isomorphism.(ii) ⇔ (iii). If we apply Theorem 4.13, then we obtain the following commutative diagram

N ⊗R EndC(Σ)⊗R R∼=

//

∼=

HomC(Σ, N ⊗R Σ)⊗R R

N ⊗R (Σ⊗C Σ†) // (N ⊗R Σ)⊗C Σ†

Since the vertical maps are ismorphisms, we find that the upper horizontal map is an isomorphismif and only if the lower horizontal map is an isomorphism.(iii) ⇔ (iv). Consider the following commutative diagram, where the first row is obtained from

(120) by cotensoring with Σ† and the lower row is obtained from (119) by tensoring with (Σ⊗CΣ†)

(F1 ⊗R Σ)⊗C Σ†

∼=

// (F2 ⊗R Σ)⊗C Σ†

∼=

// (N ⊗R Σ)⊗C Σ†

// 0

F1 ⊗R (Σ⊗C Σ†) // F2 ⊗R (Σ⊗C Σ†) // N ⊗R (Σ⊗C Σ†) // 0

The two vertical isomorphisms follow from Lemma 3.6. The lower row is exact since the func-tor − ⊗R (Σ ⊗C Σ†) is right exact. If the right vertical morphism is an isomorphism, then theupper row is also exact, and G ' − ⊗C Σ† preserves the exact row (120). If the upper row isexact, then the right vertical morphism is an isomorphism by the universal property of the cokernel.

Finally, if G preserves coequalizers or Σ† is coflat as a left C-comodule, then obviously condition(iv) is satisfied.If Σ is totally faithful, then we find by Lemma 4.24, N ⊗R ` = (N ⊗R t) (N ⊗R i) is injective,hence N ⊗R i is injective for all N ∈MR and we obtain the following diagram

(121) N ∼= N ⊗R RN⊗Ri //

νN ((RRRRRRRRRRRRR N ⊗R Σ⊗A Σ†

(N ⊗R Σ)⊗C Σ†

j

OO

where j is a canonical injection and νN is the alternative form for the unit of the adjunction (F,G)as defined in Theorem 4.12, which makes the diagram commutative. If Σ is totally faithful, thenN ⊗R ` = (N ⊗R t) (N ⊗R i) is injective, hence N ⊗R i is injective, then we obtain fromthe commutativity of diagram (121), that νN is injective as well. Let us prove that νN is also

surjective. Consider N = (N ⊗R Σ)⊗C Σ†/νN (N), and the canonical projection

π : (N ⊗R Σ)⊗C Σ† → N .


Consider an element x⊗R u ∈ N ⊗R Σ. Let x =∑

j nj ⊗R uj ⊗A gj ∈ (N ⊗R Σ)⊗C Σ†. Then∑j

ν(nj)⊗R uj ⊗A gj =∑j

nrj ⊗R er ⊗A fr ⊗R uj ⊗A gsjs

=∑j

nrj ⊗R er ⊗A fr ⊗R uj ⊗A gsj (es)fs

=∑j

nrj ⊗R er ⊗A fr ⊗R ujgsj (es)⊗A fs

=∑j

nrj ⊗R er ⊗A fr(uj)gsj ⊗R es ⊗A fs

=∑j

nrj ⊗R erfr(uj)⊗A gsj ⊗R es ⊗A fs

=∑j

nrj ⊗R r(uj)⊗A gsj ⊗R es ⊗A fs

=∑j

nrjr ⊗R (uj)⊗A gsj ⊗R es ⊗A fs

=∑j

nj ⊗R (uj)⊗A gsj ⊗R es ⊗A fs

= xr ⊗R er ⊗A fr.

Applying π to the first three tensor factors, we find

0 =∑j

π(ν(nj))⊗R uj ⊗A gj = π(xr)⊗R er ⊗A fr,

hence,

0 = π(xr)⊗R erfr(u) = π(xr)⊗R ru = π(xr)r ⊗R u = π(x)⊗R u,

so N ⊗R Σ = 0 and hence N = 0, as needed.Conversely, if F is fully faithful then the unit of the adjunction is bijective, hence νN is bijectiveand in particular injective. It follows from the diagram (121) that N ⊗R i is injective as well. Sincet is moreover a pure monomorphism, N ⊗R t is injective for all N ∈MR and therefore N ⊗R ` =is injective for all N ∈MR provided that N ⊗R i is injective.

The proof of the following technical lemma is just an easy verification.

Lemma 4.26. Let R and A be two arbitrary rings. Take any M ∈MR and Σ ∈ RMA. Thenthe following maps define an equalizer (M ⊗R Σ, e) as in diagram (122).

e : M ⊗R Σ → HomA(Σ,M ⊗A Σ)⊗R Σ,e(m⊗R u) = ϕm ⊗R u;

e1 : HomA(Σ,M ⊗A Σ)⊗R Σ → HomA(Σ,HomA(Σ,M ⊗A Σ)⊗R Σ)⊗R Σ,e1(ϕ⊗R u) = ψϕ ⊗R u

e2 : HomA(Σ,M ⊗A Σ)⊗R Σ → HomA(Σ,HomA(Σ,M ⊗A Σ)⊗R Σ)⊗R Σ,e2(ϕ⊗R u) = ξmi ⊗R ui, where ϕ(u) = mi ⊗R ui

ϕm : Σ →M ⊗R Σ,ϕm(u) = m⊗R u;

ψϕ : Σ → HomA(Σ,M ⊗A Σ)⊗R Σ,ψϕ(v) = (w 7→ ϕ(w))⊗R v

ξm : Σ → HomA(Σ,M ⊗A Σ)⊗R Σ,ξm(v) = ϕm ⊗R v


where we took arbitrary elements m ∈M , u ∈ Σ and ϕ ∈ HomA(Σ,M ⊗A Σ).(122)

M ⊗R Σ e // HomA(Σ,M ⊗A Σ)⊗R Σe1 //e2// HomA(Σ,HomA(Σ,M ⊗A Σ)⊗R Σ)⊗R Σ

Combining Theorem 4.22 and Theorem 4.25, we obtain a large number of equivalent conditionsunder which the adjunction (F,G) of (115) is an equivalence of categories. The next theorem,some of these are explicitly stated and a few more characterizations are given.

Theorem 4.27. Let R be a firm ring and C an A-coring. Let (F,G) be an adjoint pair as in(115). Denote Σ = F (R) and Σ† = G(C). The following statements are equivalent

(i) (F,G) establishes an equivalence of categories;

(ii) Σ is a firm R-C Galois comodule and −⊗R Σ : MR →MΣ†⊗RΣ induces an equivalence ofcategories;

(iii) Σ is a firm R-C Galois comodule, R is a left ideal in T , the functor − ⊗R Σ preserves theequalizers jM for all M ∈ MC and the natural map fN : N ⊗R T ⊗R R→ HomC(Σ, N ⊗RΣ)⊗R R is an isomorphism for all N ∈MR;

If R is flat as left R-module, the above statements are furthermore equivalent to(iv) Σ is a firm R-C Galois comodule, R is a left ideal in T and the following isomorphisms hold

for all M ∈MC and all N ∈MR,

N ⊗R (Σ⊗C Σ†) ∼= (N ⊗R Σ)⊗C Σ†

(M ⊗C Σ†)⊗R Σ ∼= M ⊗C (Σ† ⊗R Σ)

If moreover C is flat as a left A-module then the above statements are furthermore equivalent to(v) Σ is a firm R-C Galois comodule and a faithfully flat left R-module;(vi) Σ is an R-C comonadic-Galois comodule and a faithfully flat left R-module;(vii) Σ is an R-C comonadic-Galois comodule, −⊗R Σ : MR →MC is fully faithful and Σ is flat

as a left R-module;(viii) HomC(Σ,−)⊗R R : MC →MR is fully faithful and Σ is totally faithful as a left R-module

and firmly projective as a right A-module;(ix) Σ is a generator of MC such that −⊗R Σ : MR →MC is fully faithful;(x) Σ is a generator of MC such that −⊗R Σ : MR →MC is faithful, R is a left ideal in T ;(xi) the identical statements of (vii)-(viii), where we replace −⊗R Σ : MR →MC by −⊗R Σ :

MR →MΣ†⊗RΣ

If R is flat as a left R-module, then the flatness of C as a left A-module follows from statements(v)-(vii). If R is furthermore flat as a right R-module, all previous assertions are then furthermoreequivalent to

(xii) Σ is a firm R-C Galois comodule that is flat as a left R-module, Σ† = Σ∗ ⊗R R is coflat asa left C-comodule and R is a left ideal in T .

If −⊗R R : MR →MR reflects epimorphisms (e.g. R is faithfully flat as a left R-module) or Ris projective in MR, then Σ is a projective generator in MC.

Proof. (i) ⇔ (ii). Suppose first that (F,G) is an equivalence. Then in particular F is a rightadjoint for G. We obtain from Corollary 4.20 that Σ is R-firmly projective as a right A-module.Theorem 4.23 tells us that under this condition the fully faithfulness of G implies that Σ is a firmR-C Galois comodule. The Galois condition includes an isomorphism of corings can : Σ†⊗RΣ → C,

and this isomorphism induces an equivalence of categories CAN : MΣ†⊗RΣ →MC which makesthe following diagram commutative

(123) MΣ†⊗RΣCAN //MC

MR

F ′=−⊗RΣ

eeJJJJJJJJJ F=−⊗RΣ

<<yyyyyyyy

This implies that if Σ is an R-C Galois comodule, then F induces an equivalence of categories ifand only if F ′ induces an equivalence of categories.


(i) ⇔ (iii) ⇔ (iv). This follows directly from Theorem 4.22 and Theorem 4.25, taking into ac-

count the observation in the previous part that (i) implies that Σ is R-firmly projective as rightA-module.(i) ⇒ (vi). By Theorem 4.22 Σ must be an R-C comonadic-Galois comodule that is flat as a

left R-module. Moreover, the functor − ⊗R Σ : MR → MC is faithful. Since C is flat as a leftA-module, the forgetful functor is faithful, and thus −⊗R Σ : MR →MA is faithful. Combiningour results, we find that Σ is faithfully flat as a left R-module.(vi) ⇒ (vii). We have to show that for any right R-module M ∈MR, the unit of the adjunction

(F,G)

νM : M → HomC(Σ,M ⊗R Σ)⊗R R, νM (m) = ϕmr ⊗R r; with ϕm(u) = (m⊗R u),

is an isomorphism. Consider the following commutative diagram

M ⊗R Σ

e

νM⊗RΣ // HomC(Σ,M ⊗R Σ)⊗R Σ

jM⊗RΣ⊗RΣ

HomA(Σ,M ⊗R Σ)⊗R Σ

e1

e2

HomA(Σ,M ⊗R Σ)⊗R Σ

HomA(Σ,HomA(Σ,M ⊗R Σ)⊗R Σ)⊗R Σ

HomA(Σ,canM⊗RΣ)⊗RΣ// HomA(Σ,M ⊗R Σ⊗A C)⊗R Σ

The vertical lines in this diagram discribe equalizers. The left column is the equalizer of Lemma 4.26,the right column is an equalizer, since jM⊗RΣ is an equalizer (see (116)) and Σ is flat as a leftR-module, so it preserves equalizers. The lower horizontal morphism is an isomorphism, since Σ isa comonadic-Galois comodule. Therefore, we obtain from the universal property of the equalizer,that νM ⊗R Σ is an isomorphism. Since Σ is a faithfully flat left R-module, νM must be anisomorphism as well.(vii) ⇒ (i). We obtain from Theorem 4.22 that since Σ is flat as a left R-module, that G is afully faithful functor. Hence −⊗R Σ induces an equivalence of categories.(vi) ⇒ (viii). We already know that (vi) implies (i), so G = HomC(Σ,−) ⊗R R is fully faithful.From Proposition 2.24 we know that if Σ is faithfully flat, then it is in particular totally faithful.(viii) ⇒ (i). We know from Theorem 4.22 that Σ is an R-C comonadic Galois comodule, hencea firm R-C comonadic-Galois comodule. Then by Theorem 4.25 we obtain that F is fully faithfuland therefore an equivalence of categories.(v) ⇔ (vi). If Σ is a firm R-C Galois comodule, then it is also a comonadic Galois comodule

by Proposition 4.14. Conversely, we have already proven that (vi) is equivalent to (ii) and there-fore (vi) implies that Σ is R-firmly projective as a right A-module. By Theorem 4.15, Σ is R-Ccomonadic-Galois if and only if Σ is firm R-C Galois and we obtain the required equivalences.(i) ⇒ (ix). Clearly, F = − ⊗R Σ is fully faithful and by Lemma 4.21, the fully faithfulness of G

implies that Σ is a generator in MC.(ix) ⇒ (x). By Theorem 4.25, R is a left ideal in T if F ' −⊗R Σ is fully faithful.

(x) ⇒ (vi). By Theorem 4.22 we find that Σ is an R-C comonadic-Galois comodule and Σ is

flat as a left R-module. As in the proof of (i) implies (vi) we obtain from the faithfulness of−⊗R Σ : MR →MC that Σ is faithfully flat as a left R-module.(xi) By previous equivalences, the statements imply that Σ is a firm R-C Galois comodule. Then

the required equivalences follow from diagram (123)Suppose from now on that R is flat as a left R-module. By Theorem 4.23 we know that part (v)implies that C is flat as left A-module.(i) ⇔ (xii). If (F,G) is an equivalence of categories, then we know, since (i) implies (v), that Σ isa firm R-C Galois comodule that is flat as a left R-module. In particular, Σ is R-firmly projectiveas a right A-module and therefore G = HomC(Σ,−)⊗R R ' −⊗C Σ† by Theorem 4.13. As G isan equivalence of categories, G is in particular exact, and Σ† is coflat as a left C-comodule. Con-versely, if the conditions of statement (xii) are satisfied, then G is fully faithful by Theorem 4.22and F is fully faithful by Theorem 4.25.


Finally let us prove that Σ is a projective object in MC. Suppose first that −⊗RR reflects epimor-phisms. Consider any epimorphism N → N ′ in MC. Since (F,G) is an equivalence, in particularG preserves epimorphisms, so HomC(Σ, f) ⊗R R : HomC(Σ, N) ⊗R R → HomC(Σ, N ′) ⊗R Ris an epimorphism. By the reflection of epimorphisms by − ⊗R R, we find that HomC(Σ, f) :HomC(Σ, N) → HomC(Σ, N ′) is an epimorphism. This means exactly that Σ is projective in MC.Suppose now that R is a projective object in MR. Suppose the following diagram with exact row(epimorphism) is given in MC

Σg

N

f // N ′ // 0

After applying the functor G, which preserves epimorphisms, we can complete the diagram with amorphism h, since R is projective as a right R-module.

G(Σ) = R

G(g)

h

vvmmmmmmmmmmmmmm

G(N)G(f) // G(N ′) // 0

Apply now the functor F , then we obtain

F (R) = Σ

FG(g)=g

F (h)

ttjjjjjjjjjjjjjjjj

FG(N) = NFG(f)=f // FG(N ′) = N ′ // 0

This shows that Σ is a projective right C-comodule.

Remark 4.28. Note that in Theorem 4.27 no finiteness condition was a priori assumed on Σ,but it follows from our equivalent statements that every equivalence between categories of modulesand categories of comodules induces an R-firmly projective right C-comodule. In combination withTheorem 4.22 and Theorem 4.25 we obtain a large number of equivalent conditions to characterizeequivalences between categories of modules and comodules.

4.2.6. Applications. In this section we discuss some interesting features of the structuretheorems if we make particular choices for the firm ring R. To avoid too many repetetions, wewill only mention those results that are of particular interest and that differ from the general case.Other results and equivalent characterizations can be directly deduced from the structure theoremsin the previous Section.

Recall that an object X in a Grothendieck category C is called finitely generated if and only iffor any directed family of subobjects Xii∈I of X satisfying X =

∑i∈I Xi, there exists an i0 ∈ I

such that X = Xi0 . If X is a projective object in C, this condition is known to be equivalent tothe fact that HomC(X,−) preserves coproducts. Hence we can extend the definition of a locallyprojective module to Grotendieck categories. We call an object X in C weakly locally projective ifany diagram with exact rows of the form

(124) 0 // Ei // X

f

M

g // N // 0

and where E is finitely generated, can be extended with a morphism h : X → M such thatg h i = f i. We call X strongly locally projective if there exists a split direct systemP s : Z → Cs such that X = colimP and for all i ∈ Z, Pi is a finitely generated and projectiveobject in C.

From the definitions it is clear that if we take C = MA, with A a unital ring, we recover ourusual definitions.


Corollary 4.29. Let R be a ring with local units and C an A-coring. Suppose that we havean equivalence of categories

MR

F //MC

Goo

Put Σ = F (R). Then the following statements hold

(i) Σ is a weakly locally projective right A-module;(ii) F ∼= −⊗R Σ and G ∼= −⊗C Σ†, with Σ† = Σ∗ ⊗R R;(iii) if C is flat as a left A-module, then Σ is a weakly locally projective generator in MC;

Proof. (i). By Theorem 4.19, we know that Σ is R-firmly projective as a right A-module.Since R has local units, this implies by Theorem 2.67 that Σ is weakly locally projective as a rightA-module.(ii). Since R is a ring with local units, R is locally projective and consequently flat as a leftR-module. Now the statement follows directly from Theorem 4.13.(iii). We know from Theorem 4.22 that Σ is a generator in MC. Consider a diagram of the form

(124), with X = Σ. Let us show that G(E) is finitely generated in MR. Consider any directedfamily of subobjects Yi of G(E) such that

∑i Yi = G(E). Since F is an equivalence, we obtain

a directed family of subobjects F (Yi) of FG(E) ∼= E such that∑

i F (Yi) = E. Since E isfinitely generated, there exists an i0 such that E = F (Yi0). Therefore G(E) = GF (Yi0) ∼= Yi0 ,and E is finitely generated. If we apply the functor G on the diagram (124), we obtain

0 // G(E)G(i)// G(Σ) ∼= R

G(f)

G(M)G(g) // G(N) // 0

Then we find a morphism h : R → G(M) making this diagram commutative on the image ofG(i). If we apply the functor F to this diagram, then we see that F (h) : Σ → M satisfiesg F (h) i = f i, i.e. Σ is locally projective in MC.

Corollary 4.30. Let R be a ring with idempotent local units and C an A-coring. Supposethat we have an equivalence of categories

MR

F //MC

Goo

Put Σ = F (R). Then there exists a partially ordered set (I,≤) such that the following statementshold if we denote Z for the category associated to (I,≤)

(i) R ∼= colimR for a split directed system Rs : Z → Fks, R(i) = Ri and Ri is a ring withunit;

(ii) Σ ∼= colimP for a split directed system P s : Z →MCfgp

s, P (i) = Pi, such that Pi = F (Ri)

and EndC(Pi) = Ti;(iii) G ∼= −⊗C Σ†, with Σ† = colimP ∗, where P ∗ is the directed system of Lemma 2.43;(iv) C ∼= colimG, where G : Z → AMA, G(i) = P ∗i ⊗Ri Pi is the directed system of Proposi-

tion 3.61;(v) if C is flat as a left A-module, then Σ is a strongly locally projective generator in MC.

Proof. (i). This follows from the characterization of rings with idempotent local units givenin Theorem 2.39.(ii). We can argue as in the proof of Corollary 4.29, and conclude from Theorem 4.27 that Σ isan R-firmly projective as a right A-module. Since R has idempotent local units, we know fromTheorem 2.67 and Lemma 2.41 that Σ can be described by the colimit of a split directed system.Consider a complete set of idempotent local units ei for R, then we know that Ri = eiRei andPi = eiΣ. Since F ' −⊗RΣ (see Theorem 4.22), we find that F (Ri) ∼= eiRei⊗RΣ ∼= eiΣ = Pi.For the last statement, we know by Theorem 4.25 that R is a left ideal in T = EndC(Σ). Nowdenote Ti = EndC(Pi) and take t ∈ Ti, then eit = t = eit, since ei is the unit of Ti. Then we findfor all eirei ∈ Ri that teirei = eitrei ∈ eiRei = Ri, hence Ri is a left ideal in Ti. Since Ri is a


ring with unit ei, we conclude that Ri = Ti.(iii). Since we know by the proof of part (ii) that Σ satisfies the conditions of Lemma 2.41, we canconstruct the colimit of the system P ∗. By Lemma 2.44, we know that this colimit is isomorphicto Σ∗ ⊗R R. The statement is then a consequence of Corollary 4.29.(iv). By Proposition 3.67, the coring associated to the directed system G, is the comatrix coringassociated to the R-firmly projective module Σ. Therefore, the assertion is a consequence ofTheorem 4.27, which tells that Σ is an R-C Galois comodule.(v). We know by part (ii) that we can write Σ as a colimit of a direct system P s : Z → MC

fgps.

With notation as in Theorem 2.39, we find that for all Ri, the local unit ei and the naturalinjection βi : Ri → R constitute a finite dual basis for Ri als right R-module. As in the proof ofTheorem 4.27, we can prove that F (Ri) = Pi is projective in MC. Moreover, since Pi is finitelygenerated as a right A-module and C is flat as a left A-module, it follows easily that Pi is a finitelygenerated object in MC.

In view of Corollary 4.30, it is important to know when P ∈ RM is (faithfully) flat if R is aring with idempotent local units. We have the following results.

Proposition 4.31. Let R be a k-algebra with idempotent local units, and take P ∈ RM. Iffor every i ∈ I, there exists j ≥ i such that Pj ∈ RjM is flat, then P ∈ RM is flat.

Proof. Let f : N ′ → N be an injective map in MR, and x ∈ ker(f ⊗R P ). N ′ ⊗R P isthe colimit of the N ′

i ⊗Ri Pi, so x can be represented by∑

r n′r ⊗Ri pr with n′r ∈ N ′

i , pr ∈ Pi.∑r f(n′r)⊗Ri pr represents zero in N ⊗R P , so, replacing i by a bigger index, we can assume that∑r f(n′r)⊗Ri pr = 0 ∈ Ni ⊗Ri Pi. Replace i by a bigger index such that Pi ∈ RiM is flat. Then∑r n

′r ⊗Ri pr = 0 in N ′

i ⊗Ri Pi, and this implies that x = 0.

Proposition 4.32. Let R be a k-algebra with idempotent local units, and assume thatP ∈ RM is (faithfully) flat. If i ∈ I is such that ei is central in R, then Pi is (faithfully) flat as aleft Ri-module.

Proof. Take N ∈ MRi . We have γi : R→ Ri, making N ∈ MR via restriction of scalars.Then we claim that we have an isomorphism of k-modules

(125) N ⊗R P ∼= N ⊗Ri Pi.

Indeed, the mapf : N ⊗Ri Pi → N ⊗R P, f(n⊗Ri pi) = n⊗R pi

has an inverse g given byg(n⊗R p) = n⊗Ri eip.

g is well-defined since

g(nb⊗ p) = g(neib⊗ p) = g(neibei ⊗ p) = neibei ⊗Ri eip = n⊗Ri eibeip = g(n⊗ bp).

Assume that P ∈ RM is faithfully flat. A sequence

0 → N ′ → N → N ′′ → 0

is exact in MRi if and only if

0 → N ′ ⊗R P → N ⊗R P → N ′′ ⊗R P → 0

is exact in Mk, and, by (125), this is equivalent to exactness of the sequence

0 → N ′ ⊗Ri Pi → N ⊗Ri Pi → N ′′ ⊗Ri Pi → 0.

We remark that the condition that ei is central is fulfilled in the situation of Example 3.69.

Corollary 4.33. Let R be a ring with a enough idempotents and C an A-coring. Supposethat we have an equivalence of categories

MR

F //MC

Goo

Put Σ = F (R). Then the following statements hold


(i) Σ = ⊕P∈PP , where P ∈ P is a right C-comodule that is finitely generated and projective asa right A-module and G(C) = Σ† ∼= ⊕P∈PP ∗;

(ii) if C is flat as a left A-module, Σ = ⊕P∈PP , where P ∈ P is a right C-comodule that isfinitely generated and projective in MC;

Proof. (i). This follows from the fact that Σ is R-firmly projective as a right A-module incombination with Theorem 2.69.(ii). This is proven in the same way as part (v) of Corollary 4.30

Remark 4.34. Usualy one takes R = ⊕P,Q∈PHomC(P,Q) in the case of Corollary 4.33 and

R = colim EndC(P ) in the case of Corollary 4.30.

Corollary 4.35. Let T be a ring with unit and C an A-coring. Suppose that we have anequivalence of categories

MT

F //MC

Goo

Put Σ = F (T ). Then the following statements hold

(i) Σ is a finitely generated and projective right A-module;(ii) F ∼= −⊗T Σ and G ∼= −⊗C Σ∗;(iii) the map : T → EndC(Σ) is an isomorphism;(iv) if C is flat as a left A-module, then (F,G) is an equivalence if and only if Σ is a finitely

generated and projective generator in MC;

Proof. (i). As in the previous corollaries, we find that Σ is a T -firmly projective right A-module. Since T is a ring with unit, we obtain by Theorem 2.64 that Σ is finitely generated andprojective as a right A-module.(ii). Since T has a unit, we find Σ† ∼= Σ∗ ⊗T T ∼= Σ∗.(iii). It follows from Theorem 4.25 that T is an ideal in EndC(Σ). Since T has a unit this implies

T ∼= EndC(Σ). Explicitly, we obtain since F is fully faithful that

T ∼= GF (T ) = HomC(Σ, T ⊗T Σ) ∼= EndC(Σ).

(iv). If (F,G) is an equivalence, then one can prove in the same way as in part (v) of Corollary 4.30

that Σ is a projective generator of MC. Conversely, it follows by Theorem 4.27 that since Σ is agenerator in MC, G is fully faithful and Σ is flat as a left T -module. Moreover, F will be fullyfaithful as well if we can show that Σ is faithfully flat as a left T -module. By Proposition 2.24,it is enough to prove that JF 6= F for any proper ideal ideal J ⊂ T . Arguments similar to theones in [114, 18.4 (3)] show that for any right ideal J of T , the inclusion J ⊂ HomC(Σ, JΣ) is anequality. Details are as follows. Take g ∈ HomC(Σ, JΣ). Let u1, · · · , uk be a set of generatorsof Σ ∈ MA, since g(ui) ∈ JΣ, we can write g(ui) =

∑j fij(uj), with fij ∈ J . Let J ′ be the

subideal of J generated by the fij . Since the fij(ui) generate Im (g) as a right A-module, wehave that Im (g) ⊂ J ′Σ. We relabel the generators fij of J ′ as f1, . . . , fn. Let πi : Σn → Σand ei : Σ → Σn, i = 1, . . . , n, be the natural projections and inclusions. Consider the followingdiagram

Σg

Σn

f// J ′Σ

where the map f =∑n

i=1 fi πi : Σn → J ′Σ is surjective. Since Σ ∈ MC is projective, thereexists h : Σ → Σn such that g = f h. Using the direct sum decomposition of Σn, we can writeh =

∑nj=1 ej hj : Σ → Σk with hj ∈ T . Hence we obtain

g = f h =n∑

i,j=1

fi πi ej hj =n∑i=1

fi hi ∈ J ′ ⊂ J

where we used that J ′ is a right ideal of T . If J 6= T , then HomC(Σ, JΣ) 6= HomC(Σ,Σ), henceJΣ 6= Σ.

4.3. GALOIS THEORY IN BICATEGORIES: A UNIFYING APPROACH 139

Corollary 4.36. Let C be an A-coring and g ∈ C a grouplike element. Denote AcoC = a ∈A | ag = ga and consider any ring morphism B → A. Consider the functors

F : MB →MC, F (M) = M ⊗B A;

G : MC →MB, G(N) = N coC = m ∈M | ρN (n) = n⊗A g(1) (F,G) is a pair of adjoint functors;(2) The functor G is fully faithful and C is flat as a left A-module if and only if any of the

following conditions hold;(i) G is fully faithful and A is flat as a left B-module;(ii) can : A⊗B A→ C, can(a⊗B a′) = aga′ is an isomorphism of corings and C is flat

as a left A-module;(iii) can is an isomorphism of corings and A is flat as a left B-module;(iv) A with action induced by g is a generator in MC;

(3) The pair (F,G) is an equivalence of categories and C is flat as a left A-module if andonly if any of the following conditions hold

(i) (F,G) is an equivalence an A is flat as a left B-module;(ii) can is an isomorphism and A is faithfully flat as a left B-module;(iii) A is projective generator in MC;(iv) A is a generator in MC, F is fully faithful and B ∼= AcoC.

Proof. Follows immediately from Theorem 4.22 and Theorem 4.27.

4.3. Galois theory in bicategories: a unifying approach

In this section B denotes any bicategory. Recall from Corollary 1.6 that we can do all compu-tations in a 2-category. We keep the notation and terminology from Chapter 1, Section 1.3.

4.3.1. Push-out and pull-back functors. The first result is easy to prove.

Proposition 4.37. Let B a bicategory, and (q, α) a right comonad morphism from a comonadD to a comonad C. Then for any 0-cell Ω in B the functor

RepΩ(q) : Hom(Ω, B) → Hom(Ω, A)

induces a functorQ : Rcom(Ω,D) → Rcom(Ω,C),

defined as Q(m, ρ) = (m •B q, (m•Bα) (ρ•Bq)), for any comodule (m, ρ) ∈ Rcom(D) andQ(φ) = φ•Bq for any morphism of comodules φ ∈ Rcom(D).

Definition 4.38. Inspired by the terminology of [31], we will call the functor Q the pushoutfunctor associated to the comonad morphism (q, α).

It is a natural question to ask whether the pushout functor has a right adjoint. Generalisingresults from [31], [70] and the previous Section, we find a criterion for this in the ‘(locally) finitecase’.

Consider a left comonad morphism (p, β) from D to C. We know from Lemma 1.32 that p•B dis a C-D bicomodule. Let (n, ρ) be any comodule in Rcom(Ω,D). The the equalizer of ρ•Ap•Bdand n•Aλp•Bd, if it exists, is exactly the cotensor product of n and p •B d (see (19))

(126) n •c (p •B d)eqn // n •A (p •B d) //// n •A c •A (p •B d)

We will say that the category Rcom(Ω,D) satisfies the equalizer condition for p if and only if forall comodules (n, ρ) in Rcom(Ω,C) the equalizer of ρ•Ap•Bd and n•Aλp•Bd exists.

Proposition 4.39. Let B a bicategory, let Ω be a 0-cell in B and (p, β) a left comonadmorphism from a comonad D to a comonad C. If Rcom(Ω,D) satisfies the equalizer condition forp, then there exists a functor

P : Rcom(Ω,C) → Rcom(Ω,D),

defined as P(n, ρ) = (n •c (p •B d), ρn•c(p•Bd)) for all comodules (n, ρ) in Rcom(C).


Definition 4.40. The functor P is called the pullback functor associated to the left comonadmorphism (p, β).

Proposition 4.41. Let B be a bicategory, and consider in B a comonad morphism withadjunction (p, q, ϕ) : C → D (see Proposition 1.35). If Rcom(Ω,D) satisfies the equalizer conditionfor p, then the pullback functor P associated to p is a right adjoint to the pushout functor Qassociated to q.

Proof. First remark that by Proposition 1.35, the existence of the comonad morphismwith adjunction (p, q, ϕ) implies the existense of right and left comonad morphisms (q, α) and(p, β), in order that the pullback and pushout functors are well-defined. Denote as before p =(A,B, p, q, µ, η) for the adjoint pair in B associated to (p, q, ϕ). When we apply Theorem 1.29,we find that p induces an adjunction of categories

Hom(Ω, A)−•Ap // Hom(Ω, B)−•Bq

oo

This adjunction implies a natural isomorphism on the classes of morphisms of this categories, i.e.for any x ∈ Hom1(Ω, B) and y ∈ Hom1(Ω, A) we have that

Φ0x,y : ΩHomA

2 (x •B q, y) → ΩHomB2 (x, y •A p)

is an isomorphism.Now apply Theorem 1.26, to obtain that −•Bd defines a comonad on the category Hom(Ω, B).

We know from Proposition 3.8 that the induction functor of a comonad is the right adjoint for theforgetful functor. In the present language we obtain the following pair of adjoint functors

Rcom(Ω,D)U // Hom(Ω, B)

−•Bdoo

where U denotes the forgetful functor. The adjunction (U,− •B d) induces a second naturalisomorphism

Φ1x,y : ΩHomB

2 (x, z) → HomD(x, z •B d),where x, z ∈ Hom1(Ω, B).

Finally, we can construct the following diagram

HomC(x •B q, y) //

HomD(x, y •c (p •B d))

ΩHomA

2 (x •B q, y)

ρy−−ρx•Bq

Φ0x,y // ΩHomB

2 (x, y •A p)Φ1

x,y•Ap // HomD(x, y •A p •B d)

(ρy•Ap•Bd)−

(y•Aλp•Bd)−

ΩHomA

2 (x •B q, y •A c)Φ0

x,y•Ac

// ΩHomB2 (x, y •A c •A p)

Φ1x,y•Ac•Ap

// HomD(x, y •A •Acp •B d)

Now check that the vertical lines are equalizers and the diagram commutes (following the equallyalligned vertical lines). Since both lower horizontal arrows are isomorphisms, we find the existence ofan isomorphism in the upper horizontal line as well, by the universal property of the equalizers. Thisimplies the adjunction between the pushout functor−•Bq and the pullback functor−•c(p•Bd).

We state (without proof) the explicit form of the unit ζ and counit ν of the adjunction (Q,P).To obtain ν, consider the equalizer from (126),

y •c (p •B d)eqy // y •A (p •B d) //// y •A c •A (p •B d)

and apply the pushout functor − •B q on this exact row, then we obtain

(y •c (p •B d)) •B q

νy

**

eqy•Bq // y •A (p •B d) •B q //

y•Aε

// y •A c •A (p •B d) •B q

y


So ν is given by the formula νy = (y •A ε) (eq •B q), where we denote ε for the counit of thecomonad p •B d •B q, i.e. ε = µ (p •B εd •B q).

To obtain a formula for ζ, we calculate (126) for n = x •B q, where (x, ρ) ∈ Rcom(Ω,D),then we find the following diagram

(x •B q) •c (p •B d) // (x •B q) •A (p •B d) //// (x •B q) •A c •A (p •B d)

xζx

jjUUUUUUUUUUUUUUUUUUU

(x•Bη•Bd)ρOO

we obtain ζx by the universal property of the equalizer.

4.3.2. Weak and strong structure theorems. We will now give nessecary and sufficientconditions for the pullback and and pushout functor to be full and faithful, and thus obtain anequivalence between the categories Rcom(Ω,C) and Rcom(Ω,D). The arguments we give aredirect generalizations of the ones given in [70].

Lemma 4.42. Let C = (A, c,∆, ε) be a comonad in a bicategory B, Ω a 0-cell in B andtake any (y, ρ) ∈ Rcom(Ω,C). Then (y, ρ) is the equalizer of (ρ•Ac, y•A∆) in Rcom(Ω,C). Inparticular, this equalizer always exists.

Proof. From the coassociativity it is clear that (ρ•Ac) ρ = (y•A∆) ρ. Moreover, supposethat there exists any other (x, σ) such that (ρ•Ac) σ = (y•A∆) σ, then define τ = (y•Aε) σ :x→ y. We find

ρ τ = ρ (y•Aε) σ = ρ•AA (y•Aε) σ= (y•Ac•Aε) (ρ•Ac) σ = (y•Ac•Aε) (y•A∆) σ = σ

This proves that (y, ρ) is an equalizer.

Consider the following diagram in Rcom(Ω,C).

(127) (y •c (p •B d)) •B qνy

eqy•Bq// y •A (p •B d) •B qρy•Ap•Bd•Bq //

y•Aλp•Bd•Bq

//

y•Aϕ

y•Aε

ttjjjjjjjjjjjjjjjjjjy •A c •A p •B d •B q

y•Ac•Aϕ

y

ρy// y •A c

ρy•Ac //

y•A∆c

// y •A c •A c

As at the end of the previous section, we have denoted ε = µ (p•Bεd•Bq).

Lemma 4.43. Consider the diagram (127) and any ψ : x → y •A p •B d •B q such that(ρy•Ap•Bd•Bq)ψ = (y•Aλp•Bd•Bq)ψ, i.e. ψ equalizes the two upper horizontal arrows. Then

(y•Aϕ) ψ = ρy (y•Aε) ψ.Consequently the two squares in diagram (127) commute.

Proof. We compute

(128)ρy (y•Aε) ψ = (y•Ac•Aε) (ρy•Ap•Bd•Bq) ψ

= (y•Ac•Aε) (y•Aλp•Bd•Bq) ψ= (y•Aϕ) ψ

Here we used (4) in the first equality, the equalizing condition of ψ in the second one and one ofthe alternative formulas for ϕ in the last equality. We can conclude that the left square of (127)commutes by taking (x, ψ) = ((y •c (p •B d)) •B q, eqy •B q). The commutativity of the rightsquare follows now easily.

Theorem 4.44. Let B be a bicategory, and consider in B a comonad morphism with adjunction(p, q, ϕ) : D → C (see Proposition 1.35). Suppose Rcom(Ω,D) satisfies the equalizer conditionfor p.

Then the pullback functor P associated to p is faithfully flat if and only if ϕ is an isomorphismand the pushout functor Q preserves the equalizers of the form (126).


Proof. Suppose first that ϕ is an isomorphism and Q preserves the equalizers of the form(126). This last condition means that (y •c (p •B d)) •B q ∼= y •c (p •B d •B q). Moreover, sinceϕ is an isomorphism, we can compute further y •c (p •B d •B q) ∼= y •c c, and by Lemma 4.42we know that the equalizer y •c c = y, so we conclude (y •c (p •B d)) •B q ∼= y and the pullbackfunctor is fully faithful.

To prove the converse, take first y = c, then we see that the equalizer (c •c (p •B d), eqc) ∼=(p •B d, λp•Bd) by Lemma 4.42, and consequently (c •c (p •B d)) •B q ∼= p •B d •B q. As aconsequence, we find that

νc = (c •A ε) (eqc •B q)= (c •A ε) (λp•Bd •B q)= (c •A ε) (β •B d •B q) (p •B ∆d •B q)= (c •A ε) (ϕ •A p •B d •B q)(p •B d •B η •B d •D q) (p •B ∆d •B q)= ϕ (p •B d •B q •B ε) ∆p•Bd•Bq = ϕ

If the pullback functor is full and faithful, then ν is a natural ismorphism. So, in particular, wefind that νc = ϕ is an isomorphism.

Take any (y, ρy) in Rcom(Ω,C). We are done if we show that ((y•c (p•B d))•B q, (eqy •B q)) is

the equalizer of ρy•Ap•Bd•Bq and y•Aλp•Bd•Bq. To this end, consider the diagram (127), whereall squares are commutative by Lemma 4.43. Since we know that all vertical lines are isomorphismsand the lower horizontal line is an equalizer, we find that the upper horizontal line must be anequalizer as well.

Theorem 4.45. Let B be a bicategory, and consider in B a comonad morphism with adjunction(p, q, ϕ) : D → C (see Proposition 1.35). Suppose Rcom(Ω,D) satisfies the equalizer conditionfor p.

Then the functors (P,Q) establish an equivalence of categories between Rcom(Ω,C) andRcom(Ω,D) if and only if ϕ is an isomorphism and the pushout functor Q reflects isomorphismsand preserves the equalizers of the form (126).

Proof. First suppose that (P,Q) establishes an equivalence of categories. Then obviouslyQ reflects isomorphisms, and the other statements follow directly from Theorem 4.44.

Conversely, suppose ϕ : p •B d •B q is an isomorphism of comonads, then by Lemma 4.42, wefind the following equalizer in Rcom(Ω,C) for any (m, ρm) in Rcom(Ω,D),

m •B q // m •B q •A p •B d •B q // // m •B q •A p •B d •B q •A p •B d •B q

Furthermore, if Q preserves equalizers, we can apply −•B q on (126) in the situation n = m •B qand we obtain a second equalizer in Rcom(Ω,C). These two equalizers can be related in thefollowing diagram

m •B qζm•Bq //

((m •B q) •c (p •B d)) •B qeq•Bq

(m •B q) •A (p •B d •B q)

(m •B q) •A (p •B d) •B q

(m •B q) •A (p •B d •B q) •A (p •B d •B q)

m•Bq•Aϕ•Ap•Bd•Bq// (m •B q) •A c •A (p •B d) •B q

Since ϕ is an isomorphism, we find by the properties of the equalizers that ζm•Bq is an isomorphismas well, and since Q reflects isomorphisms, ζm must be an isomorphism. From Theorem 4.44we also know that ν is a natural isomorphism, so we find that (P,Q) is an equivalence ofcategories.

4.3.3. examples.


The bicategory of corings. If we consider B = Bim(k) the bicategory of (unital) k-algebras,bimodules and bilinear maps. Then adjoint pairs are comatrix coring contexts (A,B,Σ,Σ†, ε, η).Since B is a ring with unit, we obtain that Σ is finitely generated and projective as a right A-moduleand Σ† = Σ∗. Comonads in Bim(k) can be identified with corings. Consider a B-coring D anda comatrix coring context (A,B,Σ,Σ†, ε, η). Applying the result of Theorem 1.30, we obtain anA-coring Σ∗⊗B D⊗B Σ. This construction generalizes the construction of finite comatrix corings.

If we apply the techniques developed in the previous Sections to the present situation, thenwe recover the results of [31]. Let us state in particular the structure theorem. Remark that asufficient condition for the “equalizer condition” to be satisfied is that D is flat as a left B-module.

Theorem 4.46. Let A and B be unital rings, D a B-coring that is flat as a left B-moduleand C an A-coring. Consider Σ ∈ BMC such that Σ is finitely generated and projective as a rightA-module.

(i) We have a pair of adjoint functors (F,G)

F : MD →MC, F (M) = M ⊗B Σ

G : MC ⊗MD, G(N) = N ⊗C (Σ∗ ⊗B D)

(ii) the functor G is fully faithful if and only if can : Σ∗ ⊗B D⊗B Σ → C, can(ϕ⊗B d⊗B u) =ϕ(εD(d)u[0])u[1] is an isomorphism and (N ⊗C (Σ∗ ⊗B D))⊗B Σ ∼= N ⊗C (Σ∗ ⊗B D⊗B Σ)for all N ∈MC;

(iii) (F,G) is an equivalence of categories if and only if can is an isomorphism, −BΣ : MD →MC

reflects isomorphisms and (N⊗C (Σ∗⊗BD))⊗BΣ ∼= N⊗C (Σ∗⊗BD⊗BΣ) for all N ∈MC.

Remark 4.47. Item (iii) in the previous theorem has the following alternative formulation:(F,G) is an equivalence of categories and C is flat as a left A-module if and only if can is anisomorphism and D⊗B Σ is faithfully coflat as a left D-comodule.

Galois theory for firm comodules. Take B = Frm(k) and consider the case where D is the trivalB-coring (or, with regard to Theorem 2.11 we consider B as a coring over its Dorroh extension

B). Then we obtain the results from Section 4.2.We can combine this outcome with the results from Section 4.3.3. This leads to the construc-

tion of the bicategory of corings with firm base rings and their Galois theory. We leave the detailsof these constructions to the reader.

Coendomorphism corings. Take B = Bic(Mk), the bicatory whose 0-cells are corings thatare flat as a left and right module over their base ring, 1-cells are bicomodules and 2-cells arebicolinear maps. Then we can develop a Galois theory in this bicategory. Consider first D = Bas a trivial B-coring. If we take a comonad morphism with adjunction (p, q, ϕ) : C → B, thenin our present situation p is represented by a bicomodule P ∈ CMB and q is represented by abicomodule Q ∈ BMC. The fact the functor q = −⊗B Q has a left adjoint p, means exactly thatQ is (B,C)-quasi finite. The associated coring is given by p(Q) = Q⊗C P . This coring is knownas the coendomorphism coring associated to the quasi-finite comodule Q. We recover the theorydeveloped in [118], see also [36, Section 23].

Cotripleablility of functors. Consider now B = CAT. Then comonads are cotriples, and theGalois theory that we have developed reduces to the theory of cotripleability (or comonadicity) offunctors. In particular, we recover the famous theorem of Beck. Let (F,G) be a pair of adjointfunctors with F : B → A and G : A → B. Recall from Section 3.4.3 that we can construct acotriple C = FG on A which induces a pair of adjoint functors (FC ,GC) with FC : AC → A andGC : A → AC . Moreover there exists a unique functor K : B → AC such that F = FCK andKG = GC . From Theorem 4.44 and Theorem 4.45, we immediately obtain the following.

Theorem 4.48. If the category B has equalizers, then K has a left adjoint L : AC → B,which is fully faithful if and only if F preserves equalizers. If moreover F reflects isomorphisms,then K is an equivalence between the categories AC and B.

This part of the theory and its connection with the theory of Galois comodules over firm ringshas been discussed in [70].


Let us just remark that Galois theory in CAT applied to the situation of corings and co-modules, reduces to the theory of comonadic-Galois comodules. For any R-C bicomodule Σ,the functor − ⊗R Σ : MR → MA has a right adjoint HomA(Σ,−) ⊗R R : MA → MR.Hence we can construct the associated comonad in CAT and compare it with C by a canonicalcomonad morphism, which becomes the canonical cotriple morphism of Section 4.2.2. To applythe Galois theory in Frm(k) and obtain a firm Galois comodule, we need however a comonadmorphism with adjunction Frm(k), which means that there must exist a 2-cell in Frm(k) thatrepresents the functor HomA(Σ,−) ⊗R R. By Theorem 2.13 and Theorem 2.51, we know thatthis means exactly that Σ is R-firmly projective as right A-module. Remember that this condi-tion is satisfied once HomA(Σ,−) ⊗R R : MA → MR has a right adjoint, so in particular ifHomC(Σ,−)⊗R R : MC →MR has a right adjoint or if −⊗R Σ : MR →MC is an equivalenceof categories. In these situations, both theories coincide.

Of course, our theory can be dualized. In general, this leads to a Galois theory for monadsin a bicategory B. An interesting example of this dual theory can be the theory of tripleability offunctors, taking B = CAT.

Galois theory of matrix C-rings. Let k be a field and consider the category Bic(k). An adjointpair in this bicategory consists of two k-coalgebras C and D, a C-D bicomodule M , a D-Cbicomodule N and two bicolinear maps σ : C → N ⊗D M and τ : M ⊗C N → D, satisfyingτ ⊗D N ∼= N ⊗C σ and M ⊗C σ ∼= τ ⊗DM . Applying Theorem 1.30, we find that M ⊗D N is amonad in B and N ⊗CM is a comonad in B. If we consider the comonad, then we can apply ourgeneral theory (our in fact, the special case of the coendomorphism corings), if we consider themonad, then we have to apply the dual version of the theory. This dual Galois theory is recentlydeveloped in [35], the monad M ⊗D N is termed a matrix C-ring .

References

The outcome of Section 4.2 is a combination of results obtained in [42] (joint with S. Caenepeeland E. De Groot), [73] (joint with J. Gomez-Torrecillas), [112] and new observations. Galoiscomodules have also been studied in [29], [63] (finite type) and [70], [117] (infinite type). Theresults of Section 4.3 are part of the work in progress with J. Gomez-Torrecillas [72].

Chapter 5Morita Theory for Corings

In this Chapter we will study Galois comodules by means of Morita theory. We will startfrom an arbitrary B-C bicomodule Σ, where C is an A-coring and B is a unital ring. In view ofTheorem 4.27, this implies that once we obtain an equivalence of categories between the categoryof C-comodules and the category of B-modules, Σ is finitely generated and projective as rightA-module and the theory is reduced to the finite case. The further development of this theorywhere B is replaced by a general firm ring, is part of a current research project [23].

After some general remarks about Morita theory in the bicategory of bimodules in Section 5.1,we show in Section 5.2 how, under finiteness conditions, the Galois property of a comodule isdetermined by a dual version of the canonical map. Starting from a right C-comodule, we con-struct in Section 5.3 several Morita contexts and show how they are related and provide sufficientconditions for them to be isomorphic. In Section 5.4 we use these Morita contexts to formulatestructure theorems for the comodule to which the Morita contexts are associated. We provide twoapplications in the last section: our results can be applied to rings with a grouplike character andto corings with a grouplike element.

5.1. Finite Galois theory and Morita theory

5.1.1. Finite Galois theory. In this and the following Chapter we will only investigate ‘finite’Galois theory, in the following sense. Let Σ be a right C-comodule and denote T = EndC(Σ). Wewill investigate the Galois properties of Σ as T -C bicomodule. That is, we consider the followingparticular case of diagram (100)

(129) MT

−⊗T Σ //

GΣ=−⊗T Σ

''OOOOOOOOOOOOOOOOOOOOOOOOOO MAHomA(Σ,−)

oo

GC=−⊗AC

MC

FC

OO

HΣ=HomC(Σ,−)


Recall the explicit formulas for the unit and counit of the adjunction (GΣ,HΣ).

νN : N → HomC(Σ, N ⊗T Σ), νN (n)(u) = n⊗T u;(130)

ζM : HomC(Σ,M)⊗T Σ →M, ζM (ϕ⊗T u) = ϕ(u);(131)

145

146 CHAPTER 5. MORITA THEORY FOR CORINGS

for all N ∈ MT and M ∈ MC. Furthermore, we call Σ a comonadic-Galois comodule if theassociated natural transformation

(132) canM : HomA(Σ,M)⊗T Σ →M ⊗A C, canM (ϕ⊗T u) = ϕ(u[0])⊗T u[1]

is a natural isomorphism. We call Σ a Galois comodule if Σ is finitely generated and projective asa right A-module and the following map is an isomorphism of corings

(133) can = canA : Σ∗ ⊗T Σ → C, can(ϕ⊗T u) = ϕ(u[0])⊗T u[1].

5.1.2. Morita theory in the bicategory of bimodules. In Chapter 1, Section 1.3.1 weintroduced the notion of a Morita context in a general bicategory. In this Chapter we will usethe term ‘Morita context’ exclusively for a Morita context in the bicategory Bim = Bim(k) ofbimodules between k-algebras, where k is a commutative ring. For the classical theory of Moritacontexts we refer to [14, Chapter II.3], where Morita contexts are termed sets of (pre-)equivalencedata. One of the most important results about Morita theory is that surjectivity of the connectingmaps implies their bijectivity. A Morita context with bijective connecting maps is called strict.

A morphism of Morita contexts m : M = (A,B, P,Q, µ, τ) → M′ = (A′, B′, P ′, Q′, µ′, τ ′)consists of two algebra maps m1 : A → A′ and m2 : B → B′, an A′-B′ bimodule map m3 :P → P ′ and a B′-A′ bimodule map m4 : Q → Q′ such that m1 µ = µ′ (m3 ⊗B m4) andm2 τ = τ ′(m4 A m3).

Starting from a Morita context M = (A,B, P,Q, µ, τ), one can construct the opposite Moritacontext Mop = (Aop, Bop, Q, P, µop, τ op), where µop(q⊗ p) = µ(p⊗ q) and τ op(p⊗ q) = τ(q⊗ p).An anti-morphism of Morita contexts m : M → M′ is a morphism m : M → M′op, i.e. itconsists of two algebra maps m1 : A → A′op and m2 : B → B′op, an A′-B′ bimodule mapm3 : Q→ P ′ and a B′-A′ bimodule map m4 : P → Q′ such that m1 µ = µ′op (m4 ⊗B m3) andm2 τ = τ ′op(m3 ⊗A m4).

Observe (see e.g. [21, Remark 3.2]) that a Morita context can be identified with a k-linearcategory with two objects a and b. The algebras of the Morita contexts are End(a) and End(b),the connecting bimodules are Hom(a, b) and Hom(b, a) and multiplication and bimodule maps aregiven by composition. We denote this context as follows

(134) N(a, b) = (End(a),End(b),Hom(b, a),Hom(a, b), , •).

This can be summarised by the following diagram

aEnd(a)%%

Hom(a,b)**b

Hom(b,a)

jj End(b)zz

Recall from Example 1.27 that a sufficient condition for a Morita context to be strict is theexistence of a pair of invertible elements (j, ). This is a pair of morphisms j ∈ Hom(a, b) and ∈ Hom(b, a) such that j = a and j = b. In the present framework this implies that a andb are isomorphic objects in the considered k-linear category through the isomorphism j such that = j−1.

A particular interesting example of (134) is the following. Let A be a k-algebra and consideran A-coring C together with two right C-comodules Σ and Λ. Then we can consider the k-linearfull subcategory of MC whose two only objects are Σ and Λ. The associated Morita context readsas

N(Σ,Λ) = (EndC(Σ),EndC(Λ),HomC(Λ,Σ),HomC(Σ,Λ), , •).

Then this Morita context can be used to compare the comodule properties of Σ and Λ, in particular,it can be used to compare the Galois properties of Σ and Λ. If we denote T = EndC(Σ) and

5.2. THE DUAL OF THE CANONICAL MAP 147

S = EndC(Λ), then this idea can be visualised by the following diagram

MC

HomC(Λ,−)

!!CCC

CCCC

CCCC

CCCC

CC

HomC(Σ,−)

MT

−⊗T Σ

== −⊗T HomC(Λ,Σ) //MS

−⊗SΛ

aaCCCCCCCCCCCCCCCCC

−⊗SHomC(Σ,Λ)

oo

The diagonal arrows express the (finite) Galois properties, both for Σ and for Λ, and the horizontalarrows on the bottom express the Morita context. Moreover, we have natural transformations

−⊗T evΣΛ : −⊗T HomC(Λ,Σ)⊗S Λ → −⊗T Σ

−⊗S evΛΣ : −⊗S HomC(Σ,Λ)⊗T Σ → −⊗S Λ

compΣ : HomC(Λ,−)⊗S HomC(Σ,Λ) → HomC(Σ,−)

compΛ : HomC(Σ,−)⊗T HomC(Λ,Σ) → HomC(Λ,−)

ωΛ,Σ : −⊗T HomC(Λ,Σ) → HomC(Λ,−⊗T Σ)

ωΣ,Λ : −⊗S HomC(Σ,Λ) → HomC(Σ,−⊗S Λ)

Where compΣ and compΛ are given by composition, evΣ and evΛ are the evaluation maps, asdefined in (107) and for all M ∈MC we define

ωΛ,ΣM (m⊗T f)(λ) = m⊗T f(λ)

for all m ∈M , λ ∈ Λ and f ∈ HomC(Λ,Σ). In a similar way we can define ωΣ,Λ.In full generality this idea is now being explored in [23]. In this Chapter, we investigate only

two special situations. If the coring C is finitely generated and projective as a left A-module, then∗C is a right C-comodule, and we can construct the above Morita context for Λ = ∗C. We obtainthe following diagram

MC

J

!!CCC

CCCC

CCCC

CCCC

CC

HomC(Σ,−)

MT

−⊗T Σ

== −⊗T Σ //M∗C

∼=

aaCCCCCCCCCCCCCCCCC

−⊗∗CHom∗C(Σ,∗C)oo

We know that in this situation MC and M∗C are isomorphic, and thus the Galois properties of Σwill be completely determined by the Morita context. More general, if C is locally projective as aleft A-module, then Rat(∗C) is a right C-comodule and we can consider Λ = Rat(∗C).

5.2. The dual of the canonical map

Let A and B be two unital rings, R a firm B-ring and Σ ∈ BMA and Σ† ∈ AMB such that(Σ,Σ†, µ) is an R-firm dual pair. Then we can construct the comatrix coring D = Σ† ⊗R Σ ∼=Σ∗ ⊗R Σ.

Proposition 5.1. With notation as above, the following assertions hold.

(i) ∗D ∼= REnd(Σ)op;(ii) D∗ ∼= EndR(Σ†);(iii) EndR(Σ)op ∼= REnd(Σ†) (i.e. ∗D ∼= D∗).

Proof. (i). Take g ∈ ∗D and u ∈ Σ, then we define α : ∗D → REnd(Σ) by

α(g)(u) = erg(ϕr ⊗R ur).Conversely, we have a map β : REnd(Σ) → ∗D, defined for all ϕ⊗R u ∈ D and ψ ∈ REnd(Σ) as

β(ψ)(ϕ⊗R u) = µ(ϕ⊗R ψ(u)),


with ψ ∈ REnd(Σ). We check that α and β are inverses.

β α(g)(ϕ⊗R u) = µ(ϕ⊗R erg(ϕr ⊗R ur)) = µ(ϕ⊗R er)g(ϕr ⊗R ur)= g(µ(ϕ⊗R er)ϕr ⊗R ur) = g(ϕ · r ⊗R ur)= g(ϕ⊗R r · ur) = g(ϕ⊗R u)

α β(ψ)(u) = erµ(ϕr ⊗R ψ(ur)) = r · ψ(ur)= ψ(r · ur) = ψ(u)

Finally, let us check that α is a ring morphism. For g, f ∈ ∗D we find

α(f ∗ g)(u) = er(f ∗ g)(ϕr ⊗R ur)= erg(ϕr ⊗R esf(ϕs ⊗R urs))

α(g) α(f)(u) = α(g)(erf(ϕr ⊗R ur))= esg(ϕs ⊗R esrf(ϕr ⊗R ur))

The statement follows from the identity

er ⊗A ϕr ⊗R es ⊗A ϕs ⊗R urs = es ⊗A ϕs ⊗R esr ⊗A ϕr ⊗R ur.

(ii). This follows from left-right duality. Let us just give the explicit formula of the isomorphisms.

α′(g)(ϕ) = g(ϕs ⊗B es)ϕs and β′(ψ)(ϕ⊗B u) = µ(ψ(ϕ)⊗R u)

for all g ∈ D∗, ψ ∈ EndR(Σ†), ϕ ∈ Σ† and u ∈ Σ.(iii). By Theorem 2.51 we know that Σ† ∼= Σ∗ ⊗R R and Σ ∼= R ⊗R Σ†. Therefore, we canconstruct the following maps

α′′ : REnd(Σ) → EndR(Σ†) ∼= EndR(Σ∗ ⊗R R)α(ψ)(f ⊗R r) = f ψ ⊗R r

β′′ : EndR(Σ†) → EndR(Σ) ∼= EndR(R⊗R Σ†)α(ψ)(r ⊗R φ) = r ⊗ φ ψ

for all ψ ∈ REnd(Σ), ψ ∈ EndR(Σ†), f ∈ Σ∗, φ ∈ Σ† and r ∈ R.

Consider now an A-coring C and let Σ be any R-C bicomodule (we do not assume that Σ isR-firmly projective). The canonical map

can = canA : Σ∗ ⊗R Σ⊗ C, can(ϕ⊗R u) = ϕ(u[0])u[1],

has two dual versions,

(135) ∗can : ∗C → REnd(Σ)op, ∗can(f)(u) = u[0]f(u[1])

and

can∗ : C∗ → EndR(Σ∗), can∗(f)(ϕ) = f (ζ(ϕ)⊗ C) ρΣ

If we put Σ† = Σ∗ ⊗R R, then we obtain an obvious map

(−) : EndR(Σ∗) → EndR(Σ†), ψ(ϕ⊗R r) = ψ(ϕ)⊗R r

Combining this map with can∗ we obtain

can† : C∗ → EndR(Σ†).

A right A-module M is called reflexive if the canonical morphism i : M ⊗ ∗(M∗), i(m)(f) =f(m), for all m ∈M and f ∈M∗ is an isomorphism. If M is finitely generated and projective asa right A-module, then M is reflexive.

Proposition 5.2. With notation as above, the following assertions hold.

(i) If Σ is an R-C comonadic-Galois comodule, then can† is an isomorphism;(ii) if Σ is an firm R-C Galois comodule, then ∗can and can† are isomorphisms;(iii) let Σ be R-firmly projective, and consider the comatrix coring D = Σ∗⊗R Σ. If C and D are

reflexive and ∗can is an isomorphism, then Σ is a firm R-C Galois comodule.

5.3. MORITA CONTEXTS ASSOCIATED TO A COMODULE 149

Proof. Statements (ii) and (iii) follow easily from our previous observations. Let us justprove (i). Since − ⊗R Σ and HomC(Σ,−) ⊗R R make up a pair of adjoint functors, for anyM ∈MR and N ∈MC, we obtain an isomorphism

HomC(M ⊗R Σ, N) ∼= HomR(M,HomC(Σ, N)⊗R R).

When we apply this to the situation Σ† ∈MR and C ∈MC, we find

C∗ ∼= HomC(Σ† ⊗R Σ,C) ∼= HomR(Σ†,HomC(Σ,C)⊗R R) ∼= EndR(Σ†).

One can check that this isomorphism is exactly can†.

Remark 5.3. The reflexivity conditions in part (iii) of Proposition 5.2 are satisfied if that Cis finitely generated and projective as a left A-module and Σ is finitely generated and projectiveboth as a left B-module and a right A-module. In this case D = Σ∗ ⊗B Σ is a finite comatrixcoring.

5.3. Morita contexts associated to a comodule

5.3.1. The *-Morita context associated to a comodule. Let C be an A-coring, and Σ ∈CM. Denote T = CEnd(Σ)op.

Lemma 5.4. With notation as above, ∗Σ ∈ TM∗C and Q′ = CHom(C,Σ) ∈ ∗CMT .

Proof. Let ϕ ∈ ∗Σ, f ∈ ∗C, t ∈ T , q ∈ Q′ and u ∈ Σ. The bimodule structure on ∗Σ isdefined by

(136) (ϕ · f)(u) = f(u[−1]ϕ(u[0])) and t · ϕ = ϕ t.Let us show that the two actions commute

(t · (ϕ · f))(u) = (ϕ · f)(t(u)) = f(t(u)[−1]ϕ(t(u)[0])

)= f

(u[−1]ϕ(t(u[0]))

)= ((t · ϕ) · f)(u).

The bimodule structure on Q′ is defined by

(137) (f · q)(c) = q(c(1)f(c(2))) and q · t = t q.The two actions commute, since

((f · q) · t)(c) = t(q(c(1)f(c(2))) = (q · t)(c(1)f(c(2))) = (f · (q · t))(c).

Lemma 5.5. With notation as in Lemma 5.4, we have well-defined bimodule maps

H′ : Q′ ⊗T ∗Σ → ∗C, q H

′ ϕ = ϕ q;O′ : ∗Σ⊗∗C Q

′ → T, ϕ O′ q(u) = q(u[−1]ϕ(u[0])).

Proof. These are straightforward verifications.

Theorem 5.6. With notation as in Lemmas 5.4 and 5.5, we have a Morita context ∗M =(T, ∗C, ∗Σ, Q′, O′, H′).

Proof. We first show that H′⊗∗CQ′ = Q′ ⊗T O′. For all p, q ∈ Q′, ϕ ∈ ∗Σ and c ∈ C, we

have ((Q′ ⊗T O

′)(q ⊗T ϕ⊗∗C p))(c) = (q · (ϕ O

′ p))(c) = (ϕ O′ p)(q(c))

= p(q(c)[−1]ϕ(q(c)[0])) = p(c(1)ϕ(q(c(2)))

= ((ϕ q) · p)(c) =((H′⊗∗CQ

′)(q ⊗T ϕ⊗∗C p))(c)

∗Σ⊗∗C H′ = O′⊗T ∗Σ since((O′⊗T ∗Σ)(ϕ⊗∗C q ⊗T ψ)

)(u) = ((ϕ O

′ q) · ψ)(u) = ψ((ϕ O′ q)(u))

= ψ(q(u[−1]ϕ(u[0])) = (ψ q)(u[−1]ϕ(u[0]))

= (ϕ(ψ q))(u) =((∗Σ⊗∗C H

′)(ϕ⊗∗C q ⊗T ψ))(u)


for all q ∈ Q′, ϕ,ψ ∈ ∗Σ, u ∈ Σ and c ∈ C.

Remarks 5.7. (1) If C = A is the trivial coring, and M ∈ AM, then the Morita contextC(Σ) = (AEnd(M)op, A, ∗M,M, 4, N) is the Morita context associated to the A-moduleM , as in [14, II.4].

(2) We can also construct a Morita context associated to Σ ∈MC:

M∗(Σ) = (T = EndC(Σ),C∗, Q′ = HomC(C,Σ),Σ∗, O′, H′)

with Σ∗ ∈ C∗MT via

(f · ϕ)(u) = f(ϕ(u[0])u[1]) and ϕ · t = ϕ t,Q′ ∈ TMC∗ via

(q · f)(c) = q(f(c(1))c(2)) and t · q = t q.The connecting maps are

H′ : Σ∗ ⊗T Q′ → C∗, (ϕ H

′ q) = ϕ qO′ : Q′ ⊗C∗ Σ∗ → T, τ(q O

′ ϕ)(u) = q(ϕ(u[0])u[1])

(3) Let Σ ∈ MCfgp. Then Σ∗ ∈ C

fgpM (see Section 3.4.2). As CEnd(Σ∗)op ∼= EndC(Σ) = T ,we obtain a Morita context

(138) ∗M(Σ∗) = (T = EndC(Σ), ∗C,Σ, Q′ = CHom(C,Σ∗), O′, H′)

with Q′ ∈ ∗CMT by

(f · q)(c) = q(c(1)f(c(2))) and (q · t)(c) = q(c) tand Σ ∈ TM∗C by

t · u = t(u) and u · f = u[0]f(u[1])

and

(139) H′ : Q′ ⊗T Σ → ∗C, (q H

′ u)(c) = q(c)(u)

(140) O′ : Σ⊗∗C Q

′ → T, (u O′ q)(v) = u[0](q(u[1])(v))

5.3.2. The Morita context associated to a comodule. Let C be an A-coring, Λ ∈ CMand Σ ∈ MC. By the previous Section we know that we can construct a Morita context ∗M(Λ),connecting the algebras CEnd(Λ)op and the left dual of C, ∗C, and a Morita context M∗(Σ)connecting the algebras EndC(Σ) and the right dual C∗. If Σ is finitely generated and projectiveas a right A-module, these constructions yield a Morita context ∗M(Σ∗) connecting EndC(Σ) and∗C. The first objective of this section is to give a generalization of this last Morita context toarbitrary right C-comodules Σ.

Let C be an A-coring and consider any Σ ∈MC. We denote as before T = EndC(Σ). Definethe k-module

(141) Q : = q ∈ HomA(Σ, ∗C) | ∀x ∈ Σ, c ∈ C q(x[0])(c)x[1] = c(1)q(x)(c(2)) .In the following lemma, some properties of the k-module Q are collected, needed in order to seethat Σ and Q are bimodules connecting the algebras T and ∗C.

Lemma 5.8. For an A-coring C and its right comodule Σ, the k-module Q in (141) obeys thefollowing properties.

(i) Q is isomorphic to the k-module

(142) P : = q ∈ AHom(C,Σ∗) | ∀x ∈ Σ, c ∈ C c(1)q(c(2))(x) = q(c)(x[0])x[1] ,defined in terms of the left A-module Σ∗ : = HomA(Σ, A);

(ii) there exists a natural transformation ω : − ⊗∗C Q → HomC(Σ,−) of functors MC → MT

given by

ωM : M ⊗∗C Q→ HomC(Σ,M), ωM (m⊗∗C q)(u) = m[0]q(u)(m[1]);


(iii) Q is a k-submodule of Hom∗C(Σ, ∗C);(iv) Q is a ∗C-T bimodule, for T : = EndC(Σ), with actions

(fq)(x) : = fq(x), for f ∈ ∗C, q ∈ Q, x ∈ Σ and

(qt)(x) : = q(t(x)

), for q ∈ Q, t ∈ T, x ∈ Σ;

(v) if C is locally projective as a left A-module, then Q = Hom∗C(Σ, ∗C).

Proof. (i). The isomorphism is given by switching the arguments, that is, by the map

Q→ P, q 7→(c 7→ q(−)(c)

).

(ii). Since ∗C is an A-ring and the elements ofQ are right A-linear, the map ωM (m⊗∗Cq) : Σ →M ,

x 7→ m · q(x) is right A-linear, for q ∈ Q and m ∈ M . In order to see that it is also right C-colinear, use the right A-linearity of a C-coaction in the first equality and (141) in the second one,to conclude that, for any right C-comodule M , m ∈M , q ∈ Q and x ∈ Σ,(

m[0]q(x)(m[1]))[0]⊗A (m[0]q(x)(m[1])

)[1]

= m[0] ⊗A m[1]q(x)(m[2])

= m[0] ⊗A q(x[0])(m[1])x[1] = m[0]q(x[0])(m[1])⊗A x[1].

We leave it to the reader to verify that ω is natural in M .(iii). Using the right A-linearity of q ∈ Q and the defining identity (141), one checks that, forx ∈ Σ, f ∈ ∗C and c ∈ C,

q(xf)(c) = q(x[0]f(x[1])

)(c) = q(x[0])(c)f(x[1]) = f

(q(x[0])(c)x[1]

)= f

(c(1)q(x)(c(2))

)=(q(x)f

)(c),

where in the first equality the form of the ∗C-action in Σ has been used, and in the last one themultiplication law in ∗C.(iv). For f ∈ ∗C and q ∈ Q, fq is an element of HomA(Σ, ∗C) by the right A-linearity of q andthe fact that ∗C is an A-ring. Since q is an element of Q, so is fq as, for x ∈ Σ and c ∈ C,

(fq)(x[0])(c)x[1] =(fq(x[0])

)(c)x[1] = q(x[0])

(c(1)f(c(2))

)x[1]

= c(1)q(x)(c(2)f(c(3))

)= c(1)

(fq(x)

)(c(2)) = c(1)(fq)(x)(c(2)),

where the first and last equalities follow by the the form of the ∗C-action in Q and the second andpenultimate equalities follow by the multiplication law in ∗C. The third equality follows by (141).Since q ∈ Q and t ∈ T are right A-linear, qt is an element of HomA(Σ, ∗C). Since t ∈ T is colinearand q is an element of Q, it follows that qt ∈ Q as, for x ∈ Σ and c ∈ C,

(qt)(x[0])(c)x[1] = q(t(x)[0]

)(c)t(x)[1] = c(1)(qt)(x)(c(2)),

where the form of the T -action in Q has been used.It is straightforward to check that both actions are associative and unital and that they commute.(v). By part (iii), Q ⊆ Hom∗C(Σ, ∗C). The converse inclusion is proven as follows. Recall thatlocal projectivity of the left A-module C means that for any finite subset S ⊂ C, there exists adual basis ei ⊂ C and fi ⊂ ∗C such that c =

∑i fi(c)ei, for all c ∈ S. For an element x ∈ Σ,

fix finite sets xj ⊂ Σ and cj ⊂ C such that

x[0] ⊗A x[1] =s∑j=1

xj ⊗A cj .

Take q ∈ Hom∗C(Σ, ∗C) and c ∈ C and introduce the set S : = c1, · · · , cs, c(1)q(x)(c(2)) ⊂ C.By the assumption of local projectivity, there exists a dual basis ei ⊂ C and fi ⊂ ∗C associatedto S, hence we can write

q(x[0])(c)x[1] =∑i

q(x[0])(c)fi(x[1])ei =∑i

q(x[0]fi(x[1]))(c)ei

=∑i

q(xfi)(c)ei =∑i

(q(x)fi

)(c)ei

=∑i

fi(c(1)q(x)(c(2))

)ei = c(1)q(x)(c(2)),


where the fourth equality follows by the right ∗C-linearity of q. This shows that q belongs to thek-module Q in (141).

Remark 5.9. If the A-coring C is locally projective as a left A-module then we know fromProposition 3.38 that the image of a right C-comodule Σ under an element q ∈ Hom∗C(Σ, ∗C) lieswithin the rational part Rat(∗C) of ∗C. This implies that

Q = Hom∗C(Σ, ∗C) = Hom∗C(Σ,Rat(∗C)) = HomC(Σ,Rat(∗C)).

If C is a finitely generated and projective left A-module, with dual bases ci ⊂ C and fi ⊂∗C, then ∗C possesses a right C-comodule structure with coaction f 7→

∑i f ∗ fi⊗A ci (see (86)).

In this case Q is identical to the k-module HomC(Σ, ∗C) ≡ Hom∗C(Σ, ∗C).

In light of Lemma 5.8, there is another Morita context associated to Σ,

(143) M(Σ) = (T, ∗C,Σ, Q, O, H),

where tuf = t(u[0]f(u[1])

), for t ∈ T , u ∈ Σ and f ∈ ∗C, with connecting homomorphisms

H : Q⊗T Σ → ∗C, q Hu = q(u),(144)

O : Σ⊗∗C Q→ T, u O q(v) = u[0]q(v)(u[1]).(145)

In a symmetric way, to a left C-comodule Λ one can associate the Morita context M(Λ), connectingCEnd(Λ)op and C∗.

In the following Proposition we compare the Morita context of (143) with the Morita contextsconstructed in Section 5.3.1.

Proposition 5.10. Let C be an A-coring, Σ ∈MC and Λ ∈ CM.

(i) We have isomorphisms of Morita contexts

M∗(Σ) ∼= N(Σ,C), ∗M(Λ) ∼= N(Λ,C);

(ii) if C is locally projective as a left A-module, then Rat(C) ∈ MC and there is a morphism ofMorita contexts

M(Σ) → N(Σ,Rat(∗C)),which becomes a monomorphism if Rat(∗C) is dense in the local topology on ∗C;

(iii) if C is finitely generated and projective as a left A-module, then ∗C ∈ MC and we have anisomorphism of Morita contexts

M(Σ) ∼= N(Σ, ∗C);

(iv) if Σ is finitely generated and projective as a right A-module, then Σ∗ ∈ CM (there is nocondition on C) and

∗M(Σ∗) = M(Σ) and M∗(Σ) = M(Σ∗).

Proof. (i). Let us compute

N(Σ,C) = (EndC(Σ),EndC(C),HomC(C,Σ),HomC(Σ,C), , •)One can easily check that the isomorphisms EndC(C) ∼= C∗ and HomC(Σ,C) ∼= Σ∗ give rise to anisomorphism of Morita contexts M(Σ) → N(Σ,C). The second statement is proven in the sameway.(ii). Let us compute that

N(Σ,Rat(∗C)) = (EndC(Σ),EndC(Rat(∗C)),HomC(Rat(∗C),Σ),HomC(Σ,Rat(∗C)), , •)By Remark 5.9 we know that HomC(Σ,Rat(∗C)) ∼= Q. From the theory of rational modules (seeSection 3.3) we know that since C is locally projective as a left A-module, MC is a full subcategoryof M∗C (see Theorem 3.35). Hence EndC(Rat(∗C)) = End∗C(Rat(∗C)) and HomC(Rat(∗C),Σ) =Hom∗C(Rat(∗C),Σ). Put R = Rat(∗C), then R is a (possibly non-unital) ring. For any M ∈MR

we have a map fM : M → HomR(R,M) defined as fM (m)(r) = mr for all m ∈ M and r ∈ R.In this way we obtain morphisms Rat(∗C) → EndC(Rat(∗C)) and Σ → HomC(Rat(∗C),Σ). Weleave it to the reader to verify that these morphisms, together with the previous isomorphisminduce a morphism of Morita contexts. If Rat(∗C) is dense in the finite topology on ∗C, then, by


Proposition 3.38, we know that Rat(∗C) has local units on all right C-comodules. Take M ∈MC

and m ∈ M and suppose that the map fM (m) : R → M is zero. Since we can find a right localunit e ∈ R for m, we obtain fM (m)(e) = me = m = 0, so the map fM is injective. This impliesthat the constructed morphism of Morita contexts is injective.(iii). This is a special case of part (ii), where Rat(∗C) = ∗C. Since ∗C is a ring with unit, we obtain

Hom∗C(∗C,Σ) ∼= Σ and End∗C(∗C) ∼= ∗C. Therefore we find the required isomorphism of Moritacontexts.(iv). This follows directly from the constructions of the Morita contexts.

If Σ ∈MC, then Σ is also a right ∗C-module, and we can associate to Σ a Morita context asin [14, II.4] (see Remark 5.7(1)), namely

(146) T(Σ) = (T = End∗C(Σ), ∗C,Σ, Q = Hom∗C(Σ, ∗C), O, H),

with

H : Q⊗eT Σ → ∗C, (qHu) = q(u)(147)

O : Σ⊗∗C Q→ T , (uOq)(v) = u · q(v) = u[0](q(v)(u[1])).(148)

The following Proposition explains the relationship between the Morita contexts T(Σ) of (146)and M(Σ) of (143).

Proposition 5.11. Let C be an A-coring and Σ a right C-comodule. There exists a morphismof Morita contexts M(Σ) → T(Σ), which becomes an isomorphism if C is locally projective as aleft A-module.

Proof. There exist inclusions T = EndC(Σ) ⊂ End∗C(Σ) and, by Lemma 5.8 (3), Q ⊂Hom∗C(Σ, ∗C). Comparing (144) and (145) with (147) and (148), it is straightforward to see thatthese inclusions, together with the identity maps of ∗C and Σ, establish a morphism of Moritacontexts.

Now assume that C is locally projective as a left A-module. By Theorem 3.35, MC is a fullsubcategory of M∗C. Therefore, for any two right C-comodules M and M ′, HomC(M,M ′) =Hom∗C(M,M ′). In particular T = End∗C(Σ). By Remark 5.9, Q = Hom∗C(Σ, ∗C), whichcompletes the proof.

Proposition 5.12. Suppose that the map O of M(Σ) is surjective. Then the followingassertions hold.

(i) for all M ∈MC, we have HomC(Σ,M) ∼= Hom∗C(Σ,M);(ii) Q ∼= Q and O is surjective;(iii) the morphism of Morita contexts M(Σ) → T(Σ) of Proposition 5.11 becomes an isomorphism.

Proof. (i). Let M be any right C-comodule and take f ∈ Hom∗C(Σ,M), then we find forall g ∈ ∗C and u ∈ Σ,

f(u · g) = f(u) · g = f(u)[0]g(f(u)[1])(149)

= f(u[0]g(u[1])) = f(u[0])g(u[1]

Since O is surjective, we can a find number sets of elements ui ∈ Σ and qi ∈ Q such that∑i ui O qi = 1T . Hence we find for all x ∈ Σ

(150) f(x) = f(ui[0]qi(x)(ui[1])) = f(ui[0])qi(x)(ui[1]) = f(ui)[0]qi(x)(f(ui)[1])

where we applied (149) in the last equality. If we apply ρM on (150) then we find

f(x)[0] ⊗A f(x)[1] = f(ui)[0] ⊗A f(ui)[1]qi(x)(f(ui)[2]) = f(ui)[0] ⊗A qi(x[0])(f(ui)[1])x[1]

= f(ui)[0]qi(x[0])(f(ui)[1])⊗A x[1] = f(x[0])⊗A x[1]


Here we applied the defining property of Q in the second equation and (149) in the last equation.(ii). Consider the following diagram

Σ⊗∗C QO //

T

∼=

Σ⊗∗C Q eO // T

From the fact that T = T by part (i) and O is surjective, we easily deduce that O is surjective.

Applying classical Morita theory (see [14, Theorem II.3.4]), we find that Q ∼= Hom∗C(Σ, ∗C) ∼= Q.(iii). This follows now immediately form (i) and (ii).

5.3.3. Morita contexts associated to a bimodule. Now let A and B be rings, and Σ ∈BMA. Suppose that Σ is finitely generated and projective as a right A-module, then we canconstruct the finite comatrix coring D = Σ∗ ⊗B Σ and we have Σ ∈ MD. Consequently, we canconstruct the Morita context

M(Σ) = (T = EndD(Σ), ∗D,Σ, Q, O, H)

associated to Σ as a right D-comodule. We will compare this Morita context to the Morita contextS(Σ) associated to Σ as a left B-module (see Remark 5.7(i)). Recall that this last Morita contextis

S(Σ) = (B,S = BEnd(Σ)op,Σ, ∗Σ = BHom(Σ, B), 4, N)

with

4 : Σ⊗S ∗Σ → B, (u 4 s) = s(u)

and

N : ∗Σ⊗B Σ → S, ψ(γ Nu)(v) = γ(u)v.

Proposition 5.13. With notation as above, we have a morphism of Morita contexts

S(Σ) → M(Σ).

It is an isomorphism if Σ is totally faithful as a left B-module.

Proof. If Σ ∈ BM is totally faithful, then it follows from Theorem 4.25 that the map

ηN : N → HomD(Σ, N ⊗B Σ), η(n)(u) = n⊗B u

is an isomorphism, for every N ∈ BM. In particular, ηB : B → T = EndD(Σ) is then anisomorphism. Since Σ ∈MA is finitely generated projective, we also have an isomorphism

∗D = AHom(Σ∗ ⊗B Σ, A) ∼= S = BEnd(Σ)

We will next construct a map

λ : ∗Σ = BHom(Σ, B) → Q = DHom(D,Σ∗).

Denote by ei⊗A fi ∈ Σ⊗A Σ∗ the finite dual basis of Σ as a right A-module. A left A-linear mapϕ : D → Σ∗ belongs to Q (i.e. is left D-colinear) if and only if

(151)∑i

ϕ(f ⊗B u)⊗B ei ⊗A fi =∑i

f ⊗B ei ⊗A ϕ(fi ⊗B u).

Take γ ∈ ∗Σ = BHom(Σ, B), and define λ(γ) = ϕ by

ϕ(f ⊗B u) = fγ(u).

then λ(γ) ∈ Q since

fγ(u)⊗B ei ⊗A fi = f ⊗B γ(u)ei ⊗A fi = f ⊗B ei ⊗A fiγ(u).

If Σ ∈ BM is totally faithful, then the inverse λ of λ is

λ : DHom(D,Σ∗) → BHom(Σ, B) ∼= BHom(Σ,EndD(Σ)),


given by λ(ϕ) = β, with

(152) β(u)(v) =∑i

ei(ϕ(fi ⊗B u)(v)).

We prove that β(u) ∈ EndD(Σ). It suffices to show that

(153)∑j

β(u)(ej)⊗A fj ⊗B v =∑j

ej ⊗A fj ⊗B β(u)(v)

or A = B, where

A =∑i,j

ei(ϕ(fi ⊗B u)(ej))⊗A fj ⊗B v

B =∑i,j

ej ⊗A fj ⊗B ei(ϕ(fi ⊗B u)(v))

It follows from (151) that∑i,j

ei ⊗A ϕ(fi ⊗B u)⊗B ej ⊗A fj ⊗B v =∑i,j

ei ⊗A fi ⊗B ej ⊗A ϕ(fj ⊗B u)⊗B v,

and, after we let the second tensor factor act on the third one,

A =∑j

ej ⊗A ϕ(fj ⊗B u)⊗B v

Using (151), we also obtain that∑i,j

ej ⊗A fj ⊗B ei ⊗A ϕ(fi ⊗B u)⊗B v =∑i,j

ej ⊗A ϕ(fj ⊗B u)⊗B ei ⊗A fi ⊗B v;

letting the fourth tensor factor act on the fifth, we find

B =∑j

ej ⊗A ϕ(fj ⊗B u)⊗B v,

and (153) follows.Let us now check that λ and λ are inverses, at least if we identify B and T . Take λ ∈ BHom(Σ, B),and (λ λ)(γ) = β : Σ → EndD(Σ). Then

β(u)(v) =∑i

ei(fiγ(u))(v) =∑i

ei(fi(γ(u)(v) = γ(u)v,

as needed. Now take ϕ ∈ DHom(D,Σ∗), and put β = λ(ϕ), ψ = λ(β). Then

ψ(f ⊗B u)(v) = f(β(u)(v)) = f(∑i

ei(ϕ(fi ⊗B u)(v)))

=∑i

f(ei)(ϕ(fi ⊗B u)(v)) = ϕ(f ⊗B u)(v)

To show that we really have a morphism of Morita contexts, we first have to show that the diagram

Σ⊗S ∗Σ4 //

Σ⊗Sλ

B

ηB

Σ⊗∗D Q′

O// T

commutes. Indeed, for all γ ∈ ∗Σ, u, v ∈ Σ we find((O (Σ⊗S λ))(u⊗ γ)

)(v) = (u Oλ(γ))(v)

= u[0](λ(γ)(u[1]))(v) =∑i

ei(λ(γ)(fi ⊗B u))(v)

=∑i

eifi(γ(u)(v)) = γ(u)(v) = ((ηB 4)(u⊗ γ))(v)


Finally, we need commutativity of the diagram

∗Σ⊗B Σ N //

λ⊗BΣ

S

α

Q′ ⊗T Σ

H// ∗D

This is also straightforward: we compute for all γ ∈ ∗Σ, u, v ∈ Σ and f ∈ Σ∗ that[(H (λ⊗ Σ)

)(γ ⊗ u)

](f ⊗ v) = (λ(γ) Hu)(f ⊗ v)

= (λ(γ)(f ⊗ v))(u) = (fγ(u))(v) = f(γ(u)v)= f(ψ(γ ⊗ u)(v)) = α((γ Nu))(f ⊗ v).

5.4. Structure theorems

By standard Morita theory, if the connecting map H in the Morita context M(Σ) in (143) issurjective, then Σ is a finitely generated projective left T -module and a right ∗C-generator. If O issurjective then Σ is a finitely generated projective right ∗C-module and a left T -generator. In thissection we discuss in more detail the consequences of the surjectivity of the two connecting maps.

Proposition 5.14. Consider the Morita context M(Σ) = (T, ∗C,Σ, Q, O, H) from (143). Thefollowing statements are equivalent:

(i) O is surjective (hence bijective);(ii) for every N ∈MC, the map

ωN : N ⊗∗C Q→ HomC(Σ, N), ωN (n⊗ q)(u) = nµ(q ⊗ u)

(see Lemma 5.8) is surjective;(iii) ω establishes a the natural isomorphism between between the functors

−⊗∗C Q ∼= HomC(Σ,−) : MC →MT .

If any of these equivalent conditions are satisfied, Σ is finitely generated and projective as a rightA-module.

Proof. (i) ⇒ (iii). Choose a finite number of elements uj ∈ Σ and qj ∈ Q such that

(∑

j uj O qj) = 1T . Then we define

ψN : HomC(Σ, N) → N ⊗∗C Q′, ψN (ϕ) =

∑j

ϕ(uj)⊗∗C qj ,

where we denote ϕ ∈ HomC(Σ, N). Then ψN and ωN are inverses:

ψN (ωN (n⊗∗C q)) =∑j

ωN (n⊗∗C q)(uj)⊗∗C qj =∑j

n(q Huj)⊗∗C qj

=∑j

n⊗∗C (q Huj)qj =∑j

n⊗∗C q(uj O qj) = n⊗∗C q

and

ωN (ψN (ϕ))(u) = ωN (∑j

ϕ(uj)⊗∗C qj)(u)

=∑j

ϕ(uj)(qj Hu) =∑j

ϕ(uj)[0](qj(ϕ(uj)[1])(u))

=∑j

ϕ(uj[0])(qj(uj[1])(u)) =∑j

ϕ(uj[0](qj(uj[1])(u)))

=∑j

ϕ((uj O qj)(u))) = ϕ(u).

5.4. STRUCTURE THEOREMS 157

(iii) ⇒ (ii). Trivial.

(ii) ⇒ (i). Take N = M .For the last statement, consider again finite sets of elements uj ∈ Σ and qj ∈ Q such that(∑

j uj O qj) = 1T . We introduce the finite sets yi ⊂ Σ and ξi ⊂ Σ∗ via the requirement that∑i

yi ⊗A ξi =∑j

uj [0] ⊗A qj(−)(uj [1]).

Then we find for all u ∈ Σ that∑i

yiξi(u) =∑j

ujqj(u)(uj [1]) =∑j

(uj O qj)(u) = u,

i.e. the elements yi and ξi constitute a dual basis for Σ as a right A-module.

Consider the following diagram of functors

(154) MC

J

!!CCC

CCCC

CCCC

CCCC

CC

HomC(Σ,−)

MT

−⊗T Σ

==

⊗T Σ//M∗C

−⊗∗CQoo

In this diagram, the outer triangle commutes, and if the equivalent conditions of Proposition 5.14are satisfied, then the inner triangle commutes as well. Moreover, we know by classical Moritatheory that if O is surjective, then the functor −⊗T Σ : MT →M∗C is fully faithful. FurthermoreJ is a faithful functor and we know from Theorem 3.35 that J is full if and only if C is locallyprojective as a left A-module. Therefore, the surjectivity of O implies that −⊗T Σ : MT →MC

is a full and faithful functor provided that C is locally projective as a left A-module. The followingTheorem tells us that this last condition is redundant.

Theorem 5.15. Let C be an A-coring and Σ its right comodule. Consider the Morita contextM(Σ), associated to Σ in (143). If the connecting map O in (145) is surjective, then the functor−⊗T Σ : MT →MC is fully faithful.

Proof. The proof consists of a verification of the bijectivity of the unit of the adjunction offunctors −⊗T Σ : MT →MC and HomC(Σ,−) : MC →MT , i.e. the map

(155) ηN : N → HomC(Σ, N ⊗T Σ), n 7→ ( x 7→ n⊗T x ),

for any right T -module N . Since we know by Proposition 5.14 that HomC(Σ,M) ∼= M ⊗∗C Q forall right C-comodules M , this follows from the following composition of isomorphisms

HomC(Σ, N ⊗T Σ) ∼= N ⊗T Σ⊗∗C Q ∼= N ⊗T T ∼= N.

Let us check explicitly that this isomorphism is given by the unit of the adjunction ηN . Chooseelements xi ⊂ Σ and qi ⊂ Q such that

∑i xi O qi = 1T . The inverse of the map (155) can

be constructed as

ηN : HomC(Σ, N ⊗T Σ) → N, ζN 7→ (N ⊗T O)(∑

i

ζN (xi)⊗∗C qi),

for all ζN ∈ HomC(Σ, N ⊗T Σ). Indeed, it is obvious that ηN ηN = N . For the other equality,use the associativity of the Morita context M(Σ) to compute, for ζN ∈ HomC(Σ, N ⊗T Σ) andx ∈ Σ,(

ηN ηN)(ζN )(x) = (N ⊗T O⊗TΣ)(

∑i

ζN (xi)⊗∗C qi ⊗T x)

= (N ⊗T Σ⊗∗C H)(∑i

ζN (xi)⊗∗C qi ⊗T x)

=∑i

ζN (xi)(qi Hx) = ζN (∑i

xi(qi Hx)) = ζN (∑i

(xi O qi)x) = ζN (x),


where in the fourth equality we used the right ∗C-linearity of ζN ∈ HomC(Σ, N ⊗T Σ).

From now on, we will focus on the properties of the second connecting map H of the Moritacontext M(Σ).

Lemma 5.16. Consider the Morita context M(Σ) from (143). If C is locally projective as aleft A-module, then

Im H ⊂ Rat(∗C).

Proof. Take q ∈ Q, u ∈ Σ and compute (q Hu). For all f ∈ ∗C and c ∈ C, we have

((q Hu) ∗ f)(c) = f(c(1)(q Hu)(c(2)))

= f(c(1)q(u)(c(2))) = f(q(u[0])(c)u[1])

= q(u[0])(c)f(u[1]) = (q Hu[0])(c)f(u[1]).

Therefore we find for (q Hu) ∈ Im H, elements fi = (q Hu[0]) and ci = u[1] such that (q Hu) ∗ f =fif(ci) for arbitrary for all f ∈ ∗C and the rationality of (q Hu) follows.

Suppose that C is locally projective as a left A-module. An immediate consequence ofLemma 5.16 is that the surjectivity of H implies that ∗C = Rat(∗C), hence C is finitely gener-ated and projective as a left A-module by Theorem 3.36. In the following Lemma we give anexplicit construction of a dual basis for C as a left A-module if H is surjective, without assumingin advance that C is locally projective as a left A-module.

Lemma 5.17. Consider the Morita context M(Σ) of (143). If the connecting map H in (144)is surjective then C is a finitely generated projective left A-module.

Proof. If H is surjective then there exist finite sets qi ⊂ Q and xi ⊂ Σ such that

εC =∑i

qi Hxi ≡∑i

qi(xi).

Introduce finite sets fj ⊂ ∗C and cj ⊂ C via the requirement that∑j

fj ⊗A cj =∑i

qi(xi[0])⊗A xi[1].

We claim that they are dual bases. Indeed, for any c ∈ C,∑j

fj(c)cj =∑i

qi(xi[0])(c)xi[1] =∑i

c(1)qi(xi)(c(2)) = c(1)εC(c(2)) = c,

where the second equality follows by the definition (141) of Q. Hence C is a finitely generatedprojective left A-module, as stated.

Theorem 5.18. Consider the Morita context M(Σ) of (143). Then the surjectivity of theconnecting map H is equivalent to the fact that C is finitely generated and projective as a leftA-module together with any of the following assertions

(i) the functor HomC(Σ,−) : MC →MT is fully faithful;(ii) Σ is a generator in MC;(iii) Σ is flat as a left T -module and a comonadic Galois comodule;(iv) Σ is finitely generated and projective as a left T -module, π′ is injective and κ is surjective,

where

π : ∗C → TEnd(Σ)op, π(f)(u) = u · f,κ : Q→ THom(Σ, T ), κ(q)(u)(v) = u · q(v);

(v) Q is finitely generated and projective as a right T -module, π′ is injective and κ′ is surjective,where

π′ : ∗C → EndT (Q), π′(f)(q) = f · q,κ′ : Σ → HomT (Q,T ), κ′(u)(q)(v) = u · q(v);

5.4. STRUCTURE THEOREMS 159

(vi) Σ and Q are finitely generated and projective as a left, respectively right T -modules and themaps π, π′, κ and κ′ are bijective.

If Σ is finitely generated and projective as a right A-module, then the above statements arefurthermore equivalent to

(vii) Σ is finitely generated and projective as a left T -module and π is an isomorphism.

Proof. H ⇒ (i). Suppose that H is surjective. By Lemma 5.17 we know that C is finitely

generated and projective as a left A-module. Hence, MC ∼= M∗C and the statement follows fromclassical Morita theory. Let us just mention for all M ∈ MC that the inverse of the counit ζM ofthe adjunction is given by the map

θM : M → HomC(Σ,M)⊗M Σ, θM (m) = ωM (m⊗∗C qi)⊗T ui,where we picked an element qi ⊗T ui ∈ Q⊗T Σ such that qi Hui = εC and ω denotes the naturaltransformation of Lemma 5.8.(i) ⇒ H. Since C is finitely generated and projective as a left A-module, ∗C is a right C-comodule,

and Q ∼= HomC(Σ, ∗C) (see Remark 5.9). Now observe that H = ζ∗C.(i) ⇔ (ii). Follows directly from Lemma 4.21.

(ii) ⇔ (iii). Follows directly from Theorem 4.22.

H ⇒ (vi). Follows by classical Morita theory (see [14, Theorem II.3.4]).

(vi) ⇒ (iv). Trivial.

(iv) ⇒ H. Consider the Morita context

C(Σ) = (TEnd(Σ)op, T, THom(Σ, T ),Σ, 4, N)

associated to Σ as a left T -module (see Remark 5.7(i)). The fact that Σ is finitely generated andprojective as a left T -module means that N is surjective: a dual basis ξi⊗T ui ∈ THom(Σ, T )⊗T Σsatisfies ξi Nui = ξi(−)ui = Σ. Consider the following diagram

THom(Σ, T )⊗T Σ N //TEnd(Σ)op

Q⊗T ΣH

//

κ⊗RΣ

OO

∗C

π

OO

Take any q ⊗T u ∈ Q⊗T Σ and x ∈ Σ, let us check that this diagram commutes.

π(q Hu)(x) = x · q(u)(κ(q) Nu)(x) = κ(q)(x)(u) = x · q(u)

Since N and κ⊗T Σ are surjective and π is injective, we easily obtain that H is surjective.(v). The equivalence with (v) is proven in the same way as the equivalence with part (iv).

(iii) ⇔ (vii) If Σ is finitely generated and projective as a right A-module, then we can construct

the comatrix coring D = Σ∗ ⊗T Σ and Σ is comonadic-Galois if and only if Σ is a (finite) Galoiscomodule. Since Σ is also finitely generated and projective as a left T -module, D is reflexive asa left A-module. Furthermore, since C is finitely generated and projective as a left A-module,it is as well reflexive. Hence by Proposition 5.2 can is an isomorphism if and only if ∗can is anisomorphism. The equivalence follows if we remark that π is exactly ∗can.

Theorem 5.19. Let A and B be rings, and C an A-coring and Σ ∈ BMC. Also writeT = EndC(Σ). Then

(0) the strictness of the Morita context M(Σ) of (143) together with the bijectivity of the map` : B → T

is equivalent to the fact that C is finitely generated and projective as a left A-module togetherwith any of the following assertions:

(i) can : D = Σ∗ ⊗B Σ → C is an isomorphism and Σ ∈ BM is faithfully flat;(ii) ∗can : ∗C → BEnd(Σ)op is an isomorphism and Σ is finitely generated and projective as a

right A module and a finitely generated and projective generator in BM;


(iii) (−⊗BΣ,HomC(Σ,−)) is a pair of inverse equivalences between the categories MB and MC.

Proof. (0) ⇒ (iii). This follows directly from Theorems 5.15 and 5.18.

(iii) ⇒ (0). Since C is a finitely generated and projective left A-module, the categories MC and

M∗C are isomorphic, hence −⊗T Σ : MT →M∗C and Hom∗C(Σ,−) : M∗C →MT are inverseequivalences. The canonical strict Morita context, associated to this equivalence, is equal to M(Σ)in (143).(i) ⇔ (iii). Follows from Theorem 4.27.

(iii) ⇒ (ii). This is a consequence of Corollary 4.35 and Proposition 5.2.

(ii) ⇒ (i). Follows from Theorem 5.18 and the fact that a finitely generated and projectivegenerator is faithfully flat.

Lemma 5.20. Let C be an A-coring which is locally projective as a left A-module. If Rat(∗C)is dense in ∗C with respect to the finite topology, we have the following properties.

(i) For every M ∈MC, the map

ΩM : M ⊗∗C Rat(∗C) →M, ΩM (m⊗∗C f) = m · f = m[0]f(m[1])

is an isomorphism.(ii) The categories MRat(∗C) and MC are isomorphic.

Proof. Remark first that by Proposition 3.38, we know that RatR(∗C) acts with local unitson all M ∈MC.(i). Since R = RatR(∗C) is a ring with local units, it is in particular a firm ring and we find

M ⊗R R ∼= M for all M ∈ MC (see Lemma 2.31). Furthermore, R is a two-sided ideal in ∗C(see Proposition 3.38), hence M ⊗R R ∼= M ⊗∗C R by Lemma 2.14. The combined isomorphismM ⊗∗C R → M is given by right multiplication on M , which means it is precisely ΩM . (ii).

For M ∈ MC, we know by Proposition 3.38 that M is a firm right Rat(∗C)-module, i.e. M ∈MRat(∗C). Conversely, if M ∈ MRat(∗C), then for every m ∈ M we can find elements mi ∈ Mand gi ∈ Rat(∗C) such that m = mi · gi. For all f ∈ ∗C we then compute

m · f = (mi · gi) · f = mi · (gi ∗ f) = m · gi[0]f(gi[1]).

This means that M is a rational ∗C-module, hence M ∈MC.

We now present a generalization of Lemma 5.16.

Corollary 5.21. Consider an A-coring C which is locally projective as a left A-module andlet µ be as in the Morita context from Theorem 5.6. Then for every M ∈M∗C we have a map

rM : M ⊗∗C Q⊗T ΣM⊗H // Rat(M) ,

which is an isomorphism if Im H = Rat(∗C) and Rat(∗C) is dense in the finite topology on ∗C.

Proof. First of all, rM is well defined: pick m⊗∗C q⊗T u ∈M ⊗∗CQ⊗T Σ, then rM (m⊗∗C

q ⊗T u) = m · (q Hu). Since Im H ⊂ Rat(∗C), we find

(m · (q Hu)) · f = m · ((q Hu)) · f) = m · ((q Hu) ∗ f)

=∑i

m · (((q Hu)if(ci)) =∑i

(m · ((q Hu)i)f(ci),

so we conclude that m · (q Hu) ∈ Rat(M).If Rat(∗C) has local units, then, as Rat(M) ∈ MC, Rat(M) ∼= M ⊗∗C Rat(∗C), by Lemma 5.20.If, in addition, Imµ = Rat(∗C), then this isomorphism is exactly rM .

Corollary 5.21 provides an explicit way to construct the rational part of a ∗C-module. Remarkthat r∗C = µ.

We have seen that the Morita context M(Σ) = (T, ∗C,Σ, Q, O, H) can only be strict if C isfinitely generated and projective as a left A-module, by the surjectivity of H. Consequently, inmany cases, it is better to look to an other, restricted, Morita context. Since Im H ⊆ Rat(∗C)if C is locally projective as a left A-module, then we can restrict our context without any furtherconsequences on the connecting maps or modules to M′(Σ) = (T,Rat(∗C),Σ, Q, τ, µ).

5.5. APPLICATION: MORITA CONTEXTS ASSOCIATED TO A GROUPLIKE ELEMENT 161

If Rat(∗C) is dense in the finite topology of ∗C, it is a ring with local units and we can applyLemma 5.20. Moreover we have the following diagram of functors that extends (154).

MC

J

##FFFFFFFFFFFFFFFFFF

HomC(Σ,−)

||||

||||

||||

||||

|

MT

−⊗T Σ

==|||||||||||||||||

⊗T Σ//MRat(∗C)

Rat

ccFFFFFFFFFFFFFFFFFF−⊗∗CQoo

The outer triangle of this diagram commutes. Since (J,Rat) is an isomorphism of categories, theinner diagram commutes if O is surjective (see Proposition 5.14). In this way, the Galois propertiesof Σ are completely determined by the properties of the restricted Morita context M′(Σ).

Theorem 5.22. Let C be an A coring which is locally projective as a left A-module. Supposethat Rat(∗C) is dense in the finite topology on ∗C. Take Σ ∈ MC and consider the restrictedMorita context M′(Σ) = (T,Rat(∗C)),Σ, Q, O, H). If O is surjective, then the following statementsare equivalent.

(i) can : D = Σ∗ ⊗T Σ → C is an isomorphism and Σ is faithfully flat as a left T -module;(ii) can : D = Σ∗ ⊗T Σ → C is an isomorphism and Σ is flat as a left T -module;(iii) Σ is a generator in MC;(iv) Σ is a projective generator in MC;(v) Σ is a finitely generated and projective generator in MRat(∗C);(vi) µ is surjective (onto Rat(∗C));(vii) M′(Σ) is a strict Morita context;(viii) (−⊗T Σ,HomC(Σ,−)) is a pair of inverse equivalences between MT and MC;(ix) for allN ∈MC, the counit of the adjunction ζN : HomC(Σ, N)⊗TΣ → N is an isomorphism.

If these equivalent conditions hold, then Σ is finitely generated and projective as a right A-module.

Proof. (i) ⇔ (iv) ⇔ (viii) follow from Theorem 4.27 (and Corollary 4.35).

(ii) ⇔ (iii) ⇔ (ix) follow in the same way from Theorem 4.22.

(vi) ⇔ (vii) follows from the fact that O is surjective and general Morita theory (see also Theo-

rem 2.70).(vii) ⇒ (viii). Since Rat(∗C) is dense in ∗C, MRat(∗C)

∼= MC by Lemma 5.20. The strictness of

the Morita context M′(Σ) implies that −⊗T Σ is an equivalence between the categories MT andMRat(∗C)

∼= MC.(viii) ⇒ (ix) is trivial.

(ix) ⇒ (vii). From Remark 5.9 we know that Q ∼= HomC(Σ,Rat(∗C)) Now take N = Rat(∗C) inthe counit of the adjunction; we then find that ζRat(∗C) = H is an isomorphism, as

ζRat(∗C) : Hom∗C(Σ,Rat(∗C))⊗T Σ ∼= Q⊗T Σ → N = Rat(∗C).

(iv) ⇒ (v). We know that Σ is a generator in MC ∼= MRat(∗C). Since (iv) is equivalent to (vii),

we know that M′(Σ) is strict. From Morita theory it then follows that Σ ∈ MRat(∗C) is finitelygenerated projective.(v) ⇒ (iv) is trivial.

The last statement follows from (viii), by Corollary 4.35.

5.5. Application: Morita contexts associated to a grouplike element

5.5.1. Grouplike characters. Let A and R be rings, and i : A → R a ring morphism. Amap χ : R→ A is called a (right) grouplike character if and only if it satisfies the following threeconditions, for all r, s ∈ R:

(1) χ is right A-linear;(2) χ(χ(r)s) = χ(rs);(3) χ(1R) = 1A.


It follows from the second condition that χ2 = χ. A is a right R-module, with structure ar =χ(ar). The three conditions on the map χ are explained by the following straightforward Lemma.

Lemma 5.23. Consider R as an A-ring. A map χ : A⊗A R = R→ A makes A into a rightmodule over this algebra R if and only if it is a grouplike character.

This is the dual result of the fact that grouplike elements of an A-coring C are in one-to-onecorrespondence with right (or left) C-comodule structures on A (see Lemma 3.4).

For any right R-module M , we define

MR = m ∈M | m · r = mχ(r) ∼= HomR(A,M).

Then B = AR = b ∈ A | bχ(r) = χ(br), for all r ∈ R is a subring of A, and MR is a rightB-module. In fact we obtain a functor G = (−)R : MR → MB, which is a right adjoint ofF = −⊗B A. The unit and counit of the adjunction are ηN : N → (N ⊗B A)R, ηN (n) = n⊗B 1and ζM : MR ⊗B A→M , ζM (m⊗B a) = ma. Now consider

(156) Q = RR = q ∈ R | qr = qχ(r), for all r ∈ Rand the map ζR = µ : Q⊗B A→ R, µ(q ⊗B a) = qa. Furthermore χ(Q) ⊂ B since χ(q)χ(r) =χ(qχ(r)) = χ(qr) = χ(χ(q)r) for all q ∈ Q and r ∈ R. Also recall that R∗ = HomA(R,A) is an(A,R)-bimodule:

(a · f · r)(s) = af(rs)for all a ∈ A, r, s ∈ S and f ∈ R∗.

Lemma 5.24. (R∗)R ∼= A as a right B-module, and the counit map can = ζR∗ : A⊗BA→ R∗

is given bycan(a⊗B a′)(r) = aχ(a′r)

for all a, a′ ∈ A and r ∈ R.

Proof. First observe that f ∈ (R∗)R if and only if f(rs) = f(χ(r)s) for all r, s ∈ R. Define

j : A→ (R∗)R, j(a)(r) = aχ(r) and p : (R∗)R → A : p(f) = f(1).

It is clear that j(a) is right A-linear. Also

j(a)(rs) = aχ(rs) = aχ(χ(r)s) = j(a)(χ(r)s)

so j(a) ∈ (R∗)R. Observe that j and p are inverses, since p(j(a)) = j(a)(1) = aχ(1) = a andj(p(f))(r) = f(1)χ(r) = f(χ(r)) = f(r). Now we compute

(j(a) · a′)(r) = j(a)(a′r) = aχ(a′r).

From this formula, it follows that, for b ∈ B,

(j(a) · b)(r) = aχ(br) = abχ(r) = j(ab)(r)

so j is right B-linear.

The proof of the following result is now an easy exercise, left to the reader.

Proposition 5.25. With notation as above, A ∈ BMR andQ ∈ RMB, and we have a Moritacontext (B,R,A,Q, τ, µ). The connecting maps µ = ζR : Q⊗B A → R and τ : A⊗R Q → Bare given by

(157) µ(q ⊗B a) = qa and τ(a⊗R q) = aq = χ(aq).

Remark 5.26. Let R be a ring. Recall from [14, II.4], see also Remark 5.7 (i), that we canassociate a Morita context to any right R-module P . If we consider i : A→ R and χ : R → Aas above, then the Morita context associated to the right R-module A is isomorphic to the Moritacontext from Proposition 5.25. It suffices to observe that

B ∼= EndR(A) and Q ∼= HomR(A,R).

In this way, the Morita context from Proposition 5.25 becomes a special case of the Morita contextM(Σ) = M(A) constructed in (143) where we consider R as a trivial R-coring and A as a R-comodule. Then obviously the coring R is finitely generated and projective over its base ring R.


We will apply our general theory developed in the previous sections to determine when the Moritacontext of Proposition 5.25 is strict.

Proposition 5.27. With notation as in Proposition 5.25, the following assertions are equiv-alent:

(i) τ is surjective (hence bijective);(ii) for all M ∈MR, the map ωM : M ⊗R Q→MR, ωM (m⊗R q) = m · q is an isomorphism;(iii) there exists Λ ∈ Q such that χ(Λ) = 1;(iv) A is finitely generated and projective as a right R-module.

Proof. (i) ⇔ (ii). Follows from Proposition 5.14, taking into account Remark 5.26.

(i) ⇒ (iii). If τ is surjective, then there exist aj ∈ A and qj ∈ Q such that τ(∑

j aj ⊗R qj) =χ(∑

j ajqj) = 1. Then Λ =∑

j ajqj ∈ Q, since Q is a left ideal in R.

(iii) ⇒ (ii). For all m ∈ M and q ∈ Q, we have that (m · q) · r = m · (qr) = m · qχ(r), for all

r ∈ R, so m · q ∈MR. Consider the map

θM : MR →M ⊗R Q, θM (m) = m⊗R Λ.

For allm ∈MR, we easily compute that ωM (θM (m)) = m·Λ = mχ(Λ) = m for allm ∈MR, and,for all m ∈M and q ∈ Q, θM (ωM (m⊗R q)) = m · q⊗R Λ = m⊗R qΛ = m⊗R qχ(Λ) = m⊗R q,so θM and ωM are inverses.(i) ⇒ (iv) follows from [14, Prop. II.4.4], taking Remark 5.26 into account; we also give an easydirect proof.We know that (i) implies (iii). Let f ∈ HomR(A,R) be given by f(a) = Λa. Then for all a ∈ A,we have a = χ(Λ)a = χ(Λa) = 1f(a), hence (1, f) is a finite dual basis of A as a rightA-module.(iv) ⇒ (i) Let (aj , fj) | j = 1, · · · , n be a finite dual basis of A as a right R-module, and

take fj(1) = qj ∈ Q. Then 1 =∑

j ajfj(1) = χ(∑

j ajqj) = τ(∑

j aj ⊗R qj), hence τ issurjective.

Proposition 5.28. Consider the Morita context from Proposition 5.25, and assume that τis surjective. Then F = −⊗B A : MB →MR is fully faithful. Moreover, take Λ ∈ Q such thatχ(Λ) = 1. Then we have the following properties:

(i) Λ2 = Λ and ΛRΛ = ΛB ∼= B.(ii) B is a direct summand of A as a left B-module.

Proof. It follows from Theorem 5.15 that F is fully faithful.(i). Since χ(Λ) = 1 and Λ ∈ Q, we have Λ2 = Λχ(Λ) = Λ. For all r, s ∈ R, we have

χ(rΛ)χ(s) = χ(rΛχ(s)) = χ(rΛs) = χ(χ(rΛ)s), so χ(rΛ) ∈ B, and ΛrΛ = Λχ(rΛ) ∈ ΛB. Itfollows that ΛRΛ ⊂ ΛB.Now in the above arguments, take r = i(b), with b ∈ B. It follows that ΛbΛ = Λχ(bΛ) =Λbχ(Λ) = Λb, and ΛB ⊂ ΛRΛ. Finally, the right B-module generated by Λ is free since Λb = 0implies that 0 = χ(Λb) = χ(Λ)b = b.(ii). We define the map Tr : A→ B, Tr(a) = τ(a⊗R Λ) = χ(aΛ). Tr is left B-linear, because τ

is left B-linear. Tr is a projection, since Tr(b) = χ(bΛ) = bχ(Λ) = b, for all b ∈ B.

We recall from Morita Theory ([14, II.3.4]) that we have ring morphisms

π : R→ BEnd(A)op, π(r)(a) = ar = χ(ar);

π′ : R→ EndB(Q), π′(r)(q) = rq.

We also have an (R,B)-bimodule map

κ : Q→ BHom(A,B), κ(q)(a) = χ(aq)

and a (B,R)-bimodule map

κ′ : A→ HomB(Q,B), κ′(a)(q) = χ(aq).

If µ is surjective, then π, π′, κ and κ′ are isomorphisms, and A and Q are finitely generated andprojective as resp. a left and right B-module, and a generator as resp. a right and left R-module.


Proposition 5.29. Consider the Morita context from Proposition 5.25. The following asser-tions are equivalent:

(i) µ : Q⊗B A→ R is surjective;(ii) the functor G = (•)R : MR → MB is fully faithful, that is, for all M ∈ MR, the counit

map ζM : MR ⊗B A→M is an isomorphism;(iii) A is a right R-generator;(iv) A is finitely generated and projective as a left B-module, and π is bijective;(v) A is finitely generated and projective as a left B-module, π is injective and κ is surjective;(vi) Q is finitely generated and projective as a right B-module, π′ is injective and κ′ is surjective;(vii) A and Q are finitely generated and projective as a left, respectively right B-module and the

maps π, π′, κ and κ′ are bijective.

Proof. This is a consequence of Theorem 5.18.

5.5.2. Grouplike elements. In this Section, A is a ring and (C, x) is an A-coring with a fixedgrouplike element. Then A becomes a right C-comodule with action

ρ : A→ A⊗A C ∼= C, ρ(a) = xa,

(see Lemma 3.4) and applying the results of Section 5.3.2 we obtain a Morita context M(A) =(B′ = Endcc(A), ∗C, A,Q′ = CHom(C, A), τ ′, µ′), where

(158) µ′ : Q′ ⊗B′ A→ ∗C ; µ′(q ⊗B′ a) = q#i(a);

(159) τ ′ : A⊗∗C Q′ → B′ ; τ ′(a⊗∗C q) = a · q = q(xa).

Remark that since A is in a trivial way finitely generated and projective as a right A-module, thecontext M(A) coincides with ∗M(A) by Proposition 5.10. In this situation we can describe moreexplicitly

B′ = AcoC = b ∈ A | bx = xbQ′ = CHom(C, A) = q ∈ ∗C | q#IC = ρl q

= q ∈ ∗C | c(1)q(c(2)) = q(c)x, for all c ∈ C.(160)

Here we used the notation M coC = m ∈ M | ρN (n) = n ⊗A x ∼= HomC(A,N), as in Corol-lary 4.36.

Any right C-comodule M is also a right ∗C-module, so in particular, the grouplike elementx induces, via ρ a ∗C-action on A. Put R = ∗C. Since R-actions on A coincide with grouplikecharacters, the grouplike element x will induce a grouplike character. Indeed, consider χ : R→ A,χ(f) = f(x). Using (22), we can easily compute that χ is right A-linear, χ(i(χ(f))∗g) = χ(f ∗g),and χ(εC) = 1.

We put

B = A∗C = a ∈ A | f(xa) = af(x), for all f ∈ ∗C,

Q = (∗C)∗C

Applying the results of the previous Section, we find a Morita context (B, ∗C, A,Q, τ, µ) connectingB and ∗C.

It is easy to see that Q′ ⊂ Q and M coC ⊂ M∗C, in particular B′ ⊂ B. and the converse

inclusion holds if C is locally projective as a left A-module. Moreover, we have the following.

Theorem 5.30. With notation as above, (B′, ∗C, A,Q′, τ ′, µ′) is a Morita context, and wehave a morphism of Morita contexts

(B′, ∗C, A,Q′, τ ′, µ′) → (B, ∗C, A,Q, τ, µ),

which becomes an isomorphism if C is locally projective as a left A-module.

Proof. This is a special case of Proposition 5.11, taking Σ = A.

We now give necessary and sufficient conditions for τ ′ to be surjective.


Theorem 5.31. Consider the Morita context (B′, ∗C, A,Q′, τ ′, µ′) of Theorem 5.30. Thefollowing statements are equivalent:

(i) τ ′ is surjective (and, a fortiori, bijective);(ii) there exists Λ ∈ Q′ such that Λ(x) = 1;(iii) for every right ∗C-module M , the map ωM : M ⊗∗C Q

′ → M∗C, ωM (m ⊗∗C q) = m · q, is

bijective.

Proof. (i) ⇒ (ii). If τ ′ is surjective, then there exist aj ∈ A and qj ∈ Q′ such that

1 = τ ′(∑

j aj ⊗∗C qj) =∑

j qj(xaj) =∑

j(i(aj)#qj)(x). Now Λ =∑

j(aj)#qj ∈ Q′ because Q′

is a left ideal in ∗C.(ii) ⇒ (iii). Consider ηM : M

∗C →M⊗∗CQ′, ηM (m) = m⊗∗CΛ. It is clear that ωMηM = IM∗C .

Furthermore, for all m ∈M and q ∈ Q′,

ηM (ωM (m⊗∗C q)) = (m · q)⊗∗C Λ = m⊗∗C q#Λ = m⊗∗C q,

since q#Λ = qΛ(x) = q.(iii) ⇒ (i). ωA = τ ′ is bijective (remark that this is also a special case of Proposition 5.14).

Corollary 5.32. Consider the Morita context (B′, ∗C, A,Q′, τ ′, µ′) of Theorem 5.30. As-sume that τ ′ is surjective.Then we have:1) For all M ∈MC, M

∗C = M coC. In particular B = B′.2) Q = Q′.3) The two Morita contexts in Theorem 5.31 coincide.

Proof. This follows from Proposition 5.12 taking Σ = A.

Now let us look at the map µ′. Recall from Lemma 5.17 that the surjectivity of µ′ implies thatC is finitely generated and projective as a left A-module. As we already noticed, this implies thatMC ∼= M∗C, the two Morita contexts coincide, and we can apply Proposition 5.27. Let us statethe result, for completeness sake. From [14, II.3.4], recall that we have ring morphisms

π : ∗C → BEnd(A)op, π(f)(a) = f(xa);

π′ : ∗C → EndB(Q), π′(f)(q) = f#q.In fact π = ∗can, cf. (135). We also have a (∗C, B)-bimodule map

κ : Q→ BHom(A,B), κ(q)(a) = q(xa)

and a (B, ∗C)-bimodule map

κ′ : A→ HomB(Q,B), κ′(a)(q) = q(xa).

If µ is surjective, then π, π′, κ and κ′ are isomorphisms, and A and Q are finitely generated andprojective, respectively as a left and a right B-module. We now state some necessary and sufficientconditions for µ to be surjective.

Theorem 5.33. Assume that C is finitely generated and projective as a left A-module, andconsider the Morita context (B = B′, ∗C, A,Q = Q′, τ = τ ′, µ = µ′) of Theorem 5.30. Then thefollowing assertions are equivalent.

(i) µ : Q⊗B A→ ∗C is surjective (and, a fortiori, bijective);(ii) the functor (−)coC : MC →MB is fully faithful;(iii) A is a right ∗C-generator;(iv) A is finitely generated and projective as a left B-module and π is bijective;(v) A is finitely generated and projective as a left B-module, π is injective and κ is surjective;(vi) Q is finitely generated projective as a right B-module, π′ is injective and κ′ is surjective;(vii) Q and A are finitely generated and projective as a right respectively left B-module and the

maps κ, κ′, π and π′ are isomorphisms;(viii) A is projective as a left B-module and (C, x) is a Galois coring (i.e. A is a B-C Galois

comodule).

Proof. This is a special case of Theorem 5.18, taking Σ = A.


References

The context from Section 5.3.1 was first constructed in the author’s joint work with S.Caenepeel and E. De Groot [42]. The more general construction of Section 5.3.2 is taken formthe joint paper with G. Bohm [24]. However, many results are presented here in a more generalway. The applications about Morita theory for rings with a grouplike character and corings witha grouplike element of Section 5.5 appeared in the joint work with S. Caenepeel and ShuanhongWang [47] (compare with [4]).

Chapter 6Cleft Bicomodules

We know from Example 3.21 that bialgebras give rise to examples of corings. Hence wecan apply the Galois theory developed in Chapter 4 to the Hopf modules over a bialgebra. IfH is a Hopf k-algebra and Σ = H the trivial H-module, then EndH(H) = k and the functor− ⊗ H : Mk → MH is always an equivalence of categories. This is known as the fundamentaltheorem of Hopf algebras. However, if we choose Σ different from H or change H into a bialgebraor a coring, no fundamental theorem can be proved in general.

The key element in the proof of the fundamental theorem is the presence of the antipodeS : H → H of the Hopf algebra H. Attempts to prove more general versions of the fundamentaltheorem, in cases where Σ differs from the canonical H-module, lead to the introduction of cleftextensions of algebras, first by Hopf algebras and later by coalgebras. This theory is closely relatedto crossed products (see [21] for a general treatment). In this Chapter we want to develop ageneral framework that unifies all previous theories on cleft exensions. To this end, we introducethe notion of a cleft bicomodule, associated to a coring extension. Algebra extensions, that are cleftextensions by a Hopf algebra, a coalgebra or a Hopf algebroid, as well as cleft entwining structures(over commutative or non-commutative base rings) and cleft weak entwining structures, are shownto provide examples of cleft bicomodules.

After some notational remarks in the first section, we construct a Morita context associated toa coring extension (D : L) of (C : A) and an L-D bicomodule Σ in Section 6.2. We give sufficentand necessary conditions for this context to be strict. Section 6.3 is devoted to Galois theory for Σ.We show that strictness of the Morita context, and some mild extra conditions allow us to provestructure theorems for the comodule Σ, and this without assuming that Σ is (faithfully) flat as aleft T = EndC(Σ)-module. A special situation is treated in Section 6.4, where we introduce cleftbicomodules, by means of invertible elements in the constructed Morita context. It turns out thata bicomodule is cleft if and only if it is a Galois comodule that satisfies a normal basis property. Inthe final section we provide a number of examples, including all previously known notions of cleftextensions and provide some new examples based on partial group actions. Finally, we give a dualversion of our theory in the form of cleft factorization structures.

6.1. Some remarks on notation and coring extensions

Throughout this Chapter, k will be a fixed commutative ring, all algebras are associative unitalalgebras over k. The forgetful functor MR →Mk will be denoted by UR.

We will consider two corings D over the base k-algebra L and C over the base k-algebra A.To distinguish between the comultiplication of D and C if we make use of the Sweedler notation,we make use of the following convention. For all c ∈ C and d ∈ D we will denote (in this chapter

167

168 CHAPTER 6. CLEFT BICOMODULES

only):

∆C(c) = c(1) ⊗A c(2)(161)

∆D(d) = d(1) ⊗L d(2)(162)

i.e. we use upper indices for the comultiplication in C and lower indices for the comultiplication inD. Similarly, for all M ∈MC and N ∈MD we denote

ρM : M →M ⊗A C, ρM (m) = m[0] ⊗A m[1](163)

τN : N → N ⊗L D, τN (n) = n[0] ⊗L n[1](164)

i.e. we use the greek letter ρ and upper indices for coactions on C-comodules and the greek letterτ and lower indices for coactions on D-comodules.

Suppose that D is a right coring-extension of C (see Section 3.2.1). This means that C is aC-D bicomodule with the left regular C-coaction ∆C and some right D-coaction τC or, equivalently,there exists a k-linear functor U : MC →MD, making the following diagram commutative.

MCU //

FC

""EEE

EEEE

E MD

FD

||xxxxxxxx

MA

UA

""EEE

EEEE

E ML

UL

||yyyy

yyyy

Mk

where FD and FC denote forgetful functors. The functor U can be understood as follows. Usingthe right D-coaction τC : c 7→ c[0] ⊗L c[1], for c ∈ C, any right C-comodule M can be equippedwith a right D-comodule structure with right L-action

ml : = m[0]εC(m[1]l), for m ∈M and l ∈ L,

and D-coaction

τM : M →M ⊗L D, m 7→ m[0] ⊗L m[1] : = m[0]εC(m[1][0])⊗L m[1]

[1],

where %M : m 7→ m[0] ⊗A m[1] denotes the C-coaction in M . It is straightforward to check thatwith this definition any right C-comodule map is D-colinear. In particular, a right C-coaction,being C-colinear by coassociativity, is D-colinear.

6.2. Morita theory for coring extensions

With notation as in the previous section, let the L-coring D be an extension of the A-coringC. If Σ is an object in the category LMC of L-C bicomodules, i.e. it is a left L-module and aright C-comodule with left L-linear C-coaction, then T : = EndC(Σ) is an L-ring with unit mapL→ T , l 7→ ( x 7→ lx ). Furthermore, HomC(Σ,M) is a right L-module for any right C-comoduleM , via (ϕM l)(x) : = ϕM (lx), for ϕM ∈ HomC(Σ,M), l ∈ L and x ∈ Σ. Hence, in addition toBrzezinski’s functor U : MC →MD, we can define another k-linear functor,

V : = HomC(Σ,−)⊗L D : MC →MD.

Consider the opposite of the category of k-linear functors and their natural transformations. Thefull subcategory defined by the two objects U and V determines a Morita context

(165) (Nat(V, V )op , Nat(U,U)op , Nat(V,U) , Nat(U, V ) , , ),

where all algebra and bimodule structures are given by the opposite composition of natural transfor-mations and both connecting homomorphisms and are given by projections of the restrictionsof the opposite composition of natural transformations.

In the following proposition an equivalent description of the Morita context (165), in terms ofsets of (co)module maps, is given. In order to formulate it, we introduce the k-module

(166) Q : = q ∈ AHomL(C,Σ∗) | ∀x ∈ Σ, c ∈ C c(1)q(c(2))(x) = q(c)(x[0])x[1] .

6.2. MORITA THEORY FOR CORING EXTENSIONS 169

Proposition 6.1. Let Σ be an L-C bicomodule for a right coring extension (D : L) of (C : A).Consider the corresponding Morita context (165). There is a set of k-linear isomorphisms

α1 : LHomL(D, T )∼=−→ Nat(V, V )op,(167)

α2 : CEndD(C)op∼=−→ Nat(U,U)op,(168)

α3 : LHomD(D,Σ)∼=−→ Nat(V,U),(169)

α4 : Q∼=−→ Nat(U, V ),(170)

where T denotes the algebra (and L-ring) EndC(Σ). Moreover, the maps (167)-(170) establish anisomorphism of the Morita context (165) and the Morita context

(171) M(Σ) = (LHomL(D, T ) , CEndD(C)op , LHomD(D,Σ) , Q , , ♦).

The algebra structures, bimodule structures and connecting homomorphisms are given by thefollowing formulae.

(vv′)(d) = v(d(1))v′(d(2))(172)

(uu′)(c) = u′(u(c)

)(173)

(vp)(d) = v(d(1))(p(d(2))

)(174)

(pu)(d) = p(d)[0]εC(u(p(d)[1])

)(175)

(qv)(c) = q(c[0])v(c[1])(176)

(uq)(c) = q(u(c)

)(177)

(q p)(c) = c(1)q(c(2)[0])(p(c(2)[1])

)≡ q(c[0])

(p(c[1])

[0])p(c[1])

[1](178)

(p ♦ q)(d) = p(d)[0]q(p(d)[1]

)(−),(179)

for v, v′ ∈ LHomL(D, T ), u, u′ ∈ CEndD(C), p ∈ LHomD(D,Σ), q ∈ Q, d ∈ D and c ∈ C.

Proof. To an element v ∈ LHomL(D, T ) associate a right D-colinear map,

ΦvM : HomC(Σ,M)⊗L D → HomC(Σ,M)⊗L D, ϕM ⊗L d 7→ ϕM v(d(1))⊗L d(2),

for any right C-comodule M . This defines a k-module map α1 : LHomL(D, T ) → Nat(V, V )op,v 7→ Φv. The bijectivity of α1 is proved by constructing the inverse α−1

1 , mapping Φ ∈ Nat(V, V )to the right L-linear map

D → T, d 7→((T ⊗L εD) ΦΣ

)(1T ⊗L d).

Since Σ is an L-C bicomodule, the map Σ → Σ, x 7→ lx is right C-colinear for any l ∈ L. Henceα−1

1 (Φ) is left L-linear by the naturality of Φ. The identity α−11 α1(v) = v, for v ∈ LHomL(D, T ),

is obvious. The other identity α1 α−11 (Φ) = Φ, for Φ ∈ Nat(V, V ), is checked as follows. Since

the right D-coaction in T ⊗L D is given by T ⊗L ∆D, it follows that, for all t⊗L d ∈ T ⊗L D,

(180) (T ⊗L εD)((t⊗L d)[0]

)⊗L (t⊗L d)[1] = tεD(d(1))⊗L d(2) = t⊗L d.

Using the right D-colinearity of ΦΣ (in the second equality), (180) (in the third equality) and theright C-colinearity of ϕM together with the naturality of Φ (in the last equality), we conclude that(

α1 α−11 (Φ)

)M

(ϕM ⊗L d) = ϕM ((T ⊗L εD)(ΦΣ(1T ⊗L d(1)))

)⊗L d(2)

= ϕM ((T ⊗L εD)

(ΦΣ(1T ⊗L d)[0]

))⊗L ΦΣ(1T ⊗L d)[1]

=((ϕM ⊗L D) ΦΣ

)(1T ⊗L d) = ΦM (ϕM ⊗L d),

for ϕM ⊗L d ∈ HomC(Σ,M)⊗L D.To an element u ∈ CEndD(C) associate a map

ΞuM : M →M, m 7→ m[0](εC u)(m[1]),

for any right C-comodule M . It is checked to be right D-colinear using the relation between the Cand D-comodule structures of M , the right A-linearity of a C-coaction, the left C-colinearity and


the right D-colinearity of u and the D-colinearity of the C-coaction:(m[0] (εC u)(m[1])

)[0]⊗L(m[0](εC u)(m[1])

)[1]

=(m[0](εC u)(m[1])

)[0]εC((m[0](εC u)(m[1])

)[1][0]

)⊗L(m[0](εC u)(m[1])

)[1][1]

= m[0]εC((m[1](εC u)(m[2])

)[0]

)⊗L(m[1](εC u)(m[2])

)[1]

= m[0]εC(u(m[1])[0]

)⊗L u(m[1])[1] = m[0]

[0](εC u)(m[0][1])⊗L m[1].

This implies that we have a k-linear map α2 : CEndD(C)op → Nat(U,U)op, u 7→ Ξu. In order toprove its bijectivity, we construct the inverse α−1

2 , mapping Ξ ∈ Nat(U,U) to ΞC ∈ EndD(C). Weneed to prove that ΞC is left C-colinear. Note first that, for any right A-module N and n ∈ N ,the map C → N ⊗A C, c 7→ n ⊗A c is right C-colinear (where N ⊗A C is a right C-comodule viaN ⊗A ∆C). Hence, by naturality,

(181) ΞN⊗AC = N ⊗A ΞC.

On the other hand, the coproduct in C is right C-colinear (i.e. coassociative), hence naturalityimplies ΞC⊗AC ∆C = ∆C ΞC. Combining these two observations, we conclude that ΞC is left

C-colinear. The identity α−12 α2(u) = u, for u ∈ CEndD(C), follows easily from the C-colinearity

of u. The property α2 α−12 (Ξ) = Ξ, for Ξ ∈ Nat(U,U), follows by the commutativity of the

following diagram, for any right C-comodule M .

MΞM //

%M

M

%M

M

M ⊗A CΞM⊗AC=M⊗AΞC

// M ⊗A C

M⊗AεC

::uuuuuuuuu

Commutativity of this diagram follows by (181), the right C-colinearity (i.e. coassociativity) of aright C-coaction and the naturality of Ξ.

To an element p ∈ LHomD(D,Σ) associate the right D-colinear map,

ΘpM : HomC(Σ,M)⊗L D →M, ϕM ⊗L d 7→ ϕM

(p(d)

),

for any right C-comodule M . It defines a k-map α3 : LHomD(D,Σ) → Nat(V,U), p 7→ Θp.We prove its bijectivity by constructing the inverse α−1

3 , mapping Θ ∈ Nat(V,U) to the rightD-colinear map,

D → Σ, d 7→ ΘΣ(1T ⊗L d).Its left L-linearity follows by the right C-colinearity of the map Σ → Σ, x 7→ lx, for any l ∈ L,the naturality of Θ, and the fact that L is a subalgebra of T . The identity α−1

3 α3(p) = p,

for p ∈ LHomD(D,Σ) is obvious, and α3 α−13 (Θ) = Θ, for Θ ∈ Nat(V,U), follows by the

naturality of Θ, i.e. the identity ϕM(ΘΣ(t⊗L d)

)= ΘM (ϕM t⊗L d), for any right C-comodule

M , ϕM ∈ HomC(Σ,M) and t⊗L d ∈ T ⊗L D.

To an element q ∈ Q associate the right D-colinear map,

ΩqM : M → HomC(Σ,M)⊗L D, m 7→ m[0]q(m[1]

[0])(−)⊗L m[1][1]

≡ m[0][0]q(m[0]

[1])(−)⊗L m[1],

for any right C-comodule M . The two forms of ΩqM are equal by the right D-colinearity of the

C-coaction. It is a well-defined map by the A-L bilinearity of q and Lemma 5.8 (ii). The association

q 7→ Ωq defines a k-map α4 : Q→ Nat(U, V ). We prove its bijectivity by constructing the inverseα−1

4 , mapping Ω ∈ Nat(U, V ) to the right L-linear map

C → Σ∗, c 7→ εC((HomC(Σ,C)⊗L εD)(ΩC(c))(−)

).

By the right C-colinearity of the map C → N ⊗A C, c 7→ n ⊗A c, for any right A-moduleN and n ∈ N , and the naturality of Ω, the map ΩN⊗AC can be written as a composite of

N ⊗AΩC : N ⊗A C → N ⊗AHomC(Σ,C)⊗LD and the obvious map N ⊗AHomC(Σ,C)⊗LD →


HomC(Σ, N ⊗A C) ⊗L D. Applying this fact to the case N = A, we find the left A-linearity ofΩC, hence of α−1

4 (Ω). Consider the following commutative diagram.

CΩC //

∆C

HomC(Σ,C)⊗LDεC−⊗LD //

∆C−⊗LD

HomA(Σ,A)⊗LD

(−⊗AC)%Σ⊗LD

ooHomA(Σ,A)⊗LεD //

(−⊗AC)%Σ⊗LD

HomA(Σ,A)

(−⊗AC)%Σ

C⊗ACΩC⊗AC //

C⊗AΩC ''PPPPPPPPPPPP HomC(Σ,C⊗AC)⊗LD

(C⊗AεC)−⊗LD //HomA(Σ,C)⊗LD

(−⊗AC)%Σ⊗LD

ooHomA(Σ,C)⊗LεD // HomA(Σ,C)

C⊗AHomC(Σ,C)⊗LDC⊗AεC−⊗LD//

OO

C⊗AHomA(Σ,A)⊗LD

C⊗A(−⊗AC)%Σ⊗LD

ooC⊗AHomA(Σ,A)⊗LεD //

OO

C⊗AHomA(Σ,A)

OO

The upper left square is commutative by the right C-colinearity (i.e. coassociativity) of the coprod-uct in C and the naturality of Ω. The lower left triangle is commutative by the previous observationthat ΩN⊗AC factors through N ⊗A ΩC, for any right A-module N . The squares in the middle

column are commutative by the isomorphism of k-modules HomC(Σ, N ⊗A C) ' HomA(Σ, N),for any right A-module N , see (60). The upper line in the diagram gives an equivalent expressionfor α−1

4 (Ω). Comparing the incoming arrows from above and below in HomA(Σ,C) on the outer

right, we conclude that α−14 (Ω) is an element of Q.

The identity α−14 α4(q) = q, for q ∈ Q, follows by the right A-linearity of εC and the left

A-linearity of q. In order to prove the converse property, α4 α−14 (Ω) = Ω, for Ω ∈ Nat(U, V ),

consider the following diagram in MD, for any M ∈MC.

MΩM //

%M

''OOOOOOOOOOOOOO

τM

HomC(Σ,M)⊗L D

%M−⊗LD

M ⊗A CΩM⊗AC //

M⊗AΩC ++WWWWWWWWWWWWWWWWWWWWWWW

M⊗AτC

HomC(Σ,M ⊗A C)⊗L D

M ⊗A HomC(Σ,C)⊗L D

OO

M ⊗L DρM⊗LD// M ⊗A C⊗L D

M⊗AΩC⊗LD // M ⊗A HomC(Σ,C)⊗L D⊗L D

M⊗AHomC(Σ,C)⊗LεD⊗LD

OO

The commutativity of the upper quadrangle follows by the naturality of ΩM . The left lowerquadrangle commutes by the right D-colinearity of ρM and the right lower quadrangle does bythe right D-colinearity of ΩC and the fact that εD is the counit of D. The property that ΩM⊗AC

factors trough M⊗ΩC implies that the triangle commutes as well. Evaluating the upper and lowerincoming arrow in HomC(Σ,M ⊗A C)⊗L D, on an element m ∈M , one finds

(ρM −) ΩM (m) = m[0][0] ⊗A (HomC(Σ,C)⊗A εD)(ΩC(m[0]

[1]))⊗L m[1].

Application of (M ⊗A εC) − ⊗L D to both sides of this equation yields the required identityΩM = α4 α−1

4 (Ω)M in HomC(Σ,M)⊗L D.The proof is completed by showing that the maps α1, α2, α3 and α4 define a morphism of

Morita contexts. Indeed, it is straightforward to check that, for v, v′ ∈ LHomL(D, T ), u, u′ ∈CEndD(C), p ∈ LHomD(D,Σ) and q ∈ Q,

α1(v) α1(v′) = α1(v′v) α2(u) α2(u′) = α2(u′u)α3(p) α1(v) = α3(v p) α2(u) α3(p) = α3(p u)α1(v) α4(q) = α4(q v) α4(q) α2(u) = α4(u q)α3(p) α4(q) = α2(q p) α4(q) α3(p) = α1(p ♦ q).

This finishes the proof.


Remark 6.2. The commutative base ring k can be considered as a trivial (k-)coring, whichis a right extension of any A-coring C. A right C-comodule Σ can be viewed as a k-C bicomod-

ule for the right coring extension k of C, hence there is an associated Morita context M(Σ) =(Homk(k, T ), CEnd(C)op,Homk(k,Σ), Q ≡ P, , ♦

), as in (171). Obviously, Homk(k, T ) ∼= T

and Homk(k,Σ) ∼= Σ. Furthermore we know CEnd(C)op ∼= ∗C (see Section 3.3). By Lemma 5.8

(i), Q ∼= P ≡ Q. Composing these isomorphisms with the Morita maps in (171), one obtains the

structure maps of the Morita context (143) of Section 5.3.2. Hence the Morita context M(Σ),associated to Σ as a k-C bicomodule, coincides with M(Σ), associated to a right C-comodule Σin (143).

Lemma 6.3. Let the L-coring D be a right extension of the A-coring C and let Σ be an

object in LMC. Consider the Morita context M(Σ) in (171). If there exist finite sets of elements

j` ⊂ LHomD(D,Σ) and ` ⊂ Q such that∑

` ` j` = C (i.e. connecting map in (178) issurjective), then

(i) the identity∑

` `(c[0])(j`(c[1])

)= εC(c) holds, for all c ∈ C;

(ii) the identity∑

`m[0][0]`(m[0]

[1])(j`(m[1])

)= m holds, for any right C-comodule M and

m ∈M .

Proof. (i) This follows by applying εC to (178).

(ii) By the D-colinearity of a right C-coaction in a right C-comodule M , for any m ∈M ,∑`

m[0][0]`(m[0]

[1])(j`(m[1])

)=∑`

m[0]`(m[1][0])(j`(m[1]

[1]))

= m[0]εC(m[1]) = m,

where we used part (i) in the second equation.

Proposition 6.4. Let the L-coring D be a right extension of the A-coring C. Take Σ ∈ LMC

and consider the associated Morita context M(Σ) in (171). If both the map in (178) and thecounit εC are surjective then Σ is a generator in MA.

Proof. Choose elements j` ⊂ LHomD(D,Σ) and ` ⊂ Q such that∑

` ` j` = C, andan element c ∈ C such that εC(c) = 1A. Fix finite sets xi ⊂ Σ and ξi ⊂ Σ∗ such that∑

i

ξi ⊗L xi =∑`

`(c[0])⊗L j`(c[1]).

Then∑

i ξi(xi) = εC(c) = 1A, by Lemma 6.3 (i), which proves the claim.

A lemma by Beck (cf. a dual version of [13, 3.3 Proposition 3]) states that if F : A → B is acomonadic functor and λ is a split epimorphism in the category A, such that F (λ) has a kernel inB, then also λ has a kernel in A and F preserves this kernel. Since for an A-coring C the forgetfulfunctor MC →MA is obviously comonadic, the next lemma follows by this general result. Still,for the convenience of the reader we include a complete proof, which is a simplified version ofBeck’s arguments (as in our case the target category MA is abelian).

Lemma 6.5. The following statements about right comodules M and N for an A-coring Care equivalent.

(i) There exist finite collections of morphisms κ` ⊂ HomC(M,N) and λ` ⊂ HomC(N,M),indexed by ` = 1 . . . s, such that

∑` λ` κ` = M ;

(ii) M is a direct summand of the direct sum N s as a right C-comodule.

Proof. In terms of κ` and λ` as in (i), construct a map κ : M → N s defined byκ(m) = (κ`(m))`. Clearly, κ has a left inverse, λ : N s → M , λ((n`)`) =

∑` λ`(n`). Recall that

the category of comodules of any A-coring C has direct sums that do coincide with the direct sumsin MA (see e.g. [115, 3.3]), so λ and κ are morphisms in MC. In particular, λ and κ are rightA-linear, so there is a split exact sequence in the abelian category of right A-modules,

0 // M ′ ν // N s λ // M // 0 ,


where (M ′, ν) is the kernel of λ in MA. The proof is completed by showing that M ′ is a rightC-comodule and ν is a C-colinear section. Denoting a right A-linear retraction of ν by $, introducea right A-module map

%M′: = ($ ⊗A C) %Ns ν : M ′ →M ′ ⊗A C.

Since the A-module maps satisfy ν $+κ λ = N s, the colinearity of κ and λ implies that ν $is C-colinear. Hence

(182) (ν ⊗A C) %M ′= (ν $ ⊗A C) %Ns ν = %N

s ν.

Composing (182) by $ on both sides and using the colinearity of ν $, one obtains also %M′ $ =

($ ⊗A C) %Ns. Furthermore, using (182) (in the first, second and fourth equalities) and the

coassociativity of %Ns

(in the third equality), one checks that

(ν ⊗A C⊗A C) (%M′ ⊗A C) %M ′

= (%Ns ν ⊗A C) %M ′

= (%Ns ⊗A C) %Ns ν

= (N s ⊗A ∆C) %Ns ν = (N s ⊗A ∆C) (ν ⊗A C) %M ′

= (ν ⊗A C⊗A C) (M ′ ⊗A ∆C) %M ′.

Since ν ⊗A C⊗A C is a (split) monomorphism, this implies the coassociativity of %M′. Finally, by

the counit property of %Ns,

(M ′ ⊗A εC) %M ′= (M ′ ⊗A εC) ($ ⊗A C) %Ns ν = $ ν = M ′,

which finishes the proof of the implication (i) ⇒ (ii). The converse implication is obvious.

Since L-C bicomodules, for an algebra L and an A-coring C, can be identified with rightcomodules for the Lop ⊗k A-coring Lop ⊗k C, Lemma 6.5 applies also to L-C bicomodules.

Theorem 6.6. Let the L-coring D be a right extension of the A-coring C. Take Σ ∈ LMC

and consider the Morita context M(Σ) in (171). In particular, put T : = EndC(Σ).(i) The map in (178) is surjective if and only if Σ is a comonadic-Galois C-comodule and Σ is

a direct summand of a direct sum (T ⊗L D)s as T -D-bicomodule, for an appropriate finiteinteger s.

(ii) The Morita context M(Σ) is strict if and only if Σ is a comonadic-Galois C-comodule, Σis a direct summand of (T ⊗L D)s and T ⊗L D is a direct summand of Σz, both as T -D-bicomodules, where s and z are integers.

Proof. (i) By surjectivity of , there exist elements j` ⊂ LHomD(D,Σ) and ` ⊂ Q (cf.

(166)) such that∑

` ` j` = C. In terms of the isomorphisms α3 and α4 in Proposition 6.1, we

denote α3(j`) = J` and α4(`) = J`, for all values of `.First we check that the surjectivity of implies the Galois property of the right C-comodule

Σ. To this end, we construct the inverse of the canonical natural transformation (132). For anyright A-module N , put

ΥN : N ⊗A C → HomA(Σ, N)⊗T Σ, n⊗A c 7→∑`

n`(c[0])(−)⊗T j`(c[1]).

Since T is an L-ring, ΥN is a well defined map. It is natural in N , being a sum of composites ofnatural morphisms,

N ⊗A C( eJ`)N⊗AC // HomC (Σ, N ⊗A C) ⊗L D

∼= HomA(Σ, N)⊗T T ⊗L D

HomA(Σ,N)⊗T (J`)Σ // HomA(Σ, N)⊗T Σ .

We claim that Υ yields the inverse of the canonical natural transformation (132). Indeed, we findthat, for n⊗A c ∈ N ⊗A C,

canN (ΥN (n⊗A c)) =∑`

n⊗A `(c[0])(j`(c[1])

[0])j`(c[1])

[1]

= n⊗A∑`

(` j`)(c) = n⊗A c.


On the other hand, for φN ⊗T x ∈ HomA(Σ, N)⊗T Σ,

ΥN

(canN (φN ⊗T x)

)=

∑`

φN (x[0])`(x[1][0])(−)⊗T j`(x[1]

[1])

=∑`

φN(x[0]

[0]`(x[0][1])(−)

)⊗T j`(x[1]),

where in the second equality we used the right D-colinearity of the C-coaction in Σ and the right A-linearity of φN . Using Lemma 5.8 (ii) for the right C-comodule Σ, we conclude that x[0]`(x[1])(−)is an element of T , for all x ∈ Σ and any value of the index `. Hence∑

`

φN(x[0]

[0]`(x[0][1])(−)

)⊗T j`(x[1]) = φN ⊗T

∑`

x[0][0]`(x[0]

[1])(j`(x[1])

)= φN ⊗T x,

where the last equality follows by Lemma 6.3 (ii), applied to the right C-comodule Σ. This provesthat if the Morita map is surjective then Σ is a comonadic-Galois C-comodule.

Next we prove that Σ is a direct summand of (T ⊗L D)s as T -D-bicomodule, where s is thecardinality of the index set ` above. For any value of `, put

κ` : = (J`)Σ : Σ → T ⊗L D, x 7→ x[0][0]`(x[0]

[1])(−)⊗L x[1], and(183)

κ` : = (J`)Σ : T ⊗L D → Σ, t⊗L d 7→ t(j`(d)

).

Since the left L-action in Σ is right C-colinear by assumption, and any C-colinear map is D-colinear,both the right C, and D-coactions in Σ are left L-linear. This way κ`, being a composition of leftL-linear right D-colinear maps, is left L-linear and right D-colinear. The map κ` is obviously leftL-linear and it is right D-colinear by the colinearity of j` and t ∈ T . It follows by Lemma 6.3 (ii)that

∑` κ` κ`(x) = x. By Lemma 6.5, this implies that Σ is a direct summand of (T ⊗L D)s as

a T -D-bicomodule, where s is the cardinality of the index set `.In order to prove the converse statement, we make use of Lemma 6.5 again. Since Σ is a

direct summand of (T ⊗L D)s as T -D-bicomodule, there exist maps κ` ∈ THomD(Σ, T ⊗L D)and κ` ∈ THomD(T ⊗L D,Σ), for ` = 1 . . . s, such that

∑` κ` κ` = Σ. We can define maps j`

and ` as follows. For any value of `, put

(184) j` : D → Σ, d 7→ κ`(1T ⊗L d).

By the colinearity of κ`, j` is right D-colinear. Since κ` is left T -linear and L is a subalgebra of T ,j` is also left L-linear. Furthermore, Σ is a comonadic-Galois C-comodule by assumption, hencewe can set

(185) ` : = [Σ∗ ⊗T (T ⊗L εD) κ`] can−1A : C → Σ∗,

for ` = 1 . . . s. The map canA is left A-linear and right C-colinear. Hence it is also right D-colinear,so, in particular, right L-linear. In this way ` is a composition of left A-linear, right L-linear maps,hence it is left A-linear and right L-linear itself. Let us prove that ` ∈ Q. Consider the followingcommutative diagram.

Ccan−1

A //

∆C

HomA(Σ, A)⊗T ΣHomA(Σ,A)⊗T (T⊗LεD)κ` //

(−⊗AC)%Σ⊗T Σ

HomA(Σ, A)

(−⊗AC)%Σ

C⊗A Ccan−1

C // HomA(Σ,C)⊗T ΣHomA(Σ,C)⊗T (T⊗LεD)κ` // HomA(Σ,C)

Ccan−1

A

//

fc⊗AC

OO

HomA(Σ, A)⊗T ΣHomA(Σ,A)⊗T (T⊗LεD)κ`

//

(fc−)⊗T Σ

OO

HomA(Σ, A)

fc−

OO

Here fc denotes a right A-linear morphism from A to C, defined for an element c ∈ C as fc(a) = ca.The lower square on the left hand side commutes because of the naturality of can−1. By the explicitform (132) of can, for ξ ⊗T x ∈ Σ∗ ⊗T Σ,

canC [(−⊗A C) ρΣ ⊗T Σ](ξ ⊗T x) = ξ(x[0])x[1] ⊗A x[2] = ∆C canA(ξ ⊗T x).


Hence also the upper square on the left hand side commutes. The commutativity of the squareson the right hand side is obvious. Both the upper and lower horizontal lines express the map `.In terms of the map fc introduced above, ∆C(c) = (fc(1) ⊗A C)(c(2)). Hence the commutativityof the diagram implies, for all c ∈ C and x ∈ Σ,

`(c)(x[0])x[1] =([HomA(Σ,C)⊗T (T ⊗L εD) κ`] can−1

C ∆C(c))(x)

=([HomA(Σ,C)⊗T (T ⊗L εD) κ`] can−1

C (fc(1) ⊗A C)(c(2)))(x)

= c(1)`(c(2))(x).

This means that, ` ∈ Q, for all values of the index `. The surjectivity of (the CEndD(C)-CEndD(C)bilinear map) is proven by showing

∑` ` j` = C. Use the right D-colinearity of can−1

A (in thesecond equality), the right D-colinearity of κ` (in the third one) and the left T -linearity of κ` (inthe penultimate one) to compute the composite of the right D-coaction τC on C with

∑` `⊗L j`.

It yields ∑`

(` ⊗L j`) τC =∑`

[Σ∗ ⊗T (T ⊗L εD) κ`]⊗L j` (can−1A ⊗L D) τC

=∑`

Σ∗ ⊗T [(T ⊗L εD) κ` ⊗L j`] τΣ can−1A

=∑`

[Σ∗ ⊗T (T ⊗L εD ⊗L j`) (T ⊗L ∆D) κ`] can−1A

=∑`

[Σ∗ ⊗T (T ⊗L j`) κ`] can−1A .

= [Σ∗ ⊗T∑`

κ` κ`] can−1A = can−1

A .(186)

Note that the evaluation map Σ∗⊗T Σ → A, ξ⊗T x 7→ ξ(x) is equal to εC canA. Hence equation(186) implies∑

`

` j` =∑`

(εC canA ⊗A C) (Σ∗ ⊗T ρΣ) (` ⊗L j`) τC

= (εC canA ⊗A C) (Σ∗ ⊗T ρΣ) can−1A = (εC ⊗A C) ∆C = C,

where the third equality follows by the right C-colinearity of canA.(ii) Suppose that the Morita context (171) is strict. In view of part (i), we have to prove only

that T ⊗L D is a direct summand of Σz, for some integer z. By the surjectivity of ♦, there exist

elements hi ⊂ LHomD(D,Σ) and hi ⊂ Q such that∑

i(hi ♦ hi)(d) = εD(d)1T , for d ∈ D.Similarly to (183), for any value of i, we define left L-linear and right D-colinear morphisms,

λi : Σ → T ⊗L D, x 7→ x[0][0]hi(x[0]

[1])(−)⊗L x[1] and

λi : T ⊗L D → Σ, t⊗L d 7→ t(hi(d)

).

They satisfy, for any t⊗L d ∈ T ⊗L D,∑i

λi(λi(t⊗L d)

)=

∑i

t(hi(d))[0][0]hi

(t(hi(d))[0]

[1])(−)⊗L t(hi(d))[1]

=∑i

t(hi(d(1))

[0]hi(hi(d(1))[1])(−)

)⊗L d(2)

=∑i

t((hi ♦ hi)(d(1))

)⊗L d(2) = tεD(d(1))⊗L d(2) = t⊗L d,

where the second equality follows by C- and D-colinearity of t ∈ T , D-colinearity of hi, and rightA-linearity of t ∈ T . By Lemma 6.5, we conclude that T ⊗L D is a direct summand of Σz, wherez is the cardinality of the index set i.

Finally, we show that if Σ is a comonadic-Galois C-comodule and T ⊗LD is a direct summand

of Σz, then ♦ is surjective. Let λi ⊂ THomD(T ⊗L D,Σ) and λi ⊂ THomD(Σ, T ⊗L D) be


sets of morphisms, such that∑

i λi λi = T ⊗LD. Repeating the arguments in part (i) (cf. (184)

and (185)), we define maps hi ∈ LHomD(D,Σ) and hi ∈ Q as

hi : = λi(1T ⊗L −) and hi : = [Σ∗ ⊗T (T ⊗L εD) λi] can−1A .

For any x ∈ Σ, the association a 7→ xa defines a right A-module map A → Σ. By naturality ofthe canonical maps (132), for x ∈ Σ, c ∈ C, and any value of the index i,

xhi(c)(−) = x[HomA(Σ, A)⊗T (T ⊗L εD) λi] can−1A (c)(−)(187)

= [EndA(Σ)⊗T (T ⊗L εD) λi] can−1Σ (x⊗A c).

By (132), canΣ(Σ⊗T x) = x[0] ⊗A x[1], for x ∈ Σ. Hence (187) implies the following equality ofright A-linear endomorphisms of Σ.

x[0]hi(x[1])(−) = (T ⊗L εD) λi(x),for all x ∈ Σ and any value of the index i. Then it follows that, for d ∈ D,∑

i

(hi ♦ hi)(d) =∑i

hi(d)[0]hi(hi(d)[1]

)(−) =

∑i

(T ⊗L εD)(λi(hi(d))

)= (T ⊗L εD)

(∑i

(λi λi)(1T ⊗L d))

= εD(d)1T .

This proves that∑

i hi ♦ hi is the unit element of the algebra LHomL(D, T ), and thus the surjec-tivity of the (LHomL(D, T )-LHomL(D, T ) bilinear) map ♦.

Remark 6.7. It follows by the proof of Theorem 6.6 that the finite number s in the Theoremcan be chosen equal to the cardinality of the sets j` ⊂ LHomD(D,Σ) and ` ⊂ Q, suchthat

∑` ` j` = C. Similarly, the number z can be chosen equal to the cardinality of the sets

hi ⊂ LHomD(D,Σ) and hi ⊂ Q, such that∑

i hi ♦ hi = εD(−)1T .

Proposition 6.8. Let the L-coring D be a right extension of the A-coring C and Σ ∈LMC. Consider the associated Morita context M(Σ) in (171). If the connecting map in(178) is surjective, then Σ is a D-equivariantly L-relative projective left module of the algebraT = EndC(Σ), i.e. the left action

(188) T ⊗L Σ → Σ, t⊗L x 7→ t(x)

is a coretraction in TMD.

Proof. A retraction of the map (188) can be constructed in terms of the elements ` ⊂ Qand j` ⊂ LHomD(D,Σ), satisfying

∑` ` j` = C, as

σ : Σ → T ⊗L Σ, x 7→∑`

x[0]`(x[1][0])(−)⊗L j`(x[1]

[1]) ≡∑`

x[0][0]`(x[0]

[1])(−)⊗L j`(x[1]).

It is a well defined map by Lemma 5.8 (ii). Being a composition of right D-colinear maps, it isD-colinear. Its T -linearity follows by the fact that any t ∈ T is C-colinear and hence, in particularD-colinear and A-linear. That σ is a retraction of the map (188) is a direct consequence ofLemma 6.3 (ii).

6.3. Weak and strong structure theorems

As an L-C bicomodule Σ, for a right coring extension (D : L) of (C : A), is in particular aright C-comodule, it determines an adjunction of functors

(189) −⊗T Σ : MT →MC,

from the category of right modules for the algebra T = EndC(Σ) to the category of right comodulesfor the coring C, as in the diagonal of diagram (129), and

(190) HomC(Σ,−) : MC →MT .

In the present section we study the ‘descent theory’ of coring extensions, i.e. investigate in whatsituations the functor (190) is fully faithful or an equivalence with inverse (189).

6.3. WEAK AND STRONG STRUCTURE THEOREMS 177

Theorem 6.9 (Weak Structure Theorem). Let the L-coring D be a right extension of the

A-coring C. Take Σ ∈ LMC and consider the associated Morita context M(Σ) in (171). If themap in (178) is surjective then the functor (190) is fully faithful.

Proof. The property, that the functor (190) is fully faithful, is equivalent to the bijectivityof the counit of the adjunction,

(191) εM : HomC(Σ,M)⊗T Σ →M, ϕM ⊗T x 7→ ϕM (x),

for T : = EndC(Σ) and any right C-comodule M . Note that the restriction of the map (M ⊗AεC) canM to HomC(Σ,M)⊗T Σ is equal to εM . Furthermore, by the right C-colinearity of εM ,we have %M εM = (εM ⊗A C) (HomC(Σ,M)⊗T %Σ), which equals the restriction of canM . ByTheorem 6.6 (i), the surjectivity of implies that Σ is a comonadic-Galois C-comodule. Takingthe explicit form of can−1 in the proof of Theorem 6.6 into account, we have

can−1M %M (m) =

∑`

m[0][0]`(m[0]

[1])(−)⊗T j`(m[1]), for m ∈M,

which is an element of HomC(Σ,M)⊗T Σ, by Lemma 5.8 (ii). In light of these observations, theinverse of εM can be constructed as

can−1M %M : M → HomC(Σ,M)⊗T Σ.

Corollary 6.10. Let the L-coring D be a right extension of the A-coring C and Σ an L-C

bicomodule. Consider the Morita contexts M(Σ), associated to Σ in (171), and M(Σ), associatedto Σ as a C-comodule in (143). Suppose that the map , given in (178), is surjective. Then theconnecting map H, given in (144), is surjective if and only if C is a finitely generated and projectiveleft A-module.

Proof. If the connecting map H in (144) is surjective then C is a finitely generated projectiveleft A-module by Lemma 5.17. Conversely, if C is a finitely generated projective left A-module thenthe connecting map (144) is equal to the counit (191) for the right C-comodule ∗C (cf. Remark5.9). Then it is an isomorphism by Theorem 6.9.

Recall from Theorem 5.15 that a sufficient condition for the functor (189) to be fully faithful isthe surjectivity of the map (145). Motivated by this result, in what follows we look for conditionsunder which the map (145) is surjective.

Proposition 6.11. Let the L-coring D be a right extension of the A-coring C and take

Σ ∈ LMC. Consider the Morita contexts M(Σ), associated to the L-C bicomodule Σ in (171),and M(Σ), associated to Σ as a C-comodule in (143). In particular, denote T = EndC(Σ). If theconnecting map ♦, given in (179), is surjective and there exist elements vj ⊂ LHomL(D, T ) anddj ⊂ D, such that

∑j vj(dj) = 1T , then also the connecting map O, given in (145), is surjective

(hence Σ is a finitely generated and projective right A-module by Proposition 5.14).

Proof. By Lemma 5.8 (i), Q can be viewed as a k-submodule of Q ∼= P in (141). Hence

(identifying q ∈ Q with the corresponding element of Q), it follows by the explicit forms of

the maps ♦ and O that (p ♦ q)(d) = p(d) O q, for p ∈ LHomD(D,Σ), q ∈ Q and d ∈ D. Let

hi ⊂ LHomD(D,Σ) and hi ⊂ Q be sets of morphisms such that∑

i hi ♦ hi = εD(−)1T . Then

1T =∑j

vj(dj) =∑j

vj(dj(1))εD(dj(2)) =∑j

vj(dj(1))(∑

i

(hi ♦ hi)(dj(2)))

=∑j

vj(dj(1))(∑

i

(hi(dj(2)) O hi)

=∑i

(∑j

vj(dj(1))(hi(dj(2))))

O hi

=∑i

(∑j

(vjhi)(dj))

O hi,

where the penultimate equality follows by the left T -linearity of O and the last one follows by (174).Since O is T -T bilinear, this proves its surjectivity.


By standard Morita theory, Proposition 6.11 implies the following.

Corollary 6.12. Under the assumptions (and using the notation) of Proposition 6.11, Σis a generator of left T -modules. That is, T is a direct summand of a direct sum Σz, as a left

T -module, where z is the cardinality of the sets hi ⊂ LHomD(D,Σ) and hi ⊂ Q, such that∑i hi ♦ hi = εD(−)1T .

Remark 6.13. The assumption in Proposition 6.11 about the existence of elements vj ⊂LHomL(D, T ) and dj ⊂ D, such that

∑j vj(dj) = 1T , holds in various situations, studied in

connection with cleft entwining structures in [47, Theorem 4.5] and [4, Theorems 4.9 and 4.10].

(i) If the counit εD of D is surjective, then there exists d ∈ D such that εD(d) = 1L. Puttingv : D → T , d′ 7→ εD(d′)1T , we have v(d) = 1T .

(ii) If D contains a grouplike element then it is mapped by εD to 1L, by definition. Hence εD issurjective, being L-L bilinear, so the considerations in part (i) apply.

(iii) If D is faithfully flat as a left, or as a right L-module then εD is surjective, since εD ⊗L Dand D⊗L εD are epimorphisms, split by the coproduct ∆D. Hence this is an example of thesituation in part (i) as well.

Theorem 6.14 (Strong Structure Theorem). Let the L-coring D be a right extension ofthe A-coring C and Σ an L-C bicomodule. Denote T : = EndC(Σ). If the associated Moritacontext (171) is strict and there exist elements vj ⊂ LHomL(D, T ) and dj ⊂ D, such that∑

j vj(dj) = 1T , then the functors (189) and (190) are inverse equivalences.

Proof. This is an immediate consequence of Theorem 6.9, Theorem 5.15 and Proposi-tion 6.11.

Note that under the hypothesis of Theorem 6.14, Σ is a finitely generated and projective rightA-module (cf. Proposition 5.14), hence a (finite) Galois comodule (cf. Theorem 6.6). This followsas well from Corollary 4.35. On the other hand, under the assumptions in Theorem 6.14, Σ isnot necessarily flat as a left T -module (equivalently, C is not necessarily flat as a left A-module).This way Theorem 6.14 covers cases which are not treated by the Galois Comodule StructureTheorem Theorem 4.27, Corollary 4.35. This will be clear by the observation in Section 6.5, thatthe Fundamental Theorem of Hopf modules (for arbitrary Hopf algebras or Hopf algebroids) is aparticular instance of Theorem 6.14.

6.4. Cleft bicomodules

The results in Section 6.2 and Section 6.3 allow an application to the main case of interest,when there exist ‘invertible’ elements in the Morita context, associated to a bicomodule of a coringextension, in the following sense. Let D be an L-coring which is a right extension of an A-coring

C. For an L-C bicomodule Σ consider the associated Morita context M(Σ) =(LHomL(D, T ),

CEndD(C)op, LHomD(D,Σ), Q, , ♦)

in (171), where T = EndC(Σ) and Q is the k-module (166).

Definition 6.15. An object Σ of LMC is called a weak cleft bicomodule for the right coringextension (D : L) of (C : A) provided there exist elements j ∈ LHomD(D,Σ) and ∈ Q suchthat j = C.

An object Σ of LMC is called a cleft bicomodule for the right coring extension (D : L) of

(C : A) provided there exist elements j ∈ LHomD(D,Σ) and ∈ Q such that j = C and, inaddition, j ♦ = εD(−)1T .

Note that if Σ ∈ LMC is a (weak) cleft bicomodule for a right coring extension (D : L) of (C :A), with morphisms j ∈ LHomD(D,Σ) and ∈ Q as in Definition 6.15, then, by Proposition 6.1,the natural transformation α4() in Proposition 6.1 is a (left) inverse of α3(j) there.

In standard Hopf Galois theory, cleft extensions are Galois extensions with additional ‘normalbasis property’. In order to derive a similar result for coring extensions, we impose the followingdefinition.

6.5. EXAMPLES 179

Definition 6.16. An L-C bicomodule Σ for a right coring extension (D : L) of (C : A) issaid to obey the weak normal basis property if it is isomorphic to a direct summand of T ⊗L D,as a T -D bicomodule, for T = EndC(Σ).

An L-C bicomodule Σ for a right coring extension (D : L) of (C : A) is said to obey the normalbasis property if it is isomorphic to T ⊗L D, as a T -D bicomodule.

Note that the normal basis property of an L-C bicomodule Σ, for a right coring extension(D : L) of (C : A), implies the isomorphism of the cotensor product Σ2DW to T ⊗L W , as aleft T : = EndC(Σ)-module, for any left D-comodule W . This way, properties (like projectivity orfreeness) of the left L-module W are inherited by the associated left T -module Σ2DW .

It is immediately clear from the definition that the Morita context (171), associated to a cleftbicomodule of a coring extension, is strict. Hence the proof of Theorem 6.6 and Remark 6.7 leadto the following relation between cleft bicomodules and Galois comodules which satisfy the normalbasis property.

Corollary 6.17. (i) Σ ∈ LMC is a weak cleft bicomodule for the right coring extension(D : L) of (C : A) if and only if Σ is a comonadic-Galois C-comodule and satisfies the weaknormal basis property;

(ii) Σ ∈ LMC is a cleft bicomodule for the right coring extension (D : L) of (C : A) if and onlyif Σ is a comonadic-Galois C-comodule and satisfies the normal basis property.

Corollary 6.12 has the following consequence.

Corollary 6.18. Let the L-coring D be a right extension of the A-coring C and let Σ ∈ LMC

be a cleft bicomodule. Put T : = EndC(Σ). If there exist elements vj ⊂ LHomL(D, T ) anddj ⊂ D such that

∑j vj(dj) = 1T then Σ contains the left regular T -module as a direct

summand.

Theorems 6.9 and 6.14 imply the following structure theorems.

Corollary 6.19. For the functors (189) and (190), associated to a cleft L-C bicomodule Σof a right coring extension (D : L) of (C : A), the following assertions hold.

(i) (Weak Structure Theorem) The functor (190) is fully faithful.(ii) (Strong Structure Theorem) The functors (189) and (190) are inverse equivalences provided

that there exist elements vj ⊂ LHomL(D, T ) and dj ⊂ D, such that∑

j vj(dj) = 1T ,

where the notation T = EndC(Σ) is used. It follows that in this situation Σ is finitelygenerated and projective as a right A-module.

6.5. Examples

6.5.1. Cleft entwining structures. Let (A,D, ψ) be a (right,right) entwining structure overa k-algebra A and a k-coalgebra D entwined by the map ψ : D ⊗ A → A ⊗ D, where theunadorned tensor product denotes the tensor product over k. As we know from Section 3.2.1,C = A ⊗ D is an A-coring such that D is a right extension of C and MC coincides with thecategory of entwined modules MD

A(ψ). Suppose that A is a right entwined module with theregular right A-module structure and right D-coaction ρ : A→ A⊗D. Then it follows from (71)

that ρ(a) = a[0] ⊗ a[1] = 1A[0]aψ ⊗ 1ψA[1] for all a ∈ A. Moreover, the A-coring C has a grouplike

element g = 1A[0] ⊗ 1A[1]. If we define

B = b ∈ A | ρ(a) = 1A[0]aψ ⊗ 1ψA[1] = a1A[0] ⊗ 1A[1] ∼= EndC(A)

then we obtain the pair of adjoint functors

MDA(ψ)

(−)coC //MB−⊗BA

oo

where MC = m ∈ M | m[0] ⊗m[1] = m1A[0] ⊗ 1A[1] ∼= HomC(A,M) for all M ∈ MDA . We

define Q as the k-submodule of Hom(D, A) conisting of all those maps satisfying

(192) q(d(2))ψ ⊗ dψ(1) = q(d)1A[0] ⊗ 1A[1],


for all d ∈ D. A particular example of this situation is obtained if there exists a grouplike elementx ∈ D. Then we A has a right D-coaction of the form a 7→ ψ(x ⊗ a). Note that in this case

g = 1A ⊗ x and q ∈ Q if and only if q(d(2))ψ ⊗ dψ(1) = q(d) ⊗ x for all d ∈ D and the C- and

D-coinvariants of all M ∈MDA(ψ) coincide.

Proposition 6.20. Let (A,D, ψ) be an entwining structure and suppose that A ∈MDA(ψ).

Assume that λ : D → A is convolution invertible, with convolution inverse λ. Then the followingassertions are equivalent:1) λ ∈ Q;2) for all d ∈ D, we have

(193) λ(d(1))λ(d(3))ψ ⊗ dψ(2) = εD(d)1A[0] ⊗ 1A[1];

3) for all d ∈ D, we have

(194) λ(d(1))⊗ d(2) = 1A[0]λ(d)ψ ⊗ 1ψA[1].

Note that condition 3) means that λ is right D-colinear.

Proof. 1) ⇒ 2).

λ(d(1))λ(d(3))ψ ⊗ dψ(2) = λ(d(1))λ(d(2))1A[0] ⊗ 1A[1] = εD(d)1A[0] ⊗ 1A[1]

2) ⇒ 3).

1A[0]λ(d)ψ ⊗ 1ψA[1] = ε(d(1))1A[0]λ(d(2))ψ ⊗ 1ψA[1]

(193)= λ(d(1))λ(d(3))ψ′λ(d(4))ψ ⊗ dψ

′ψ(2)

(67)= λ(d(1))

(λ(d(3))λ(d(4))

)ψ⊗ dψ(2)

= λ(d(1))(εD(d(3))ψ)⊗ dψ(2)(70)= λ(d(1))⊗ d(2)

3) ⇒ 1).

λ(d(2))ψ ⊗ dψ(1) = λ(d(1))λ(d(2))λ(d(4))ψ ⊗ dψ(3)(194)= λ(d(1))1A[0]λ(d(2))ψ′ λ(d(3))ψ ⊗ 1ψ

′ψA[1]

(67)= λ(d(1))1A[0]

(λ(d(2))λ(d(3))

)ψ⊗ 1ψA[1] = λ(d)1A[0] ⊗ 1A[1]

If the conditions of Proposition 6.20 are satisfied, then we call (A,D, ψ) a cleft entwiningstructure.

Proposition 6.21. An entwining structure (A,D, ψ) is cleft if and only if A (with the rightregular A-module structure) is a cleft bicomodule for the coring extension D of C : = A⊗k D.

Proof. Let us assume first that (A,D, ψ) is a cleft entwining structure. In this case A is anentwined module, i.e. a right C-comodule, by assumption. Let λ : D → A be a right D-colinearmap, with convolution inverse λ. Put j : = λ and : C → A, a⊗k d 7→ aλ(d). We need to prove

that is an element of the appropriate bimodule (166) in the Morita context M(A), associated toA as in (171), that is, of

Q ' q ∈ AHom(C, A) | ∀d ∈ D, a ∈ A ψ(d(1) ⊗k q(1A ⊗k d(2))a

)= q(1A ⊗k d)a[0] ⊗k a[1],

(which is equal to CHom(C, A), cf. Proposition 5.10 (iv)). Using the assumption that A is anentwined module (in the second equality), Proposition 6.20 (in the third one), and property (67)of entwining structures (in the fourth one), one checks that, for d ∈ D and a ∈ A,

(1A ⊗k d)a[0] ⊗k a[1] = λ(d)a[0] ⊗k a[1] = λ(d)1A[0]aψ ⊗k 1A[1]ψ = λ(d(2))ψ′aψ ⊗k d(1)

ψ′ψ

= ψ(d(1) ⊗k λ(d(2))a

)= ψ

(d(1) ⊗k (1A ⊗k d(2))a

),(195)

6.5. EXAMPLES 181

that is, ∈ Q. By the assumption that λ is left convolution inverse of λ, for a ∈ A and d ∈ D,

( j)(a⊗k d) = aλ(d(1))λ(d(2))[0] ⊗k λ(d(2))[1](196)

= aλ(d(1))λ(d(2))⊗k d(3) = aεD(d(1))⊗k d(2) = a⊗k d,

where the second equality follows by the colinearity of λ. Similarly, since λ is also a right convolutioninverse of λ, for d ∈ D,

(j ♦ )(d) = λ(d)[0]λ(λ(d)[1]

)= λ(d(1))λ(d(2)) = εD(d)1A.

This proves that A is a cleft bicomodule, as stated.Conversely, assume that A is a cleft bicomodule for the coring extension D of C. Then it is, in

particular, an entwined module. Let j ∈ HomD(D, A) and ∈ Q be elements of the bimodules in

the Morita context M(A), associated to A as in (171), such that j = C and j ♦ = 1AεD(−).Then λ : = j : D → A is right D-colinear and λ : d 7→ (1A ⊗k d) is its convolution inverse.

Let (A,D, ψ) be an entwining structure. We say that A satisfies the normal basis property for(A,D, ψ) if A ∈MD

A(ψ) and there exists a left B-linear, right D-colinear isomorphism B⊗D → A.

Theorem 6.22. Let (A,C, ψ, x) be an entwining structure with a fixed grouplike element.The following assertions are equivalent:

(i) (A,D, ψ) is cleft;(ii) The functor (−)coC : MD

A(ψ) →MB is an equivalence of categories and A satisfies the rightnormal basis property for (A,D, ψ);

(iii) The functor (−)coC : MDA(ψ) →MB is fully faithful and A satisfies the right normal basis

property for (A,D, ψ);(iv) (A,C, ψ, x) is Galois, A satisfies the right normal basis property for (A,D, ψ);(v) the map

∗can : #(C,A) → EndB(A)op, ∗can(f)(a) = a[0]f(a[1]) = 1A[0]aψf(1A[1])

is bijective and A satisfies the right normal basis property for (A,D, ψ).

Proof. One can easily observe that the notion of the normal basis property over an entwiningstructure coincides with the notion of normal basis property defined in the previous section, if weconsider the entwining structure as a coring extension (see Section 3.2.1). Therefore we canapply the results of the previous sections by Proposition 6.21. By Corollary 6.19 (i) implies (ii).Obviously, (ii) implies (iii) and, by Theorem 4.22, (iii) implies (iv). It follows from Proposition 5.2that (iii) implies (iv). Therefore, we are done if we prove 4) ⇒ 1). From the right normal basisproperty, we know that there exists a left B-linear, right D-colinear isomorphism h : B⊗D → A.We consider the maps λ : D → A, λ(d) = h(1⊗ d) and j = (B ⊗ εD) h−1 : A→ B. Clearlyλ is right D-colinear and j is left B-linear. Take a ∈ A, and write h−1(a) =

∑i bi ⊗ di. Then∑

i

bih(1⊗ di(1))⊗ di(2) =∑i

h(bi ⊗ di(1))⊗ di(2)

= (h⊗D)ρ(∑i

bi ⊗ di) = ρ(h(∑i

bi ⊗ di)) = ρ(a) = 1A[0]aψ ⊗ 1ψA[1].

Apply j ⊗D to both sides:

j(1A[0]aψ)⊗ 1ψA[1] =∑i

bi(j h)(1⊗ di(1))⊗ di(1)(197)

=∑i

bi(B ⊗ εD)(1⊗ di(1))⊗ di(1) =∑i

bi ⊗ di = h−1(a).

Now let q = (∗can)−1(j). We are done if we can show that λ is the convolution inverse of q, byTheorem 6.22. The fact that λ is right D-colinear means

(198) λ(d(1))⊗ d(2) = 1A[0]λ(d)ψ ⊗ 1ψA[1]


and we compute, for all d ∈ D,

(λ ∗ q)(d) = λ(d(1))q(d(2)) = 1A[0]λ(d)ψq(1ψA[1]) = ∗can(q)(λ(d))

= j(λ(d)) = ((B ⊗ εD) h−1 h)(1⊗ d) = εD(d)1Aas needed. For all a ∈ A, we have

∗can(q ∗ λ)(a) = 1A[0]aψ(q ∗ λ)(1ψA[1]) = 1A[0]aψq((1ψA[1])(1))λ((1ψA[1])(2))

(69)= aψψ′q(1

ψ′

A[1])λ(1ψA[2]) = ∗can(q)(1A[0]aψ)λ(1ψA[1])

= j(1A[0]aψ)λ(1ψA[1]) = j(aA[0]aψ)h(1⊗ 1ψA[1])

= h(j(1A[0]aψ)⊗ 1ψA[1])(197)= h(h−1(a)) = a.

This proves that ∗can(q ∗ λ) = A = ∗can(ηA εD), and q ∗ λ = ηA εD by the injectivity of ∗can.Thus λ is the convolution inverse of q, as needed.

6.5.2. Cleft extensions of algebras by a coalgebra. Let D be a coalgebra over k and A ak-algebra and a right D-comodule. In [27] A has been termed a D-cleft extension of the subalgebra

T : = t ∈ A | ∀a ∈ A (ta)[0] ⊗k (ta)[1] = ta[0] ⊗k a[1] ,if it is a D-Galois extension, i.e. the canonical map

(199) A⊗T A→ A⊗k D, a⊗T a′ 7→ aa′[0] ⊗k a′[1]

is bijective, and there exists a convolution invertible right D-colinear map λ : D → A.Recall that for any D-Galois extension A of T there exists a (unique) entwining structure

(A,D, ψ) such that A is an entwined module (cf. [36, 34.6]). On the other hand, the canonicalmap (199) is bijective for any cleft entwining structure (A,D, ψ) by [4, Proposition 4.8 1]. Thismeans that cleft extensions of algebras by a coalgebra are in one-to-one correspondence with cleftentwining structures. Combining this observation with Proposition 6.21, we conclude that A isa D-cleft extension of T if and only if A is a cleft bicomodule for the coring extension D ofC : = A⊗k D.

6.5.3. Cleft extensions of algebras by a Hopf algebra and the fundamental theorem.Let H be a bialgebra over k, and consider the entwining structure associated to H as in Exam-ple 3.21. This entwining structure is cleft if and only if there exist a map λ : H → H which isright H-colinear and has a convolution inverse in the algebra Hom(Hc,Ha), where we denote Hc

to emphasize the coalgebra structure on H and Ha to emphasize the algebra structure. A Hopfalgebra can now be defined as a bialgebra for which the associated entwining structure is cleft withright H-colinear cliving map λ = H. Explicitly, H is a Hopf algebra if and only if there exists amap S : H → H such that for all h ∈ H,

S(h(1))h(2) = h(1)S(h(2)) = ε(h)1H .

Since EndH(H) ∼= k and 1H is a grouplike element for H, we can apply Corollary 6.19 to obtain

Theorem 6.23 (fundamental theorem for Hopf algebras). Let H be a Hopf algebra, then thefunctor −⊗H : Mk →MH

H is an equivalence of categories.

Let D be a Hopf algebra over k and A a right comodule algebra. The algebra A and thecoalgebra underlying D are entwined by the map

ψ : D⊗k A→ A⊗k D, d⊗k a 7→ a[0] ⊗k da[1].

Since 1D is a grouplike element in D, 1A⊗k1D is a grouplike element in the A-coring C : = A⊗kD,associated to the entwining structure (A,D, ψ). Hence A is an entwined module.

A is called a D-cleft extension of its D-coinvariant subalgebra if and only if there exists aconvolution invertible right D-colinear map λ : D → A (see e.g. [92, Definition 7.2.1]), i.e. if andonly if (A,D, ψ) is a cleft entwining structure. (Note that in this way a cleft extension of algebrasby a Hopf algebra is a cleft extension by the underlying coalgebra.) By Proposition 6.21, this isequivalent to A being a cleft bicomodule for the coring extension D of C : = A⊗k D.

6.5. EXAMPLES 183

6.5.4. Cleft weak entwining structures. A weak entwining structure [39] consists of a k-algebra A, a k-coalgebra D and a k-linear map ψ : D⊗kA→ A⊗kD, such that the compatibilityconditions (67) and (69) hold true, while (68) and (70) are replaced by

ψ (D⊗k 1A) = (e⊗k D) ∆D, and(200)

(A⊗k εD) ψ = µA (e⊗k A),(201)

respectively, where e : = (A⊗k εD) ψ (D⊗k 1A) : D → A.To a weak entwining structure (A,D, ψ) one can associate an A-coring C : = a1Aψ ⊗k

dψ a⊗kd∈A⊗kD (cf. [28, Proposition 2.3] or [36, 37.4]). The left A-module structure is given byleft multiplication in the first tensorand, and the right A-module structure is given by (a1Aψ ⊗kdψ)a′ = aa′ψ ⊗k dψ. The coproduct is given by the restriction of A⊗k ∆D, i.e. by

∆C : C → C⊗A C,

a1Aψ ⊗k dψ 7→ (a1Aψ ⊗k dψ(1))⊗A (1A ⊗k dψ(2)) = (a1Aψ ⊗k d(1)ψ)⊗A (1Aψ′ ⊗k d(2)

ψ′).

The counit is given by the restriction of A⊗k εD, i.e. by

εC : C → A, a1Aψ ⊗k dψ 7→ a1AψεD(dψ) = ae(d).

C is a C-D bicomodule with the left regular C-coaction ∆C and right D-coaction, given by therestriction of A⊗k ∆D, i.e.

(202) τC : C → C⊗k D, a1Aψ ⊗k dψ 7→ (a1Aψ ⊗k dψ(1))⊗k dψ(2) = (a1Aψ ⊗k d(1)ψ)⊗k d(2),

where the equality of the two forms of τC follows by (200) and the coassociativity of ∆D. Thatis, D is a right extension of C. Right C-comodules are called weak entwined modules and theycan be characterized as a right D-comodules M , that are right A-modules as well such that thecompatibility condition (71) holds true.

By [6, Definition 1.9], a weak entwining structure (A,D, ψ) is cleft if A (with the right regularA-module structure) is a weak entwined module and there exists a right D-colinear map λ : D → Aand a k-linear map λ : D → A, satisfying (192) and

(203) 1Aψλ(dψ) = λ(d) and λ(d(1))λ(d(2)) = e(d), for d ∈ D.

(The first condition in (203) can be read as a convenient normalisation. Indeed, if there existsλ ∈ Homk(D, A), satisfying (192) and the second condition in (203), then it can be replaced bythe (non-zero) map d 7→ 1Aψλ(dψ).)

Proposition 6.24. A weak entwining structure (A,D, ψ) is cleft if and only if A (with theright regular A-module structure) is weak cleft bicomodule for the right coring extension D of theA-coring C, associated to the weak entwining structure (A,D, ψ).

Proof. Let us assume first that (A,D, ψ) is a cleft weak entwining structure. We constructelements j and in the bimodules of the Morita context (171), associated to A, such that j = C.Put j : = λ and

: C → A, a1Aψ ⊗k dψ 7→ a1Aψλ(dψ) = aλ(d),where λ : D → A is a right D-colinear map and λ : D → A is a k-linear map, satisfying (192)and (203). Analogously to (195) and (196), assumption (192) implies that is left C-colinear, i.e.an element of

Q ' q ∈ AHom(C, A) | ∀d ∈ D, a ∈ Aψ(d(1) ⊗k q(1Aψ ⊗k d(2)

ψ)a)

= q(1Aψ ⊗k dψ)a[0] ⊗k a[1] ,and (203) implies j = C.

Conversely, assume that A is a weak cleft bicomodule, i.e. there exist elements ∈ Q and

j ∈ HomD(D, A) in the bimodules of the Morita context M(A), associated to A in (171), suchthat j = C. The map λ : = j : D → A is right D-colinear. Together with the map λ : d 7→(1Aψ ⊗k dψ), for d ∈ D, they satisfy (203) and (192). Indeed, the first condition in (203) followsby the left A-linearity of and (67). The second one follows by j = C as, for d ∈ D,

e(d) = 1AψεD(dψ) = (A⊗k εD)(( j)(1Aψ ⊗k dψ)

)= (1Aψ ⊗k d(1)

ψ)j(d(2)) = λ(d(1))λ(d(2)),


where the third equality follows by the forms (178) of the map and (202) of the D-coaction inC. Condition (192) is easily seen to follow by the assumption that is an element of the bimodule

Q.

Note that the first condition in (203) and (192), imposed on a weak entwining structure in[6], gain an explanation by Proposition 6.24. They mean that λ ∈ Homk(D, A) corresponds to an

element of Q ⊆ AHom(C, A) in Proposition 6.24, via the isomorphism

AHom(C, A) ' ν ∈ Homk(D, A) | ∀d ∈ D 1Aψν(dψ) = ν(d) .

6.5.5. Cleft extensions by partial group actions. Extending the definition of (idempotent)partial actions of finite groups on commutative algebras in [59] and [60], Caenepeel and De Grootintroduced in [40] idempotent partial actions of finite groups G on arbitrary algebras A, as follows.An idempotent partial G-action on A consists of a collection eσσ∈G of central idempotents inA and a collection ασ : Aeσ−1 → Aeσσ∈G of isomorphisms of ideals, satisfying the conditions

A1 = A and α1 = A, for the unit element 1 of G, andασ(ατ (aeτ−1)eσ−1

)= αστ (aeτ−1σ−1)eσ, for σ, τ ∈ G, a ∈ A.

They constructed an A-coring for such a partial action, eσ, ασσ∈G of G on A, as a k-moduleC : = ⊕σ∈GAeσ with A-A bimodule structure

a1(aνσ)a2 = a1aασ(a2eσ−1)νσ,

for a1, a2, a ∈ A, and elements νσ of C, taking the value eσ in the component σ and 0 everywhereelse, for σ ∈ G. The coproduct and the counit are inherited from the coalgebra k(G), the k-dualof the group algebra. Explicitly,

∆C(aνσ) =∑τ∈G

aντ ⊗A ντ−1σ and εC(aνσ) = aδσ,1, for aνσ ∈ C.

Note that the coalgebra (k-coring) k(G) is a right extension of the A-coring C. That is, thereexists a left C-colinear right k(G)-coaction in C,

τC : C → C⊗k k(G), aνσ 7→∑τ∈G

aντ ⊗k uτ−1σ,

where uσσ∈G is the k-basis for k(G), dual to the basis σσ∈G of the group algebra. Since Cpossesses a grouplike element,

(204)∑σ∈G

νσ,

(cf. [40, Lemma 2.3]), A possesses a right C-comodule structure (and hence a right k(G)-comodulestructure).

By a plausible definition we call A a cleft extension of its G-invariant subalgebra a ∈ A | ∀σ ∈G ασ(aeσ−1) = aeσ if there exists a convolution invertible right colinear map from the rightregular k(G)-comodule to A.

Proposition 6.25. Let G be a finite group with an idempotent partial action eσ, ασσ∈Gon an algebra A. Let C be the associated A-coring. Then A is a cleft extension of its G-invariantsubalgebra if and only if A is a cleft bicomodule for the coring extension k(G) of C.

Proof. The proof is surprisingly similar to that of Proposition 6.21.Assume first that A is a cleft bicomodule, that is, there exist elements ∈ Q and j ∈

Homk(G)(k(G), A) in the bimodules of the Morita context M(A), associated to A as in (171),such that j = C and j ♦ = 1Aεk(G)(−). Put λ : = j : k(G) → A. It is right colinear. Weclaim that it is also convolution invertible.

Using the notation as introduced earlier in this Section, the k(G)-coaction in A, determinedby the grouplike element (204), comes out as

τA : A→ A⊗k k(G), a 7→∑σ∈G

ασ(aeσ−1)⊗k uσ.

6.5. EXAMPLES 185

Then, since k(G) is a free k-module of finite rank, the colinearity of λ means that

(205) ατ(λ(uσ)eτ−1

)= λ(uστ−1), for σ, τ ∈ G.

Condition (205) implies, in particular, that λ(uσ)eτ = λ(uσ), for any σ, τ ∈ G. (Hence there existno non-trivial right k(G)-comodule maps k(G) → A if the ideals Aeσσ∈G have no non-trivialintersection.)

Now put λ(uσ) : = (νσ), for σ ∈ G. By j = C, λ is left convolution inverse of λ. Similarly,it follows by j ♦ = 1Aεk(G)(−) that∑

τ∈Gατ(λ(uσ)eτ−1

)λ(uτ ) = δσ,1 1A for σ ∈ G.

Using the colinearity of λ, i.e. the identity (205), we conclude that λ is also a right convolutioninverse of λ.

Conversely, assume that there exists a right k(G)-comodule map λ : k(G) → A with convo-

lution inverse λ. We construct elements j ∈ Homk(G)(k(G), A) and ∈ Q such that j = Cand j ♦ = 1Aεk(G)(−). Put j : = λ : k(G) → A. Since λ is convolution inverse of a rightk(G)-comodule map λ, its range is in the intersection of the ideals Aeσσ∈G. Hence we can put

: C → A, aνσ 7→ aλ(uσ).

The conditions j = C and j ♦ = 1Aεk(G)(−) follow easily by the assumptions that λ is left,and right convolution inverse of λ, respectively, and the colinearity condition (205). Furthermore,using that λ is left convolution inverse of λ (in the second equality), the colinearity condition (205)(in the third one) and the assumption that λ is left convolution inverse of λ (in the last one), wededuce that

ατ(λ(uτ−1σ)aeτ−1

)=

∑ω∈G

δω,τατ(λ(uω−1σ)aeτ−1

)=

∑ω,µ∈G

λ(uµ)λ(uµ−1ωτ−1)ατ(λ(uω−1σ)aeτ−1

)=

∑ω,µ∈G

λ(uµ)ατ(λ(uµ−1ω)eτ−1

)ατ(λ(uω−1σ)aeτ−1

)=

∑ω,µ∈G

λ(uµ)ατ(λ(uµ−1ω)λ(uω−1σ)aeτ−1

)= λ(uσ)ατ (aeτ−1),

for a ∈ A and σ, τ ∈ G. Using the forms of the coproduct ∆C in C and the C-coaction (determinedby the grouplike element (204)) in A, it is straightforward to check that this is equivalent to the

property that is an element of the bimodule Q, associated to A as in (166).

6.5.6. Cleft entwining structures over arbitrary base. An entwining structure over analgebra L consists of an L-ring A, an L-coring D and an L-L bilinear map ψ : D⊗LA→ A⊗LD,satisfying conditions (67-70), with the only modification that k-module tensor products are replacedby L-module tensor products. Just as in the case of commutative base rings, C : = A ⊗L Dpossesses an A-coring structure (cf. [19, Example 4.5]) such that D is a right extension of C.

Recall that, for an L-ring A and an L-coring D, the set of bimodule maps LHomL(D, A) isan algebra with the convolution product (fg)(d) = f(d(1))g(d(2)) and unit εD(−)1A. In completeanalogy with Proposition 6.21 one proves the following.

Proposition 6.26. Let (A,D, ψ) be an entwining structure over an algebra L and let C : =A⊗L D be the associated A-coring. A (with the right regular A-module structure) is a cleft L-Cbicomodule for the coring extension D of C if and only if the following assertions hold.

(a) A (with the right regular A-module structure) is an entwined module, i.e. it is a rightD-comodule such that the compatibility condition

(aa′)[0] ⊗L (aa′)[1] = a[0]a′ψ ⊗L a[1]

ψ

holds true, for all a, a′ ∈ A;

(b) The right D-coaction a 7→ a[0] ⊗L a[1] in A is left L-linear;


(c) There exists a convolution invertible morphism λ ∈ LHomD(D, A) ⊆ LHomL(D, A).

If conditions (a)-(c) in Proposition 6.26 hold, we call the L-entwining structure (A,D, ψ) cleft.Note that if the coring D possesses a grouplike element x, then the left L-linearity of the

D-coaction a 7→ ψ(x⊗L a) in A is equivalent to the element x to be central in the L-L bimoduleD.

6.5.7. Cleft extensions of algebras by a Hopf algebroid. A Hopf algebroid H consists of aleft bialgebroid structure HL, over a base algebra L, and a right bialgebroid structure HR, over abase algebra R, on the same total algebra H, and a k-linear map S : H → H, called the antipode,relating the two bialgebroid structures [22],[18]. For a Hopf algebroid H, we denote by γL and πL(resp. γR and πR) the coproduct and the counit of the bialgebroid HL (resp. HR).

The category of right comodules for the right bialgebroid HR is a monoidal category suchthat the forgetful functor to the bimodule category RMR is strict monoidal. Right HR-comodulealgebras are defined as monoids in the category of right HR-comodules, hence they are in particularR-rings [102]. A right HR-comodule algebra A determines an entwining structure over R withR-ring A and R-coring (H, γR, πR), underlying the bialgebroid HR (cf. [17, (3.17)]). Hence thereexists a corresponding A-coring C : = A⊗R H, with coproduct A⊗R γR, inherited from HR. Bythe definition of a Hopf algebroid, the coproduct γR in HR is (left and right) HL-colinear, henceC possesses a C-HL bicomodule structure with the left regular C-coaction and right HL-coactionA⊗R γL. That is, the L-coring (H, γL, πL), underlying the bialgebroid HL, is a right extension ofthe A-coring C = A⊗R H. Note that this coring extension does not correspond to any entwiningstructure.

Under the additional assumption that the right HR-comodule algebra A is also an L-ringwith left L-linear HR-coaction, one can associate to it a Morita context like in [21, Remark 3.2(1)]. It is formulated in terms of the two convolution products, (f, g) 7→ µA (f ⊗L g) γL, forf ∈ HomL(H,A) and g ∈ LHom(H,A) on one hand, and (f ′, g′) 7→ µA (f ′ ⊗R g′) γR, forf ′ ∈ HomR(H,A) and g′ ∈ RHom(H,A) on the other hand. These two convolution productsdefine the convolution algebras LHomL(H,A) and RHomR(H,A), respectively, and also theirbimodules LHomR(H,A) and RHomL(H,A). The connecting homomorphisms of the Moritacontext are defined as projections of the appropriate convolution product. Note that, under theassumption imposed on A, it is in particular an L-C bicomodule. The precise relation of theMorita context described above to the one associated to A as in (171), is formulated in thefollowing lemma.

Lemma 6.27. Let H = (HL,HR, S) be a Hopf algebroid and A a right HR-comodule algebra.Denote the associated A-coring A⊗RH by C. Assume that A is also an L-ring and itsHR-coaction

is left L-linear. Then the Morita context M(A), associated to the L-C bicomodule A as in (171),is isomorphic to a sub-Morita context of

(206) (LHomL(H,A) , RHomR(H,A) , LHomR(H,A) , RHomL(H,A) , , ♦♦),

where the algebra and bimodule structures are given by the respective convolution product andthe connecting homomorphisms and ♦♦ are defined as projections of the convolution products.

Proof. The endomorphism algebra T = EndC(A) can be identified with the subalgebra of C(equivalently, HR) -coinvariants in A, via T 3 t 7→ t(1A) cf. [36, 28.4]. This injection T → A ofL-rings defines an injection of convolution algebras

ι1 : LHomL(H,T ) → LHomL(H,A).

The algebra of left C-colinear right HL-colinear endomorphisms of A ⊗R H can be injected intothe other convolution algebra RHomR(H,A) via the map

ι2 : CEndHL(C) → RHomR(H,A), u 7→[h 7→

((A⊗R πR) u

)(1A ⊗R h)

].

Right HL-colinear maps are right R-linear by [21, Theorem 2.1], so we have an obvious injection

ι3 : LHomHL(H,A) → LHomR(H,A).

Finally, by standard hom-tensor relations, we have an inclusion

ι4 : Q → AHomL(A⊗R H,A) ' RHomL(H,A).

6.5. EXAMPLES 187

It is left to the reader as an easy exercise to check that the four injections constructed define amorphism of Morita contexts.

In [20, Example 3.11] a right HR-comodule algebra A (with R-ring structure ηR : R → A)for a Hopf algebroid H = (HL,HR, S) has been called an H-cleft extension of its HR-coinvariantsubalgebra B if the following conditions are satisfied.

(a) A is an L-ring (with unit morphism ηL : L→ A) and B is an L-subring of A;

(b) there exists a left L-linear right HR-colinear map λ : H → A which is invertible in the Moritacontext (206), i.e. for which there exists a left R-linear right L-linear map λ : H → A suchthat

λ ♦♦ λ ≡ µA (λ⊗R λ) γR = ηL πL and λ λ ≡ µA (λ⊗L λ) γL = ηR πR.

For more details on cleft extensions by Hopf algebroids, in particular for their characterization ascrossed products, we refer to [21]. This definition, cited from [20], can be reformulated using theterminology of the present paper as follows. The proof is a simple generalization of the one ofProposition 6.21, using [21, Lemmas 3.6 and 3.7].

Proposition 6.28. Let H = (HL,HR, S) be a Hopf algebroid and A a right HR-comodulealgebra. Let C be the associated A-coring A ⊗R H, with coproduct inherited from HR. A is anH-cleft extension of its HR-coinvariant subalgebra if and only if the following hold.

(a) A is an L-ring;

(b) A (with the left L-module structure in (a), the right regular A-module structure and thegiven HR-comodule structure) is a cleft L-C bicomodule for the coring extension HL of C.

It should be emphasized that cleft extensions of algebras by a Hopf algebroid are not examplesof the kind discussed in Section 6.5.6. As explained, one can associate an R-entwining structure– with R-ring A and R-coring underlying HR – to a right HR-comodule algebra A, for a Hopfalgebroid H = (HL,HR, S). It is not true, however, that this R-entwining structure was cleftfor an H-cleft extension. The right HR-coaction in A is determined by the grouplike element 1H(cf. last paragraph in Section 6.5.6), which is not central in the R-R bimodule HR. Hence theright HR-coaction in A is not left R-linear in the sense of a cleft R-entwining structure (i.e. ofProposition 6.26 (b)).

As a particular example of anH-cleft extension, consider the right regularHR-comodule algebrafor a Hopf algebroid H = (HL,HR, S). It is an L-ring via the source map of HL. The coinvariantsubalgebra is the image of the base algebra R under the target map of HR, which coincides withthe image of the base algebra L under the source map of HL. The (left L-linear right HR-colinear)identity map H → H possesses an inverse in the associated Morita context (206), the antipode.Hence Rop ⊆ H is an H-cleft extension. That is, by Proposition 6.28, H is a cleft L-(H ⊗R H)bicomodule. In light of this observation, the Fundamental Theorem of Hopf modules for a Hopfalgebroid [18, Theorem 4.2] is a special instance of Corollary 6.19 (ii) (note the existence of agrouplike element 1H in HL, cf. Remark 6.13 (i) and (ii)). This explains why the FundamentalTheorem of Hopf modules can be proven without assuming H to be a faithfully flat R-module(unlike the Galois Coring Structure Theorem [36, 28.19]).

6.5.8. Cleft factorization structures. Our results can be dualized, leading to cleft bimodulesfor ring extensions. In this section we discuss a particular case of this dual situation.

Let (A,S, ρ) be a factorization structure, and χ : S → k an algebra map. Then the map

X : R = A#ρS → A, X(a#s) = χ(s)a

is a left grouplike character, i.e. X is left A-linear, X(rX(s)) = X(rs), and X(1) = 1. We cantherefore apply the results of Section 5.5.1. In particular, we obtain that A is a left R-module:

(a#s) b = X((a#s)b) = X(abρ#sρ) = χ(sρ)abρ.

We put

B = AR = b ∈ A | χ(sρ)bρ = χ(s)b, for all s ∈ S;

Q = ∑i

ai#si ∈ A#R |∑i

aiR#tρsi = χ(t)∑i

ai#si for all t ∈ S.


Then have a Morita context (B,A#S,A,Q, τ, µ) with

µ : A⊗B Q→ A#S, µ(a⊗B (∑i

ai#si)) =∑i

aai#si;

τ : Q⊗ρ A→ B, τ(∑i

ai#si)⊗ρ a) =∑i

aiaρχ(siρ).

Take q =∑

i ai#si ∈ Q, and assume that q is invertible in Aop ⊗ S, i.e. there existsq =

∑j aj#sj ∈ A#S such that

(207)∑i,j

aiaj#sjsi =∑i,j

ajai#sisj = 1A#1S .

Proposition 6.29. Let q =∑

i ai#si ∈ A#S be invertible in Aop ⊗ S, with inverse q =∑j aj#sj . Then the following assertions are equivalent:

(i) q ∈ Q;(ii)

∑i,j(aj)ρai#sitρsj = χ(t)1A#1S , for all t ∈ S;

(iii)∑

j χ(tρ)(aj)ρ#sj =∑

j aj#sjt, for all t ∈ S.

In this situation, we call (A,S, ρ, χ) a cleft factorization structure.

Proof. (i) ⇒ (ii). Using the defining property of Q and (207), we find, for all t ∈ S:∑i,j

(aj)ρai#sitρsj = χ(t)∑i,j

ajai#sisj = χ(t)1A#1S .

(ii) ⇒ (iii). For all t ∈ S, we compute∑j

χ(tρ)(aj)ρ#sj =∑j

χ(tρ)(aj)ρ1A#1Ssj =∑i,j,k

(aj)ρ(ak)rai#sitRrsksj

(76)=

∑i,j,k

(ajak)ρai#sitRsksj(207)=∑i

(1A)ρai#sitρ1S(73)=∑i

ai#sit,

where we applied part (ii) to obtain the second equality. (iii) ⇒ (i).∑i

(ai)ρ#tρsi =∑i,j,k

(ai)ρajak#sksjtρsi =∑i,j,k

χ(tRr)(ai)ρ(aj)rak#sksjsi

(76)=

∑i,j,k

χ(tR)(aiaj)ρak#sksjsi(207)=∑k

ak#sk.

Here we used part (iii) in the second equality.

Proposition 6.30. Assume that (A,S,R, χ) is cleft. Then we have an equivalence of cate-gories

F : BM→ ρM, F (N) = A⊗B N ; G : ρM→ BM, G(M) = RM.

Consequently the map can : A⊗B A→ Hom(S,A), can(a⊗ a′)(s) = a′ρχ(sρ)a is bijective.

Proof. We first prove that the functor F is fully faithful. This follows from Proposition 5.28after we show that the map τ from the Morita context from Proposition 5.25 is surjective. Itsuffices to show that there exists Λ ∈ Q with (IA ⊗ χ)(Λ) = 1 (Proposition 5.27).Take q ∈ Q as in Proposition 6.29. Then

∑j χ(tρ)χ(sj)(aj)ρ =

∑j χ(t)χ(sj)aj , which means

that∑

j χ(sj)aj ∈ B. Q is a left B-module, so

Λ =∑

i,j χ(sj)ajai#si ∈ Q, and it follows from (207) that

(IA ⊗ χ)(∑i,j

χ(sj)ajai#si) =∑i,j

χ(sj)χ(si)ajai = 1.

Now we show that G is fully faithful, or, equivalently, the counit of the adjunction (F,G) is anisomorphism. Recall that, for M ∈ ρM, ζM : A ⊗B RM → M is given by ζM (a ⊗m) = am.Take q ∈ Q as in Proposition 6.29, and m ∈M . Then q ·m ∈ RM since

(1#t)q ·m =∑i

(1#t)(qi#si) ·m = (∑i

(qi)ρ#tρsi) ·m = χ(t)q ·m,

6.5. REFERENCES 189

where we used the defining property of Q in the last equality. Now let γM : M → A ⊗B RMbe defined by γM (m) =

∑j aj ⊗B qsjm For all m ∈ M , we have, by (207), that ζM (γM (m)) =∑

j ajqsjm = m. Finally, for all b ∈ A and m ∈ RM :

γM (ζM (b⊗B m)) = γM (bm) =∑j

aj ⊗B qsjbm

=∑j

aj ⊗B X(qsjb)m =∑j

ajX(qsjb)⊗B m

=∑i,j

ajaiχ(sisj)b⊗B m = b⊗B m,

where we used the fact that X(qsjb) ∈ Im(τ) ⊂ B.

References

Most of the results of this section are deduced from the author’s joint work with G. Bohm[24]. The results from Section 6.5.1 are inspired by the joint work with S. Caenepeel and S. Wang[47] and [4], Section 6.5.8 is obtained form [47].

IIIPart III :

Separable and FrobeniusFunctors

And now for something completely different.– Monthy Python

Chapter 7Separable Functors and relative

Cohomology

Separable functors are known to play an important role in coring theory. If a the forgetfulfunctor MC → MA is separable, then the coring C is called coseparable and this has importantimplications for the Galois theory of the coring C (see e.g. Chapter 9, Section 9.1). The aim ofthis Chapter is to characterize cotriples with a separable forgetful functor by means of cohomologygroups using cointegrations into bicomodules.

In the first section we recall several definitions on separability and relative injectivity. In Section7.2, we discuss in detail the category of bicomodules over cotriples and proof that the forgetfulfunctor of a cotriple is separable if and only of the forgetful functor for the category of bicomodulesis a Maschke functor if and only if the comultiplication of the cotriple splits in the category ofbicomodules (see Theorem 7.7). In Section 7.3 we define coderivations and cointegrations, whichare tools to develop in Section 7.4 the relative cohomology for bicomodules. We present twoapplications in the last Section: the characterization of coseparable corings stated in [76], and thecharacterization of coseparable coalgebra coextensions stated in [94].

7.1. Separability and relative injectivity

Let A and B be two categories. Any covariant functor F : A → B leads to a (bi)functor

HomB(F (−), F (−)) : Aop ×A → Set.

In particular, the identity functor 11A : A → A gives rise to

HomA(−,−) : Aop ×A → Set.

So we find a natural transformation induced by F ,

F : HomA(−,−) → HomB(F (−), F (−));

defined by FX,X′(f) = F (f), for any morphism f : X → X ′ in A. Recall from [100] (see [95]for the original definition) that the functor F is called separable if and only if F has a left inverse,i.e. there exists a natural transformation

P : HomB(F (−), F (−)) → HomA(−,−)

such that P F = 11HomA(−,−). The following well-known theorem, known as Rafael’s Theoremfor separable functors, was first proven in [100]

193

194 CHAPTER 7. SEPARABLE FUNCTORS AND RELATIVE COHOMOLOGY

Theorem 7.1 (M.D. Rafael). Let F : A → B be a covariant functor with a right adjointfunctor G : B → A. Denote by η : 11A → GF the unit of the adjunction (F,G), then F isseparable if and only if there exists a natural transformation µ : GF → 11A such that µ η = 11A.

Let F : A → B be a covariant functor. Recall from [45], that an object M ∈ A is calledrelative injective (or F -injective) if and only if for every morphism i : X → X ′ in A, such thatF (i) : F (X) → F (X ′) has a left inverse j in B (i.e. F (i) is a split monomorphism) and for everyf : X →M in A we can find a morphism g : X ′ →M in A such that g i = f . The functor F issaid to be a Maschke functor if every object of A is F -injective. The following version of Rafael’sTheorem for Maschke functors was proven in [45, Theorem 3.4].

Theorem 7.2. Let F : A → B be a covariant functor with a right adjoint functor G : B → A.Denote by η : 11A → GF the unit of the adjunction (F,G), then an object M ∈ A is F -injectiveif and only if ηM has a left inverse. In particular F is a Maschke functor if and only if for everyobject M ∈ A, ηM has a left inverse.

Assume that a preadditive category A is given. Following [62, pages 3-4], a sequence

E : Xi // X ′ j // X ′′

(i.e. j i = 0) is said to be co-exact if i has a cokernel and if in the commutative diagram

Xi // X ′ j //

ic

X ′′

Coker(i)`

88pppppp

` is a monomorphism. If in addition ` is a split-mono, then the sequence E is said to be cosplit.The exact and split sequence are dually defined by using kernels. The notions of sequence,coexact sequence, cosplit sequence,... are extended to long diagrams simply by applying them toeach consecutive pair of morphisms. One can prove that the above notions of exact and coexactsequences coincide with the usual meaning of exact sequences in abelian categories. In case of adiagram of the form

E′ : 0 // X // X ′ // X ′′ // 0

(i.e. short sequence) in the category A , we have by [80, Lemma 2.1], that E′ is cosplit if andonly if it is split.

Let E be a class of sequences in A, then an object X ∈ A is said to be E -injective ifHomA(E,X) is an exact sequence of abelian groups, for every sequence E in E . The class of allE -injective objects is denoted by IE . Conversely, given I a class of objects of A, a sequence E ofmorphisms in A is said to be I -exact if HomA(E, Y ) is an exact sequence of abelian groups, forevery object Y in I . The class of all I -exact sequences is denoted by EI . A class of sequencesE in A is said to be closed whenever E coincides with EIE

. An injective class is a closed classof sequences E such that, for every morphism X → X ′, there exists a morphism X ′ → Y withY ∈ IE and with X → X ′ → Y in E . If in addition the category A possesses cokernels, thenone can check that the class E0 of all cosplit sequences form an injective class and IE0 is exactlythe class of all objects of A. Consider any adjunction (F,G), where F : A → B and a class ofsequences E ′ in B. Denote by E = F−1(E ′) the class of sequences E in A such that F (E) is inE ′. The Eilenberg-Moore Theorem [80, Theorem 2.9] asserts that E is an injective class wheneverE ′ is.

7.2. Bicomodules and separability

Let β : F → G be a natural transformation between two functors F,G : A → B. Consider twoother functors H : B → C, and I : D → A. Then βI denotes the natural transformation definedat each object Z ∈ D by βI(Z) : FI(Z) → GI(Z), while Hβ denotes the natural transformationdefined at each object X ∈ A by H(βX) : HF (X) → HG(X).

7.2. BICOMODULES AND SEPARABILITY 195

Let A and B two Grothendieck categories, we denote by Funct(A, B) the class of all rightcontinuous covariant functors F : A → B. Recall from Example 1.1 (4) that this means thatF preserves cokernels and commutes with direct sums. Moreover, the natural transformationsbetween two objects of the class Funct(A, B) form a set. Henceforth, Funct(A, B) is a Hom-setcategory (or Set-category).

Let F = (F, δ, ξ) be a cotriple on A with F ∈ Funct(A, A). This means F is a coalgebra inFunct(A, A).

Recall from Chapter 1 that Funct(B, A) is a right monoidal category on Funct(A, A). Conse-quently, we can consider F-comodules in Funct(B, A). The category ComodFunct(B,A)-F will be

called the category of B-F bicomodules and denoted as BMF. This category is explicitly describedby the following data:

• Objects: A B-F bicomodule is a pair (M,m) consisting of a functor M ∈ Funct(B, A) and anatural transformation m : M → FM satisfying

(208) δM m = Fm m, ξM m = M.

• Morphisms: A morphism f : (M,m) → (M ′,m′) is a natural transformation f : M → M ′

satisfying

(209) m′ f = Ff m.

It is easily seen that (FM, δM ) is an object of the category BMF, for every object M ∈Funct(B, A). This fact induces a functor GF : Funct(B, A) → BMF. Similar to Proposition 3.8GF is the right adjoint of the forgetful functor FF : BMF → Funct(B, A).

In the same way, we can define the category FMB of F-B bicomodules, using the objects ofthe category Funct(A, B).

Remark 7.3. Consider any adjoint pair of functors (M,N), where M : A → B, with counit ζand unit η. Then [70, Proposition 1.1] (compare with Theorem 4.7 and Theorem 4.9) establishesa one-to-one correspondence between natural transformations m : M → FM satisfying equation(208) and homomorphisms of cotriples from (MN,MηN , ζ) to F, and a natural transformationss : N → NF satisfying the dual version of equation (208). When N and M are both right exactand preserve direct sums, then the previous correspondence can be interpreted in our terminologyas follows: There are bijections between the F-B bicomodule structures on M , the B-F bicomodulestructures on N , and the homomorphisms of cotriples from (MN,MηN , ζ) to F.

Consider now a second cotriple G = (G,ϑ, ς) on B with G ∈ Funct(B, B). We can constructthe category of G-F bicomodules as the category G-ComodFunct(B,A)-F. We will denote this

category shortly by GMF and it can be described as follows:

• Objects: A G-F bicomodule is a three-tuple (M,m, n) consisting of a functor M ∈ Funct(B, A)and two natural transformations m : M → FM , n : M →MG such that (M,m) ∈ BMF and(M, n) ∈ GMA, that is

(210) δM m = Fm m, ξM m = M and Mϑ n = nG n, Mς n = M

with compatibility condition

(211) mG n = Fn m.

In other words m is a morphism of GMA, equivalently, n is a morphism of BMF, where(FM,Fn) ∈ GMA and (MG,mG) ∈ BMF.

• Morphisms: A morphism f : (M,m, n) → (M ′,m′, n′) is a natural transformation f : M →M ′

such that f : (M,m) → (M ′,m′) is a morphism of BMF and f : (M, n) → (M ′, n′) is amorphism of GMA, that is

(212) n′ f = fG n and m′ f = Ff m.

It is clear that 11BMF = BMF and GM11A = GMA, where 11A and 11B are endowed with atrivial cotriple structure.

By the observation (see Section 1.3) that the bicomodules as introduced above coincide withcertain 1-cells in a 2-category, we can state the following well-known Lemma.


Lemma 7.4. Let A (respectively B) be a Grothendieck category, and F = (F, δ, ξ) (respec-tively G = (G,ϑ, ς)) a cotriple in A (respectively in B) whose underlying functor F (respectivelyG) is right exact and preserves direct sums. The category of G-F bicomodules GMF is a pread-ditive category with cokernels and arbitrary direct sums.

Consider the categories of bicomodules BMF and GMF. There are two functors connect-ing those categories. The left forgetful functor S : GMF → BMF, which sends any (G,F)-bicomodule (M,m, n) to the (B,F)-bicomodule (M,m) and which is the identity on the mor-phisms. Secondly, the functor T : BMF → GMF which sends (M ′,m′) → (M ′G,m′

G,M′ϑ) and

f → fG. These functors form an adjunction, more precisely, we have

Lemma 7.5. For every pair of objects((N, r, s), (M,m)

)of GMF×BMF, there is a natural

isomorphism

HomGMF

((N, r, s), T (M,m)

) ΦN, M // HomBMF

(S(N, r, s), (M,m)

)f // Mς f

gG s g.oo

Otherwise stated, S is a left adjoint functor to T .

Let X be the discrete one-object category, then the category XMF can be described as follows.A functor X : X → A is completely determined by the image X of the single object in X . Anatural transformation x : X → FX is completely determined by a morphism dX : X → F (X).In this way, we can identify an object in XMF with a pair (X, dX) consisting of an object X ∈ Aand a morphism dX : X → F (X) satisfying

δX dX = F (dX) dX , ξX dX = X.

Similarly, a morphism f : (X, dX) → (X ′, dX′) in XMF is completely determined by a morphism

f : X → X ′ of A such that

dX′ f = F (f) dX .

Under this identification, we will denote this category by AF. Denote by S : AF → A the forgetfulfunctor and T : A → AF,T(Y ) = (F (Y ), δY ), T = F (f), for every object Y and morphism f ofA. Then we obtain an adjoint pair (S,T), with ST = F satisfying a universal property, see [61,Theorem 2.2].

Remark 7.6. It is well known that AF is an additive category with direct sums and cokernels,admitting (F (U), δU ) as a sub-generator, whenever U is a generator of A. However, AF is notnecessarily a Grothendieck category. But, if we assume that F is an exact functor and that Apossesses a generating set of finitely generated objects, then one can easily check that AF becomesa Grothendieck category.

The main result of this section is the following

Theorem 7.7. Let A be a Grothendieck category. Consider a cotriple F = (F, δ, ξ) in Awhose functor F preserves cokernels and commutes with direct sums. The following are equivalent

(i) S : AF −→ A is separable functor;(ii) S : FMF −→ AMF is a Maschke functor;(iii) δ : (F, δ, δ) −→ (F 2, δF , F δ) is a split monomorphism in the category FMF.

Proof. (i) ⇒ (iii). The unit of the adjunction (S,T) is given by

(213) η(X, dX) : (X, dX) dX// TS(X, dX) = (F (X), δX) ,

for every object (X, dX) of AF. By hypothesis there is a natural transformation ψ : TS → 11AF

such that ψ η = 11AF . Let us denote by ∇ : F 2 → F the natural transformation given by thecollection of morphisms ∇X = S(ψ(F (X), δX)), where X runs through the class of objects of A.

By construction ∇ δ = F and ∇ : (F 2, δF ) → (F, δ) is a morphism of the category AMF. Since

7.2. BICOMODULES AND SEPARABILITY 197

ψ is a natural transformation and δX : (F (X), δX) → (F 2(X), δF (X)) is a morphism in AF, wehave the following commutative diagram

F 3SψF2 // F 2

F 2SψF //

Fδ

OO

F

δ

OO

Therefore δ ∇ = ∇F Fδ, which means that ∇ : (F 2, δF , F δ) → (F, δ, δ) is a morphism in thecategory FMF. Thus δ is a split monomorphism in the category FMF.(iii) ⇒ (ii). Let us denote by Λ : (F 2, δF , F δ) → (F, δ, δ) the left inverse of δ : (F, δ, δ) →(F 2, δF , F δ), i.e. Λ δ = F , in the category FMF. Let (M,m, n) be any F-bicomodule. Theunit of the adjunction (S, T ) stated in Lemma 7.5, at this bicomodule is given by

(214) Θ(M,m, n) : (M,m, n) n // T S(M,m, n) = (MF,mF ,Mδ).

Consider the natural transformation defined by the following composition

υ : MFnF // MF 2 MΛ // MF

Mξ // M .

It is easily seen that υ n = M . The implication will be established if we show that υ is amorphism in the category of bicomodules FMF. We can compute

m υ = m Mξ MΛ nF

= FMξ mF MΛ nF , m− is natural= FMξ FMΛ mF 2 nF , m− is natural= FMξ FMΛ FnF mF , by (211)

= F(Mξ MΛ nF

)mF

= Fυ mF ,

which proves that υ is a morphism in AMF. On the other hand, we have

n υ = n Mξ MΛ nF

= MFξ nF MΛ nF , n− is natural= MFξ MFΛ nF 2 nF , n− is natural= MFξ MFΛ MδF nF , by (210)

= MFξ Mδ MΛ nF , by (212)

= MΛ nF ,

and

υF Mδ = MξF MΛF nF 2 Mδ

= Mξ MΛF MFδ nF , n− is natural= MξF M(ΛF Fδ) nF

= MξF Mδ MΛ nF , by (212)

= MΛ nF .

Therefore υF Mδ = nυ and υ is a morphism of F-bicomodules. Hence S is a Maschke functor.(ii) ⇒ (i). Given an F-bicomodule (M,m, n), we denote by

Γ(M,m, n) : T S(M,m, n) = (MF,mF ,Mδ) // (M,m, n)

the splitting morphism of Θ(M,m,n) in the category of F-bicomodules. Here Θ− is the unit of theadjunction (S, T ). Since (F, δ, δ) is F-bicomodule, we put γ := Γ(F, δ, δ), thus γ δ = F . For any

object (X, dX) of the category AF, we consider the composition

φ(X, dX) : F (X)F (dX) // F 2(X)

γX // F (X)ξX // X .


We claim that φ− is a natural transformation which satisfies φ− η− = 11AF , where η− is theunit of the adjunction (S,T) given in (213). First of all, we have

φ(X, dX) η(X, dX) = ξX γX F (dX) dX

= ξX γX δX dX

= ξX dX = (X, dX),

for every object (X, dX) of AF. To see that φ(X, dX) is a morphism in AF, we can compute onone hand

dX φ(X, dX) = dX ξX γX F (dX)

= ξF (X) F (dX) γX F (dX), ξ− is natural

= ξF (X) γF (X) F 2(dX) F (dX), γ− is natural

= ξF (X) γF (X) F (δX) F (dX)

= ξF (X) δX γX F (dX), by (212) applied to γ

= γX F (dX)

and on the other hand,

Fφ(X, dX) δX = FξX FγX F 2(dX) δX= FξX FγX δF (X) F (dX), δ− is natural

= FξX δX γX F (dX), by (212) applied to γ

= γX F (dX).

Therefore, Fφ(X, dX) δX = dX φ(X, dX). Finally, if we consider a morphism f : (X, dX) →(Y, dY ) in AF, then

f φ(X, dX) = f ξX γX F (dX)

= ξY F (f) γX F (dX), ξ− is natural

= ξY γY F 2(f) F (dX), γ− is natural

= ξY γY F (dY ) F (f)= φ(Y, dY ) F (f),

which shows that φ− is a natural transformation.

In view of Remark 7.3, condition (iii) in Theorem 7.7 means that F is coseparable as acoalgebra in the monoidal category Funct (A,A) (see [8, 9]).

7.3. Coderivations and cointegrations

Let F = (F, δ, ξ) be a cotriple in A with underlying functor F ∈ Funct(A, A). Considera bicomodule (M,m, n) ∈ FMF. A coderivation from M to F is a natural transformationg : M −→ F such that

(215) δ g = Fg m + gF n.

The set of all coderivations from (M,m, n) is an additive group which we denote by Coder(M,F ).A coderivation g ∈ Coder(M,F ) is said to be inner if there exists a natural transformationλ : M → 11A such that

(216) g = λF n − Fλ m.

The sub-group of all inner coderivations will be denoted by InCoder(M,F ).Let (M,m, n) and (M ′,m′, n′) be two F-bicomodules. A left cointegration from (M,m, n) into

(M ′,m′, n′) is a natural transformation h : M → M′F which satisfies

(217) m′F h = Fh m, M ′δ h = n′F h + hF n.

The first equality means that h : S(M,m, n) = (M,m) → ST S(M,m, n) = (M′F,m′F ) is a

morphism in the category AMF. Right cointegrations are defined in a similar way. Since we are only

7.3. CODERIVATIONS AND COINTEGRATIONS 199

concerned with the left ones, we will not mention the word “left” before cointegration. The additivegroup of all cointegrations from (M,m, n) into (M ′,m′, n′) will be denoted by Coint(M,M ′).A cointegration h ∈ Coint(M,M ′) is said to be inner if there exists a natural transformationϕ : M →M ′ which satisfies

(218) m′ ϕ = Fϕ m, h = ϕF n − n′ ϕ.The first equality means that ϕ : (M,m) → (M ′,m′) is a morphism in the category AMF. Thesub-group of all inner cointegrations will be denoted by InCoint(M,M ′). The following propositionwas first stated for bimodules over ring extension in [84] and for bicomodules over corings in [76].For the sake of completeness, we give the proof.

Proposition 7.8. For any F-bicomodule (M,m, n), there is a natural isomorphism of additivegroups

Coint(M,F ) ∼ // Coder(M,F )

h // ξF h

Fg m goo

whose restriction to the inner sub-groups gives again an isomorphism

InCoint(M,F ) ∼= InCoder(M,F ).

Proof. We only show that the mutually inverse maps are well defined. Let h ∈ Coint(M,F ),and put g := ξF h. We have

δ g = δ ξF h= ξF 2 Fδ h, δ− is natural

= ξF 2 (δF h+ hF n

)= (ξF δ)F h + ξF 2 hF n

= h + ξF 2 hF n

and

FξF Fh m + ξF 2 hF n = FξF δF h + ξF 2 hF n = h+ ξF 2 hF n.

This shows that g ∈ Coder(M,F ). Conversely, given g ∈ Coder(M,F ), we put h = Fg m. Wefind

δF h = δF Fg m

= F 2g δM m, δ− is natural= F 2g Fm m, by (210)

= Fh m,

which shows the first equality of equation (217). Now,

Fδ h = Fδ Fg m

= F (δ g) m

= F(Fg m + gF n

)m

= F 2g Fm m + FgF Fn m, by (210) and (211)

= F 2g δM m + FgF mF n

= δF Fg m +(Fg m

)F n, δ− is natural

= δF h+ hF n,

which proves that h = Fg m ∈ Coint(M,F ).

Following [76], we will give in the next step the notion of universal cointegration and that ofuniversal coderivation.

Given (M,m, n) any F-bicomodule, consider the F-bicomodule (MF,mF ,Mδ), which is theimage of (M,m, n) under the functor T S. We call it the bicomodule induced by M . Since


n : (M,m, n) → (MF,mF ,Mδ) is a morphism of F-bicomodules, we obtain by Lemma 7.4 thefollowing sequence of F-bicomodules

(219) 0 // (M,m, n) n // (MF,mF ,Mδ)nc// (K(M), u, v) // 0 ,

where (K(M), u, v), nc denotes the cokernel of n in the category FMF. Note that this is stilla cokernel in the category AMF, after applying the forgetful functor S. Consider the naturaltransformation

w′ := MF − n Mξ : MF −→ MF.

It is easily checked that mF w′ = Fw′ mF , thus w′ is a morphism in the category AMF. Also,w′ satisfies w′ n = 0. So, by the universal property of cokernels, there exists a morphism in thecategory AMF, w : (K(M), v) → (MF,mF ) which makes the following diagram commutative

(220) Mn // MF

nc//

w′

K(M)

wwwo o o o o o

MF

Thus w nc = w′, and so nc w nc = nc. Hence nc w = K(M), since nc is an epimorphism.A universal cointegration into M is a cointegration u from K(M) into M such that every

cointegration into M factors through u. That is, u satisfies the following universal property:for every F-bicomodule (M ′,m′, n′) and every cointegration h ∈ Coint(M ′,M), there exists amorphism of F-bicomodules f : (M ′,m′, n′) → (K(M), u, v) such that h = u f .

Proposition 7.9. The morphism w is a cointegration into M (i.e. w ∈ Coint(K(M),M))which satisfies the following universal property: for every F-bicomodule (M ′,m′, n′) and everycointegration h ∈ Coint(M ′,M), there exists a morphism of F-bicomodules f : (M ′,m′, n′) →(K(M), u, v) such that h = w f . Moreover, the following are equivalent

(i) The sequence

0 // (M,m, n) n // (MF,mF ,Mδ) nc// (K(M), u, v) // 0

splits in the category of bicomodules FMF;(ii) the universal cointegration w : K(M) →MF is inner.

Proof. For the first statement, it is enough to show that w′ is cointegration into M , sincenc is an epimorphism. By construction, w′ satisfies the first equality in (217). The second equalityin (217), is obtained as follows

Mδ w′ = Mδ −Mδ n Mξ = Mδ − nF n Mξ

and

nF w′ + w′F Mδ = nF − nF n Mξ +Mδ − nF MξF Mδ

= Mδ − nF n Mξ

= Mδ w′

The fact that w is universal follows from the following isomorphism of additive groups

(221) Hom FMF

(M ′, K(M)

)∼ // Coint(M ′,M)

ϕ // w ϕnc h h

oo

The proof is an easy computation. Now we check the equivalence of the statements.(i) ⇒ (ii). Let us denote by λ : (K(M), u, v) → (MF,mF ,Mδ) the right inverse of nc in the

category FMF, i.e. nc λ = K(M). Consider the composition

ϕ : K(M) λ // MFMξ // M .

7.3. CODERIVATIONS AND COINTEGRATIONS 201

Then we have

m ϕ = m Mξ λ = FMξ mF λ = FMξ Fλ u = F(Mξ λ

) u = Fϕ u,

which entails that ϕ is a morphism in AMF. The cointegration w is inner by ϕ. Namely,

ϕF v− n ϕ = MξF λF v− n Mξ λ= MξF Mδ λ− n Mξ λ= λ− n Mξ λ

=(MF − n Mξ

) λ

= w nc λ = w.

(ii) ⇒ (i). Suppose that there exists a morphism β : K(M) → M in AMF such that w =βF v− n β. Consider the natural transformation

Γ : K(M) v // K(M)FβF // MF .

Then we find nc Γ = nc βF v = nc w+ nc n β = nc w = K(M). Furthermore, Γ isa morphism in the category of bicomodules FMF, as the following commutative diagrams show

K(M) v //

u

K(M)FβF //

uF

MF

mF

FK(F )

Fv// FK(M)F

FβF

// FMF

K(M) v //

v

K(M)FβF //

K(M)δ

K(M)F

Mδ

K(M)FvF

// K(M)F 2βF2

// MF 2

Therefore the listed sequence splits in the category FMF.

From now on, w denotes the universal cointegration into the F-bicomodule (F, δ, δ). That isw : K(F ) → F 2 with properties w δc = F 2 − δ Fξ and δc w = K(F ), where

0 // (F, δ, δ) δ // (F 2, δF , F δ)δc// (K(F ), u, v) // 0

is the canonical sequence. Consider the natural transformation d : K(F ) → F defined by d : =Fξ w − ξF w.

Lemma 7.10. The morphism d is a coderivation with the following universal property: forevery F-bicomodule (M,m, n) and every coderivation g ∈ Coder(M,F ), there exists a naturaltransformation g′ : M → K(F ) such that d g′ = g.

Proof. First, observe thatd δc = Fξ − ξF

as wδc = F 2−δFξ. Now, since δc is an epimorphism, in order to get that d is a coderivation,it is enough to check that e := Fξ − ξF is a coderivation and in fact we have

Fe δF + eF Fδ = F 2ξ δF − (Fξ δ)F + F (ξF δ)− ξF 2 Fδ= F 2ξ δF − ξF 2 Fδ= δ Fξ − δ ξF = δ e.

Take g ∈ Coder(M,F ). By Proposition 7.8, Fg m ∈ Coint(M,F ) so we can apply Proposition7.9 to obtain a morphism of F-bicomodules f : M → K (F ) such that Fg m = w f . We have

d f = e w f= e Fg m

= Fξ Fg m− ξF Fg m

= F (ξ g) m− ξF (δ g − gF n)= F (ξ g) m− ξF δ g − ξF gF n

= F (ξ g) m− g − (ξ g)F n.

Once we observe that ξ g = 0 (since g ∈ Coder (M,F )), it is clear that we can set g′ = −f .


Corollary 7.11. Let F = (F, δ, ξ) be a cotriple on a Grothendieck category A such thatF is right exact and commutes with direct sums. Consider the universal cointegration w and theuniversal coderivation d associated to the F-bicomodule (F, δ, δ). The following statements areequivalent

(i) The sequence

0 // Fδ // F 2 δc

// K(F ) // 0

is a split sequence in the category of bicomodules FMF;(ii) the universal cointegration w is inner;(iii) the universal coderivation d is inner.

Proof. The equivalence (i) ⇔ (ii) is consequence of Proposition 7.9. Let us check the

equivalence between (ii) and (iii).(ii) ⇒ (iii). We know there exists a morphism ϕ : K(F ) → F in AMF such that w = ϕF v−δ ϕ. We have

ξF w = ξF ϕF v− ϕ =(ξ ϕ

)F v− ϕ

Fξ w = Fξ ϕF v− ϕ.

Hence

d = Fξ ϕF v− ϕ−(ξ ϕ

)F v + ϕ

= ϕ K(F )ξ v−(ξ ϕ

)F v

= ϕ−(ξ ϕ

)F v.

But F(ξ ϕ

) u = ϕ, as ϕ is a morphism in AMF, which proves that d is inner by −ξ ϕ.

(iii) ⇒ (ii). Let us denote by λ : K(F ) → 11A the natural transformation which satisfies d =λF v− Fλ u. We define the map ψ as the composition ψ = Fλ u : K(F ) → FK(F ) → F .Then ψ satisfies

δ ψ = δ Fλ u = F 2λ δK(F ) u = F 2λ Fu u = Fψ u,

that is, ψ is a morphism in AMF. The universal cointegration is inner by −ψ, as the followingcomputations show

ψF v− δ ψ =(Fλ u

)F v− δ Fλ u

= FλF uF v− δ Fλ u

= FλF Fv u− δ Fλ u

= F(λF v

) u− δ Fλ u

= F((

Fξ − ξF

) w + Fλ u

) u− δ Fλ u

= F 2ξ Fw u− FξF Fw u + F 2λ Fu u− δ Fλ u

= F 2ξ δF w − FξF δF w + δ Fλ u− δ Fλ u

= F 2ξ δF w −(Fξ δ

)F w

= δ Fξ w − w =(δ Fξ − F 2

) w = −w δc w = −w.

7.4. Cohomology for bicomodules

The following Lemma, which will be used in the sequel, was in part proven in [45, Theorem3.4].

7.4. COHOMOLOGY FOR BICOMODULES 203

Lemma 7.12. Let A and B be two preadditive categories with cokernels, and F : A → Ba covariant functor with right adjoint functor G : B → A. Denote by χ and θ respectively, thecounit and unit of this adjunction. Let E0 be the injective class of all cosplit sequences in B, andput E = F−1(E0). For every object M ∈ A, the following are equivalent

(i) M is F -injective;(ii) M is E -injective;(iii) the unit at M , θM : M → GF (M), is a split-mono in A.

In particular every object of the form G(N) is E -injective, for every object N ∈ B. Moreover thefunctor F is Maschke if and only if the class of E -injective objects coincides with the class of allobjects of A.

Proof. (i) ⇒ (iii). We know by properties of the adjunction that χF (M) F (θM ) = F (M).Since M is F -injective, θM has a left inverse.(iii) ⇒ (ii). Let us denote by γ : GF (M) →M the left inverse of θM . For any sequence

E : Xi // X ′ j // X ′′

in E , we need to prove that its corresponding sequence of abelian groups

HomA(X ′′,M) // HomA(X ′,M) // HomA(X,M)

is exact (in the usual sense). Given such E in E , we have a commutative diagram in B

F (X)F (i) // F (X ′)

F (j) //

F (i)c

F (X ′′)

Coker(F (i))l

77ooooooooooo

where ` is split as monomorphism by `′. Let τ : X ′ →M be a morphism in A, such that τ i = 0.Then there exists a morphism g : Coker(F (i)) → F (M) in B such that g F (i)c = F (τ). Thisleads to the composition

X ′′

θX′′

α //_________ M

GF (X ′′)G(g`′) // GF (M)

γ

OO

The morphism α satisfies

α j = γ G(g `′) θX′′ j= γ G(g `′) GF (j) θX′

= γ G(g `′ F (j)

) θX′

= γ G(g `′ ` F (i)c

) θX′

= γ G(g F (i)c

) θX′

= γ GF (τ) θX′

= γ θM τ = τ

which proves the exactness of the sequence of abelian groups.(ii) ⇒ (i) Let i : X → X ′ be a morphism of A such that F (i) has a left inverse. The latter condi-

tion means that 0 // F (X)F (i) // F (X ′) is a cosplit sequence in B. Thus 0 // X

i // X ′

is a sequence in E . Therefore, the corresponding sequence of abelian groups

HomA

(X ′, M

)// HomA

(X, M

)// 0

is exact. Whence HomA

(i, M

)is surjective and so M is F -injective.


Consider in the category of bicomodules AMF the class E0 of all co-split sequences. This isan injective class, as AMF is an additive category with cokernels. As we have mentioned, thecorresponding class of E0-injective objects coincides with the class of all objects of AMF. Denote

by E := S−1(E0

)the class of sequences E in the category FMF such that S(E) is a sequence

in E0, as we have pointed out E is also an injective class.

Proposition 7.13. Let (M,m, n) be an F-bicomodule. The following statements are equiv-alent

(i) (M,m, n) is E -injective;(ii) (M,m, n) is S-injective;(iii) the unit Θ(M,m, n) of the adjunction (S, T ) at (M,m, n), stated in (214), is a split monomor-

phism.

In particular every bicomodule of the form T (N, r) is E -injective, for every bicomodule (N, r) ∈AMF, and so is every induced F-bicomodule T S(M,m, n) = (MF,mF ,Mδ).

Proof. Follows immediately from Lemma 7.12.

Fix a cotriple F = (F, δ, ξ) on a Grothendieck category A with F ∈ Funct(A, A). Forevery F-bicomodule (M,m, n) and each i ≥ 1, we consider the i-th induced F-bicomodule(MF i,mF i ,MF i−1δ).

Proposition 7.14. Let (M,m, n) be any F-bicomodule. The following sequence in thecategory of F-bicomodules

(222) 0 // Mn // MF

d0// MF 2 d1

// · · · // MFn+1 dn// MFn+2 // · · ·

where d0 = Mδ − nF and recursively

(223) dn+1 = dnF + (−1)n+1MFn+1δ, n = 0, 1, 2, . . .

defines an E -injective resolution for (M,m, n).

Proof. Let us denote by E(M) the sequence defined in (222). One can easily check thatthe family of morphisms

un := (−1)n+1MFnξ : MFn+1 −→MFn

in AMF, defines a contracting homotopy for S(E(M)). This implies by [80, Lemma 2.4] thatS(E(M)) is sequence in E0. Hence E(M) is in E .

Let (N, r, s) be another F-bicomodule and denote by ExtE

(N, M

)the homology of the

complex

(224) 0 // HomFMF

(N, MF

)// HomFMF

(N, MF 2

)// · · ·

obtained by applying the functor HomFMF

(N, −

)to the E -injective resolution of M given in

(222). Using the natural isomorphism stated in Lemma 7.5, we can show that the complex (224)is isomorphic to

(225) 0 // HomAMF

(N, M

)∂0// HomAMF

(N, MF

)∂1// · · ·

where

∂0(f) = fF s− n f,∂1(f) = Mδ f − fF s− nF f,

∂n(f) =n−1∑i=0

(−1)iMF iδFn−i−1 f + (−1)nfF s− nFn f, n = 2, 3, ...

7.4. COHOMOLOGY FOR BICOMODULES 205

In particular, we have

Ker (∂1) = f : (N, r) → (MF,mF )|Mδ f = fF s + nF fIm (∂0) = f : (N, r) → (MF,mF )| f = ϕF s− n ϕ, for some ϕ : (N, r) → (M,m)

This means that the 1-cocycle are cointegrations and the 1-coboundaries are inner cointegrations.Thus

(226) Ext1E(N, M

)∼= Coint(N,M)/InCoint(N,M).

The pair (T S,Θ−) form a resolvent pair in the sense of [80, Proposition 2.10] (dual to [62,Corollary 2.3,page 16]) for the injective class E . Since FMF has cokernels, [80, Lemma 2.11]implies that the cokernels constructed in (219) lead to a functor

K : FMF → FMF,

and a natural transformationT S → K.

Furthermore, K(E) is a sequence in E , whenever E is a sequence in E . By the isomorphism

given in (221), we find that HomFMF

(N, K(E)

)∼= Coint(N,E) is an exact sequence of abelian

groups, for every E -projective F-bicomodule N and every sequence E in E . On the other hand,given an E -injective F-bicomodule M , then K(M) is clearly E -injective. Thus Coint(E,M),which by (221) is isomorphic to HomFMF

(E, K(M)

), is an exact sequence of abelian groups.

This proves that the E -derived functor of the bifunctor Coint(−,−) can be constructed. For twoF-bicomodules N and M , let H∗(N,M) be this E -derived functor which can be computed usingthe E -injective resolution given in Proposition 7.14. Using this time the natural isomorphisms of(221) and the fact that T S(M) are E -injective for every F-bicomodule M , we can easily showthat

ExtnE(N, K(M)

)∼= Hn(N,M), n ≥ 0(227)

Extn+1E

(N, M

)∼= ExtnE

(N, K(M)

), n ≥ 1.(228)

By Propositions 7.13 and 7.9, and the isomorphisms given in (226), (227), and (228), we have

Corollary 7.15. For a F-bicomodule (M,m, n), the following are equivalent

(i) M is E -injective;(ii) M is S-injective;(iii) the sequence

0 // Mn // MF

nc// K(M) // 0

splits in the category of bicomodules FMF;(iv) the universal cointegration from K(M) into M is inner;(v) every cointegration into M is inner.

Now we can formulate a characterization of cotriples with a separable forgetful functor bymeans of the cohomology groups of their bicomodules.

Theorem 7.16. Let A be a Grothendieck category and F = (F, δ, ξ) a cotriple on A with

universal cogenerator the adjunction S : AF // A : Too . If F is right exact and preserves direct

sums, then the following are equivalent

(i) S : AF → A is a separable functor;(ii) S′ : FA → A is a separable functor;(iii) S : FMF → AMF is a Maschke functor;(iv) S ′ : FMF → FMA is a Maschke functor;(v) δ : F → F 2 is a split monomorphism in the category of bicomodules FMF;(vi) (F, δ, δ) is E -injective F-bicomodule;(vii) the universal coderivation from K(F ) into F is inner;(viii) every coderivation into F is inner;


(ix) all cointegrations between F-bicomodules are inner;

(x) ExtnE(−, −

)= 0 for all n ≥ 1;

(xi) Hn(N,F ) = 0 for all F-bicomodule N and all n ≥ 1.

Proof. Corollary 7.15, Proposition 7.8, and properties of Ext give the equivalences (iii) ⇔(ix), (iii) ⇔ (x), (vi) ⇔ (xi), (vi) ⇔ (viii). Proposition 7.13 gives the equivalence (vi) ⇔ (v),and finally Theorem 7.7 gives the equivalences (i) ⇔ (iii) ⇔ (v). The equivalence with (ii) and(iv) follows from the left-right symmetry in (v).

7.5. Applications

We present in this section two different applications of Theorem 7.16. The first one is devotedto coseparable corings [76], where of course the cotriple is defined by the tensor product over analgebra. The second deals with coalgebra coextensions over fields, and the cotriple is defined usingcotensor product. Here we obtain Nakajima’s results [94] without requiring the cocommutativity ofthe base coalgebra. This condition is however replaced, in our case, by assuming that the extendedcoalgebra is left coflat.

7.5.1. Coseparable corings. Let k be commutative ring with unit. In what follows all algebrasare k-algebras, and all bimodules over algebras are assumed to be central k-bimodules.Let R be an algebra. In this sub-section the unadorned symbol − ⊗ − between R-bimodulesand R-bilinear maps denotes the tensor product − ⊗R −. Consider an R-coring (C,∆, ε). Wedenote as usual by CMC the category of C-bicomodules. The objects are three-tuples (M,%M , λM )consisting of an R-bimodule M and two R-bilinear maps %M : M → M ⊗ C (right C-coaction),λM : M → C⊗M (left C-coaction).

Recall from Theorem 3.5 that F := (F, δ, ξ) where F = −⊗C : MR →MR, δ = −⊗∆, andξ = −⊗ε, is a cotriple on the category of right R-modulesMR. Moreover, F ∈ Funct(MR,MR),since F has a right adjoint HomR(C,−), by the Eilenberg-Watts Theorem 2.13.

Given any F-bicomodule (M,m, n) we can use the Eilenberg-Watts Theorem to find a naturalisomorphism

(229) kM− : M −→ −⊗M(R)

satisfying (− ⊗ ψR) kM− = kM ′− ψ for every natural transformation ψ : M → M ′ where

(M ′,m′, n′) is another F-bicomodule. With the help of this natural isomorphism we can establisha functor

G : FMF // CMC

(M,m, n) //(M(R), %M(R), λM(R)

)f // fR

where the C-coactions are defined by %M(R) = mR and λM(R) = kMF (R) nR.

Conversely, given any C-bicomodule (M,%M , λM ), we clearly obtain an F-bicomodule defined

by the three-tuple

(−⊗M,−⊗ %M ,

(kMF)−1

(−⊗ λM ))

. This entails an inverse functor, up

to the natural isomorphisms k−−, of the functor G . Henceforth, G is an equivalence between

the categories FMF and CMC. It is then obvious that δ is a split-mono in the category ofF-bicomodules if and only if ∆ is a split-mono in the category of C-bicomodules. It is wellknown (see [36]) that this later condition is satisfied if and only if the right coaction forgetfulfunctor is separable. Since a coring is said to be coseparable if ∆ is split-mono in the categoryof C-bicomodules, this means that C is coseparable if and only if the functor forgetting the rightcoaction is separable.

Consider two C-bicomodules (M,%M , λM ) and (N, %N , λN ). Following [76], a R-bilinear mapg : M → C is said to be coderivation if it satisfies

∆ g = (g ⊗ C) %M + (C⊗ g) λMThe coderivation g is said to be an inner coderivation if there exists a R-bilinear map γ : M → Rsuch that g = (C ⊗ γ) λM − (γ ⊗ C) %M . We denote by CoderC(M,A) the abelian group of

7.5. APPLICATIONS 207

all coderivations from M to C. A (left) cointegration from N into M is an R-bilinear morphismf : N → C⊗M such that

(∆⊗ C) f = (C⊗ λM ) f + (C⊗ f) λNThe cointegration f is said to be an inner cointegration if there exists an R-bilinear map ϕ : N →M satisfying

%M ϕ = (ϕ⊗ C) %N , and f = (C⊗ ϕ) λN − λM ϕThe abelian group of all cointegrations from N into M will be denoted by CointC(N,M).

Cointegrations and coderivations in both categories of bicomodules FMF and CMC are con-nected by the following isomorphisms of abelian groups

Coder(M,F )∼= // CoderC

(M(R),C

)g // gR

(−⊗ g) kM− goo

and

Coint(N,M)∼= // CointC

(N(R),M(R)

)f // kMF (R) fR(

kMF (−)

)−1 (−⊗ f) kN− foo

where the isomorphism F (R) ∼= C was used as an isomorphism of R-corings. The restrictions ofthe above isomorphisms to the sub-groups of inner coderivations or inner cointegrations are alsoisomorphisms.

Applying Theorem 7.16 to this situation, we obtain immeadiately the following.

Corollary 7.17 ([76, Theorem 3.10]). For any R-coring (C,∆, ε), the following are equiv-alent

(i) the forgetful functor S : MC →MR from the category of right C-comodules to the categoryof right R-modules is a separable functor;

(ii) the forgetful functor S : CM→ RM from the category of right C-comodules to the categoryof right R-modules is a separable functor;

(iii) the forgetful functor CMC → CMR is a Maschke functor;(iv) the forgetful functor CMC → RMC is a Maschke functor;(v) the short exact sequence

0 // C∆ // C⊗ C

∆c// Ω(C) // 0

splits in the category of bicomodules CMC;(vi) C is E -injective, where E is the injective class in CMC whose sequences split in the category

of R-bimodules RMR;(vii) the universal coderivation from Ω(C) into C is inner;(viii) every coderivation into C is inner;(ix) all cointegrations between C-bicomodules are inner;

(x) ExtnE(−, −

)= 0 for all n ≥ 1;

(xi) Hn(N,C) = 0 for all C-bicomodule N and all n ≥ 1;

7.5.2. Coseparable coalgebra co-extensions. In what follows k is assumed to be a field.The unadorned symbol ⊗ between k-vector spaces means the tensor product ⊗k. Let A, C aretwo k-coalgebras, and consider φ : A → C a morphism of k-coalgebras. This defines a pair ofadjoint functors (O,−⊗C A),

MC−⊗CA //MA

Ooo


between the categories of right comodules and where −⊗C − is the cotensor product over C. Inthe remainder, we denote this bifunctor by −− := −⊗C −. Notice that −− is associative (upto natural isomorphism), as C is a k-coalgebra and k is a field. From now on, we assume that−A : MC →MA is right exact, and thus exact. Put F := O(−A) : MC →MC , since MC

is a Grothendieck category we can construct the category Funct(MC ,MC), and we have in thiscase that F ∈ Funct(MC ,MC). Let us denote by ∆ : A → AA the resulting map from theuniversal property of kernels. This is in fact an A-bicomodule map, and thus a C-bicomodule mapby applying O. Furthermore, we have

(A∆) ∆ = (∆A) ∆(φA) ∆ = (Aφ) ∆ = A (up to isomorphism).

Using these equalities, we can easily check that there is a cotriple F := (F, δ, ξ) in the categoryof right C-comodules MC , where δ and ξ are defined by the following commutative diagrams ofnatural transformations

−A−∆ // −AA,

Fδ //_______ F 2

−A−φ // −C

∼=

Fξ //_____ 11MC

Given any F-bicomodule (M,m, n), we know that M : MC →MC is right exact and preservesdirect sums. If M is assumed to be left exact, then by [69, Theorem 3.5], M(C) := M is a C-bicomodule, and there is a natural isomorphism

(230) ΥM− : M

∼= // −M,

which satisfies (−βC) ΥM− = ΥN

− β, for every natural transformation β : M → N with

N ∈ Funct(MC ,MC) and N an exact functor. The natural transformations m and n induce bythis isomorphism a structure of A-bicomodule on M . The right and left A-coactions are given by

MmC //

%M ''OOOOOOO MA

eqkM, A

M ⊗A

MnC //

λM,,YYYYYYYYYYYYYYYYYY MF (C)

ΥMF (C)

∼=// F (C)M ∼= AM

eqkA, M

A⊗M

where eqkX, Y is the equalizer map, that is the kernel of eqX,Y : X ⊗ Y //X ⊗ C ⊗ Y// defined

by eqX,Y = %X ⊗ Y − X ⊗ λY for every right C-comodule X and left C-comodule Y . Thecounitary conditions of these new A-coactions are easily verified, while the coassociativity andcompatibility conditions need routine and long computations using properties of cotensor productsover coalgebras over fields. Let us denote by FEF the full subcategory of FMF whose objects areF-bicomodules (M,m, n) such that M : MC → MC is an exact functor which commutes withdirect sums. For instance (F, δ, ξ) is an object of this category.

The above arguments establish a functor from the subcategory of F-bicomodules FEF to thecategory of A-bicomodule, sending

(231) F : FEF −→ AMA,(M,m, n

)→(M,%M , λM

), f → fC .

For every F-bicomodule M ∈ FEF, it is clear that F (M) = M is a coflat left C-comodule.Conversely, given any A-bicomodule (N, %N , λN ) such that the underlying left C-comodule

CN is coflat, then we have a functor −N : MC → MC which is exact and preserves directsums together with two natural transformations

−N−λ′N // −AN, −N

−%′N // −NA,

7.5. APPLICATIONS 209

where λ′N and %′N are C-bicolinear defined by universal property

NλN //

λ′N ''OOOOOOO A⊗N

AN

eqkA, N

OO N%N //

%′N ''OOOOOOO N ⊗A

NA

eqkN, A

OO

By definition and the properties of the cotensor product, λ′N and %′N satisfy the following equalities

(∆N) λ′N = (Aλ′N ) λ′N , (φN) λ′N = N (up to isomorphism)(N∆) %′N = (%′NA) %′N , (Nφ) %′N = N (up to isomorphism)(A%′N ) λ′N = (λ′NA) %′N .

Consider the obtained three-tuple (N, r, s), where N := −N : MC →MC is a functor, and r :=−%′N : N → FN , s := −λ′N : N → NF are two natural transformation. Since N is assumedto be a coflat left C-comodule, the previous equalities show that (N, r, s) is actually an objectof the category FEF, whose image by F is isomorphic to the initial A-bicomodule (N, %N , λN ),via the natural isomorphisms Υ−

−. Now, given an A-bicolinear morphism g : (N, %N , λN ) →(N ′, %N ′ , λN ′), we get an F-bicomodule morphism g := −g : N → N ′. This shows that theabove constructions are functorial.

In conclusion, we have shown that the functor F defined in (231) establishes an equivalenceof categories between FEF and ACA, where the latter is the full subcategory of the categoryof A-bicomodules AMA whose objects are coflat left C-comodules after forgetting the right C-coaction.

Recall from [94] that A is said to be a separable C-coalgebra if the A-bicolinear map ∆ :A→ AA is a split monomorphism in the category of A-bicomodules. By [69, Theorem 5.6] thisis equivalent to saying that the forgetful functor O is a separable functor. Using the equivalenceof categories established above, it is easy to check that δ is a split monomorphism in FEF (orequivalently in FMF) if and only if ∆ is a split monomorphism in ACA (or equivalently in AMA).

Given two A-bicomodules (M,%M , λM ) and (N, %N , λN ), a C-bicolinear map g : M → A issaid to be C-coderivation if it satisfies

∆ g = (gA) %′M + (Ag) λ′M

The C-coderivation g is said to be an inner C-coderivation if there exists a C-bicolinear mapγ : M → C such that g = (Aγ) λ′M − (γA) %′M . We denote by CoderC(M,A) the abeliangroup of all C-derivations from M to A. A (left) C-cointegration from N into M is a morphismof C-A bicomodules f : N → AM such that

(∆A) f = (Aλ′M ) f + (Af) λ′N

The C-cointegration f is said to be an inner C-cointegration if there exists a C-bicolinear mapϕ : N →M satisfying

%′M ϕ = (ϕA) %′N , and f = (Aϕ) λ′N − λ′M ϕ

The abelian group of all C-cointegrations from N into M will be denoted by CointC(N,M).Let (M,m, n) and (N, r, s) be two F-bicomodules in FEF and consider their associated A-

bicomodule via the above equivalence of categories F :

(M(C) := M,%M , λM ) and (N(C) := N, %′N , λ′N ).

We have an abelian group isomorphism

Coder(M,F )∼= // CoderC

(M(C), A

)g // ιA gC

(−g) ΥM− goo


where ι− : C− → 11MC is the obvious natural isomorphism. The isomorphism between thegroups of cointegrations is given by

Coint(N,M)∼= // CointC

(N(C),M(C)

)f // (ιAM(C)) ΥM

F (C) fC(ΥMF (−)

)−1 (−f) ΥN

− foo

Of course the restrictions of those isomorphisms to the subgroups of inner cointegrations orinner coderivations are also group isomorphisms. Applying Theorem 7.16, we arrive at the followingresult.

Corollary 7.18 (compare to [94, Theorem 1.2]). Let φ : A → C be a morphism ofk-coalgebras over a field k. Assume that CA is a coflat left C-comodule. The following areequivalent

(i) A is a separable C-coalgebra;(ii) for any A-bicomodule M such that CM is coflat, every C-coderivation from M to A is inner;(iii) for any pair of A-bicomodules M and N such that CM and CN are coflat, every C-

cointegration from M into N is inner.

References

All results in this chapter are deduced from the author’s joint work with L. El Kaoutit [67].

Chapter 8Co-Frobenius Corings and related

Functors

In this Chapter, we study co-Frobenius and, more generally, quasi co-Frobenius corings overarbitrary base rings and over PF base rings in particular. We generalize some results about (quasi)co-Frobenius coalgebras to the case of non-commutative base rings and give several new charac-terizations for co-Frobenius and quasi co-Frobenius corings, some of them are new even in thecoalgebra situation.

After recalling some elementary results and definitions in the first Section, we define in Sec-tion 8.2 the notion of an action of a class of natural transformations on a category. In this way,we can weaken the definition of a pair of adjoint functors, leading to the notion of a pair of lo-cally adjoint functors. In Section 8.3, we study the induction functor of a coring and its left andright adjoint. We give several descriptions of the sets of natural transformations between thosefunctors. These results are then used in the final section, where we construct Morita contexts thatdescribe the Frobenius properties of a coring and a second kind of Morita contexts that discribethe adjointness of functors. Comparing both Morita contexts by means of the results obtained inSection 8.3, we obtain our main result that characterizes Frobenius corings, co-Frobenius coringsand quasi co-Frobenius corings.

8.1. Elementary results

8.1.1. Direct sums and direct products. Let I be any index set and M an object in acategory with products and coproducts. Troughout this chapter, we will denote M (I) for thecoproduct (direct sum) and M I for the product of a set of copies of M indexed by I.

Let us state two elementary results from module theory, which will be useful for our theory.We give a short proof for sake of completeness.

Lemma 8.1. Let A be any category and A and B two objects in A. Let I be any index setand suppose that A(I) and BI exist. Then

Hom(A(I), B) ∼= Hom(A,BI) ∼= (Hom(A,B))I .

211

212 CHAPTER 8. CO-FROBENIUS CORINGS AND RELATED FUNCTORS

Proof. Denote by ι` : A → A(I) the canonical monomorphisms and by π` : BI → B thecanonical epimorphisms, where ` ∈ I. Consider the following diagram.

Af //

ι`

f`

''NNNNNNNNNNNNNNN BI

π`

A(I)

f// B

Any morphism f ∈ Hom(A,BI) as well as any morphism f ∈ Hom(A(I), B) is completelydetermined by the family of morphisms f` : A→ B, ` ∈ I.

Lemma 8.2. Consider a ring morphism B → A and take any M ∈ AMB, then there is an

isomorphism AHomB(A,M)ξM∼= MB := m ∈M | bm = mb, for all b ∈ B.

Proof. For any f ∈ AHomB(A,M), we obtain bf(1A) = f(b) = f(1A)b, consequentlyf(1A) ∈ MB. Conversely for any x ∈ MB, define fx ∈ AHomB(A,M) as fx(a) = ax. One caneasily check that this correspondence is bijective.

8.1.2. Frobenius corings. An A-coring is said to be Frobenius if and only if C and ∗C areisomorphic in the category of A-∗C bimodules. It is well known that this notion is left-rightsymmetric, i.e. C is Frobenius if and only if C and C∗ are isomorphic in the category of C∗-Abimodules (see also Corollary 8.37).

Generalizing the notion of co-Frobenius and quasi co-Frobenius coalgebras in a natural way tothe setting of corings, one obtains the following defintions.

Definition 8.3. An A-coring is called left co-Frobenius if and only if there exists a monomor-phism j : C → ∗C in the category of A-∗C bimodules. Similarly, one introduces right co-Frobeniuscorings as those corings for which there exists a monomorphism C → C∗ of C∗-A bimodules.

We say that an A-coring C is left quasi co-Frobenius if and only if there exists an A-∗C bimodulemonomorphism j : C → (∗C)I for some index set I. The coring C is right quasi co-Frobenius ifthere exists a C∗-A bimodule monomorphism C → (C∗)I for some index set I.

Remark 8.4. Usually one defines a left quasi co-Frobenius coalgebra over a field k as a k-coalgebra C such that there exists a monomorphism of left C∗-modules j : C → (C∗)(I). If weconsider C as a k-coring C, then notice first that the multiplication in C∗ for the coalgebra Cis opposite to the multiplication in ∗C for the coalgebra C if we use the convention introduced inExample 1.46. Secondly, it was proven in [71, Theorem 1.3] that the existence of a left C∗-linearmonomorphism j : C → (C∗)(I) is equivalent to the existence of a left C∗-linear monomorphismj′ : C → (C∗)I .

A pair of functors

CF // DG

oo

is said to be Frobenius if and only if F is at the same time a left and right adjoint for G.

8.1.3. Morita contexts. Recall from Chapter 6 (see also [93, Remarks p 389, Examples 1.2],[21, Remark 3.2]) that a Morita context can be identified with a k-linear category with two objectsa and b. The algebras of the Morita contexts are End(a) and End(b), the connecting bimodulesare Hom(a, b) and Hom(b, a) and multiplication and bimodule maps are given by composition.We denote this context as follows

N(a, b) = (End(a),End(b),Hom(b, a),Hom(a, b), , •).If j ∈ Hom(a, b) and ∈ Hom(b, a) are such that j = a and j = b, then we call (j, ),as in Chapter 6, a pair of invertible elements. This means that j and are inverse isomorphismsbetween a and b. As a Morita context M with an invertible pair (j, ) is always strict, we say thatM is strict by (j, ).

A morphism of Morita contexts

m : M = (A,B, P,Q, µ, τ) → M′ = (A′, B′, P ′, Q′, µ′, τ ′)

8.2. LOCALLY ADJOINT FUNCTORS 213

consists of two algebra maps m1 : A → A′ and m2 : B → B′, an A′-B′ bimodule map m3 :P → P ′ and a B′-A′ bimodule map m4 : Q → Q′ such that m1 µ = µ′ (m3 ⊗B m4) andm2 τ = τ ′(m4 ⊗A m3).

There are two canonical ways to construct new Morita contexts out of an existing one, withoutadding or removing any information.

(i) The opposite of a Morita context M = (A,B, P,Q, µ, τ) is the Morita context Mop =(Aop, Bop, Q, P, µop, τop), where µop(q⊗Bop p) = µ(p⊗B q) and τop(p⊗Aop q) = τ(q⊗A p).An anti-morphism of Morita contexts m : M → M′ is a morphism m : M → M′op. Itconsists of two algebra maps m1 : A→ A′op and m2 : B → B′op, an A′-B′ bimodule mapm3 : Q→ P ′ and a B′-A′ bimodule map m4 : P → Q′ such that m1 µ = µ′op (m4⊗B m3)and m2 τ = τ ′op(m3 ⊗A m4).

(ii) The twisted of a Morita context M = (A,B, P,Q, µ, τ) is the Morita context Mt = (B,A,Q, P, τ, µ).

8.2. Locally adjoint functors

8.2.1. Action of a set of natural transformations on a category. Let F : C → D be afunctor and consider a semigroup of natural transformations Φ ⊂ Nat(F, F ). We define for allα ∈ Φ, C,C ′ ∈ C and f : F (C) → F (C ′) in D,

α · f := αC′ f : F (C) → F (C ′).

This defines an action of Φ on HomD(F (C), F (C ′)), indeed for α, β ∈ Φ, C,C ′ ∈ C and f :F (C) → F (C ′) in D, we have

(α β) · f = (α β)C′ f = αC′ βC′ f = α · (β · f).

Since this action exists for all choices of C,C ′ ∈ C, we will say that Φ acts on C. We say that Φacts unital on C, if there exists an element e ∈ Φ such that for all f : F (C) → F (C ′) in D withC,C ′ ∈ C, we have e · f = f . We say that Φ acts with local units on C if an only if there exists agenerating subcategory E ⊂ C, such that for all f : F (E) → F (C) in D with C ∈ C and E ∈ E ,there exists an eE ∈ Φ such that eE · f = f .

Example 8.5 (action of a ring with local units). Let R be a (non-unital) B-ring, and Mbe a full subcategory of MR. Consider the forgetful functor U : M→MB. Then we have a mapRB → Nat(U,U). Indeed take any M ∈ M, e ∈ RB and define αeM (m) = m · e for all m ∈ M ,then αeM is a right B-linear map that is natural in M . Now consider a subcategory N ⊂ MB.Following the procedure of this section, RB, viewed as a subset of Nat(U,U), has the followingaction on N . For any right B-linear map f : N → M with N ∈ N and M ∈ M we definee · f = αeM f . Suppose now that N generates M in MB. Then it follows from Theorem 2.34that R acts with local units on all objects in M if and only if RB (as natural transformations)acts unitally on N , in other words, if and only if RB (as natural transformations) acts with localunits on M.

Consider now a functor G : D → C and let Γ be a semigroup of natural transformationsΓ ⊂ Nat(G,G). We define for all α ∈ Γ and f : C → G(D) with C ∈ C and D ∈ D,

α · f := αD f : C → G(D).

One can easily check that defines an action of Γ on HomC(C,G(D)). Since this action exists forall choices of C, we will say that Γ acts on C. We say that Γ acts unital on C, if there exists anelement e ∈ Γ such that for all f : C → G(D) in C with C ∈ C and D ∈ D, we have e · f = f .We say that Γ acts with local units on C if an only if there exists a generating subcategory E ⊂ C,such that for all f : E → G(D) in C with E ∈ E and D ∈ D, there exists an eE ∈ Γ such thateE · f = f .

Example 8.6. Let R be a ring (with unit), and MR the category of right R-modules. Denoteby 1 the category with a unique object ∗ and a unique (endo)morphism. Consider a functorG : 1 →MR. Then G is completely determined by G(∗) = M , and furthermore Nat(G,G) ∼= EndR(M).Therefore, for any semigroup Γ of endomorphisms of M , we have a natural map Γ → Nat(G,G).We know that R is a generator for MR and HomR(R,G(∗)) = HomR(R,M) ∼= M . The action


Γ (considered as set of natural transformations) on M ∈MR coincides with the canonical actionof Γ (considered as set of endomorphisms) on M .

8.2.2. Locally adjoint functors. We now consider CAT, the bicategory whose 0-cells arecategories, 1-cells are functors and 2-cells are natural transformations. Then CAT is in fact evena 2-category. To avoid set-theoretical problems, we will consider a sub-bicategory B of CAT, suchthat the natural transformations between each pair of functors form a set. In other words, B,being enriched over Set, fits into the setting of Section 1.4. Recall from Example 1.1 that theusual composition of functors is opposite to the convention we used to write the composition of1-cells in a bicategory, which has some important implications if we apply the general results ofSection 1.4 to our sub-bicategory B of CAT.

Take two categories C and D, and two functors F : C → D and G : D → C, and consider theMorita context from Proposition 1.47.

(232) M(F,G) = (Nat(G,G),Nat(F, F )op,Nat(D, FG),Nat(GF, C), ♦, ).

The connecting maps are given by the following formulas,

(α ♦β)D = βGD GαD and (β α)C = FβC αFC ,where α ∈ Nat(D, FG), β ∈ Nat(GF, C), C ∈ C and D ∈ D. By Theorem 1.48, (G,F ) is anadjoint pair if and only if there exists a pair of invertible elements for the Morita context M(G,F ),i.e. if and only if we can find elements η ∈ Nat(D, FG) and ε ∈ Nat(FG, C) such that η ♦ ε = Gand ε η = F . Formulas (17-18) can be derived from this.

Applying left-right symmetry, we can construct a second Morita context, that describes theadjunction of the pair (F,G):

(233) M(F,G) = (Nat(F, F ),Nat(G,G)op,Nat(C, GF ),Nat(FG,D), ♦, ),

where

(α♦β)C = βFC FαC and (β α)D = GβD αGD,for α ∈ Nat(C, GF ) and β ∈ Nat(FG,D).

We will now introduce the notion of a pair of locally adjoint functors.

Definitions 8.7. Consider functors F : C → D, G : D → C, and let E be a generatingsubcategory for C.

We call G an E-locally left adjoint for F , if and only if, there exists a natural transformationε ∈ Nat(GF, C) and for all morphisms f : E → GD in C, with E ∈ E and D ∈ D, we can find a

natural transformation ηf ∈ Nat(D, FG), such that

(234) f = (ηf ε)D f = εGD GηfD f.(In other words, the set Nat(D, FG) ε ⊂ Nat(F, F )op acts with local units on C.)

We call F an E-locally right adjoint for G, if and only if, there exists a natural transformationε ∈ Nat(GF, C) and for all morphisms f : FE → FC in D, with E ∈ E and C ∈ C, we can find

a natural transformation ηf ∈ Nat(D, FG), such that

(235) f = (ε ♦ ηf )C f = FεC ηfFC f.(In other words, the set ε ♦ Nat(D, FG) ⊂ Nat(G,G) acts with local units on C.)

If F an E-locally right adjoint for G and G is an E-locally left adjoint for F , then we call(G,F ) an E-locally adjoint pair. If (F,G) is an adjoint pair and (G,F ) is an E-locally adjoint pair,then we call (F,G) an E-locally Frobenius pair.

Definition 8.8. (compare to [75, Definitions 2.1 and 2.2]) We use the same notation asin Definition 8.7. Suppose the category D has coproducts and consider the functor S : D →D, S(D) = D(I), where I is a fixed index set.

We call G a left E-locally quasi-adjoint for F if and only if G is a left E-locally adjoint for SF .We call F a right E-locally quasi-adjoint for G if and only if F is a right E-locally adjoint for GS.

We call (G,F ) an E-locally quasi-adjoint pair if and only if G is a left E-locally quasi-adjointfor F and at the same time F is a right E-locally quasi-adjoint for G. We call (F,G) a E-locallyquasi-Frobenius pair if (F,G) is an adjoint pair and (G,F ) an E-locally quasi-adjoint pair.

8.3. THE INDUCTION FUNCTOR 215

8.3. The induction functor

8.3.1. Adjunctions. Let C be an A-bimodule. It is well-known (see e.g. [36, 18.28]) thatC is an A-coring if and only if the functor − ⊗A C : MA → MA is a comonad. This comonadfunctor induces a functor

G = GC : MA →MC, GC(N) = N ⊗A C;

where we denote N ∈ MA. The induction functor GC has both a left adjoint FC (the forgetfulfunctor) and a right adjoint HC. These are given by

F = FC : MC →MA, FC(M) = M ;

H = HC : MC →MA, HC(M) = HomC(C,M);

here we denote M ∈MC.The unit and counit of these adjunctions are given by

ηM : M → GF(M) = M ⊗A C, ηM (m) = m[0] ⊗A m[1];

εN : FG(N) = N ⊗A C → N, εN (n⊗A c) = nεC(c);

and

λN : N → HG(N) = HomC(C, N ⊗A C), λN (n)(c) = n⊗A c;κM : GH(M) = HomC(C,M)⊗A C →M, κM (f ⊗A c) = f(c);(236)

for all M ∈MC, N ∈MA.Recall that a functor is said to be Frobenius if it has a right adjoint that is at the same time

a left adjoint. Since adjoint functors are unique up to natural isomorphism, the study of theFrobenius property of the induction functor is related to the description of the sets (k-modules)

V = Nat(F ,H) and W = Nat(H,F).

Proposition 8.9. There exist isomorphisms of k-modules

Nat(GF , 11MC) ∼= V = Nat(F ,H) ∼= Nat(11MC ,GH);(237)

Nat(11MA,FG) ∼= W = Nat(H,F) ∼= Nat(GH, 11MA

).(238)

Proof. The isomorphisms (237) follow directly form the adjunctions (F ,G) and (G,H) if weapply (16).

To prove (238), take any α ∈ Nat(H,F), and define α′ ∈ Nat(11MA,FG) as

α′N = αGN λN .

Conversely, for any β ∈ Nat(11MA,FG) we define β′ ∈ Nat(H,F) by

β′M = FκM βHM .

If we compute α′′, we find α′′M = FκM αGHM λHM . By the naturality of α, we know thatFκM αGHM = αM HκM . Applying adjointness identity (18) on the adjunction (G,H), weobtain HκM λHM = HM . Combining both identities, we find that α′′M = αM .

Similarly, we find β′′ = β, making use of (17) on the adjunction (G,H) and the naturality ofβ. Finally, Nat(H,F) ∼= Nat(GH, 11MA

) follows in the same way from the adjunction (F ,G).

8.3.2. Description of sets of natural transformations. To give a further description of V ,let R be an additive subset of ∗C such that AR ⊆ RA and consider the k-modules

V1 = CHomC(C⊗ C,C);V2 = θ ∈ AHomA(C⊗A C, A) | c(1)θ(c(2) ⊗A d) = θ(c⊗A d(1))d(2);V3 = AHom∗C(C, ∗C);V ′3 = C∗HomA(C,C∗);

V4 = AHomC(C,RatR(∗C)), only if C is R-locally projective as a left A-module;

V5 = AHomC(C, ∗C), only if C is finitely generated and projective as a left A-module.


Proposition 8.10. Let C be an arbitrary A-coring, then we have an isomorphism of k-modules

V3 = AHom∗C(C, ∗C) ∼= V ′3 = C∗HomA(C,C∗).

Proof. Take any ϕ ∈ AHom∗C(C, ∗C), then we can easily construct a map

ϕ : C → C∗;c 7→ (d 7→ ϕ(c)(d) = ϕ(d)(c)).

It is straightforward to check that switching the arguments as above corresponds in an isomorphismV3∼= V ′3 .

By [46, section 3.3] (see also Section 8.3.4 of this paper for a more general setting), V ∼=V1∼= V2 for all corings, and V ∼= V5 if C is finitely generated and projective as a left A-module.

We extend this result.

Lemma 8.11. Let C be an A-coring which is R-locally projective as a left A-module. Thenthe following identity holds for all f ∈ RatR(∗C) and c ∈ C :

c(1)f(c(2)) = f[0](c)f[1].

Proof. For all g ∈ R, we have

g(c(1)f(c(2))) = (f ∗ g)(c)= (f[0]g(f[1]))(c)= f[0](c)g(f[1])= g(f[0](c)f[1]).

Let now∑ci ⊗A gi ∈ C ⊗A R be a local basis for the elements c(1)f(c(2)) and f[0](c)f[1], then

c(1)f(c(2)) =∑cigi(c(1)f(c(2))) =

∑cigi(f[0](c)f[1]) = f[0](c)f[1].

Proposition 8.12. Let C be an arbitrary A-coring. Then there exists a map α2 : V ∼= V2 →V3. If C is R-locally projective as a left A-module, then V ∼= V3

∼= V4. In particular, V ∼= V5 if Cis finitely generated and projective as a left A-module.

Proof. Let C be any A-coring; then we can define a map

α2 : V2 → V3;θ 7→ (c 7→ (d 7→ θ(d⊗A c))).

We verify that α2 is well-defined. First check α2(θ) = ϕ is an A-bimodule map.

(aϕ(c))(d) = ϕ(c)(da) = θ(da⊗A c)= θ(d⊗A ac) = ϕ(ac)(d);

(ϕ(c)a)(d) = (ϕ(c)(d))a = θ(d⊗A c)a= θ(d⊗A ca) = ϕ(ca)(d).

Next, we prove ϕ is also a right ∗C-module map. Take f ∈ ∗C, then we find

ϕ(c · f)(d) = ϕ(c(1) · f(c(2)))(d) = ϕ(c(1))(d)f(c(2))= f(ϕ(c(1))(d)c(2)) = f(θ(d⊗A c(1)c(2)))= f(d(1)θ(d(2) ⊗A c)) = f(d(1)ϕ(c)(d(2)))= (ϕ(c) ∗ f)(d).

Conversely, we can define a map

α′2 : V3 → AHomA(C⊗A C, A);ϕ 7→ (d⊗A c 7→ ϕ(c)(d)).

We demonstrate that α′2 is well-defined. Take ϕ ∈ V3, then θ = α′2(ϕ) is an A-bimodule map:

θ(ad⊗A c) = ϕ(c)(ad) = a(ϕ(c)d)= aθ(d⊗A c);

θ(d⊗A ca) = ϕ(ca)(d) = (ϕ(c)a)(d)= (ϕ(c)(d))a = θ(d⊗A c)a.

Now suppose that C is R-locally projective as a left A-module. We prove that the image of α2 lieswithin V2. Since ϕ is a ∗C-module map, it follows from the theory of rational modules (see [48]


and [115]) that ϕ(d) ∈ RatR(∗C) and ϕ is also a C-comodule map between C and RatR(∗C). Wecan compute

c(1)θ(c(2) ⊗A d) = c(1)ϕ(d)(c(2))= ϕ(d)[0](c)ϕ(d)[1]= ϕ(d(1))(c)d(2)

= θ(c⊗A d(1))d(2).

The second equation follows by Lemma 8.11 and the third one by the C-colinearity of ϕ.All the other implications are now straightforward.

We will now describe the set W . Consider the following k-modules.

W1 = AHomA(A,C);

W2 = z ∈ C | az = za = CA;W3 = AHom∗C(∗C,C);W ′

3 = C∗HomA(C∗,C);W r

3 = AHom∗C(RatR(∗C),C), only if C is R-locally projective over A;

W4 = AHomC(∗C,C), only if C is finitely generated and projective over A;W5 = AHomA(∗C, A);W r

5 = AHomA(RatR(∗C), A), only if C is R-locally projective over A.

Again by [46, section 3.3] (or Section 8.3.4 of this paper), we know W ∼= W1∼= W2 for

arbitrary corings and W ∼= W4 if C is finitely generated and projective as a left A-module. Thiscan be easily generalized in the following way.

Proposition 8.13. Let C an A-coring, then W ∼= W3∼= W ′

3. If C is R-locally projective asa left A-module, then W r

3∼= W r

5 . Consequently, if C is finitely generated and projective as a leftA-module, (W ∼=)W3

∼= W4∼= W5.

Proof. By Lemma 8.2 we immediately obtain that W3∼= W2

∼= W ′3. Suppose that C is R-

locally projective as a leftA-module. By rationality properties we find thatW r3∼= AHomC(RatR(∗C),C),

and from the adjunction between the forgetful functor AMC → AMA and − ⊗A C we find that

AHomC(RatR(∗C),C) ∼= W r5 .

To finish this section, we will describe the following classes of natural transformations

X = Nat(F ,F) and Y = Nat(G,G).(239)

Proposition 8.14. Let C be an A-coring and consider the classes of natural transformationsX and Y as in (239). Then the following isomorphisms hold,

X ∼= Nat(MC,GF) ∼= Nat(FG,MA) ∼= Y

∼= CHomC(C,C⊗A C) ∼= AEndC(C) ∼= CEndA(C)∼= ∗C∗ ∼= (∗C)A ∼= (C∗)A

∼= AEnd∗C(∗C) ∼= C∗EndA(C∗) ∼= Z = Nat(H,H).

In particular, X, Y and Z are sets.

Proof. The isomorphisms X ∼= Nat(MC,GF) and Y ∼= Nat(FG,MA) follow directly as anapplication of (16) as (F ,G) is an adjoint pair.

Take any α ∈ X and (M,ρM ) ∈ MC. For any m ∈ M , the map fm : C → M ⊗A C, c 7→m⊗A c is right C-colinear. We obtain by naturality of α that αM⊗AC = M ⊗A αC. The naturalityof α implies as well that ρM αM = αM⊗AC ρM . This way we find

αM = (M ⊗A εC) ρM αM= (M ⊗A εC) αM⊗AC ρM

= (M ⊗A εC) (M ⊗A αC) ρM .


We conclude that α is completely determined by αC. By definition αC ∈ HomA(C,C) and bythe naturality of α we find that αC is left C-colinear as well. One can now easily see that thecorrespondence we obtained between X and CEndA(C) is bijective.

Now take β ∈ Y . In a similar way as above, one can prove that βN = N ⊗ βA for allN ∈ MA. Observe that by definition βA ∈ EndC(C) and βA is left A-linear by the naturality ofβ. We conclude on the isomorphism Y ∼= AEndC(C).

The isomorphism EndC(C) ∼= ∗C restricts in a straightforward way to an isomorphism AEndC(C) ∼=∗C∗ and similarly CEndA(C) ∼= ∗C∗.

Take f ∈ (∗C)A, then for all c ∈ C, we find f(ca) = (af)(c) = (fa)(c) = f(c)a, i.e. f is rightA linear. This way we find that ∗C∗ ∼= (∗C)A and dually ∗C∗ ∼= (C∗)A.

Furthermore, for any γ ∈ AEndC(C), define θ ∈ CEndC(C,C ⊗A C) as θ(c) = c(1) ⊗A θ(c(2))and conversely γ = (ε⊗A C) θ.

Consider the map ν : (∗C)A → AEnd∗C(∗C), ν(f)(g) = f∗g, which has an inverse by evaluatingat ε.

Finally, take f ∈ ∗C∗. Then we define γ ∈ Z as follows γM : HomC(C,M) → HomC(C,M), γM (ϕ)(c) =ϕ(f · c). One easily checks that γM is well-defined and natural in M . In this way we obtaina map z : ∗C∗ → Z. Conversely, for γ ∈ Z, take γC(C) ∈ HomC(C,C). Then by natu-rality of γ one can easily check that γC(C) is left A-linear. This way, we can define a mapz′ : Z → ∗C∗, z′(γ) = εC γC(C). Let us check that z and z′ are each other inverses. For allc ∈ ∗C∗, z′ z(f)(c) = ε(f · c) = f(c). For all γ ∈ Z, M ∈MC and ϕ ∈ HomC(C,M) we find

z z′(γM )(ϕ)(c) = ϕ((ε γC(C)) · c)= ϕ(γC(C)(c)) = (γM (ϕ))(c),

where the last equation follows from the naturality of γ, applied to the morphism ϕ ∈ HomC(C,M).

Remark 8.15. The above theorem only states isomorphisms of modules. However, some ofthese objects have an additional ring structure. All stated (iso)morphisms are also ring morphismsfor those objects that posses a ring structure, but sometimes one has to consider the oppositemultiplication. For sake of completeness, we state the correct isomorphisms, but we leave theproof to the reader.

Nat(F ,F)op ∼= Nat(G,G) ∼= Nat(H,H)op

∼= CEndA(C)op ∼= AEndC(C) ∼= ∗C∗

∼= AEnd∗C(∗C) ∼= C∗EndA(C∗)op

8.3.3. The Yoneda-approach.

Lemma 8.16. (1) Let N be a C∗-A bimodule. Then the following assertions hold(a) C∗HomA(N,C∗) ∈M∗C∗ ;(b) C∗HomA(N,C) ∈M∗C∗ .

(2) Let M be a C-A bicomodule and R ⊂ C∗. Then the following assertions hold(a) CHomA(M,C) ∼= AHomA(M,A) ∈M∗C∗ ;(b) C∗HomA(M,C∗) ∼= C∗HomA(M,RRat(C∗)) ∼= CHomA(M,RRat(C∗)) ∈M∗C∗ ;

where the R-rational part of C∗ is only considered if C is R-locally projective as a right A-module.

Proof. (1a) Take any N ∈ C∗MA, for any f ∈ C∗HomA(N,C∗), g ∈ ∗C∗ and n ∈ N , wedefine

(240) (f ∗ g)(n) = (f(n)) ∗ g.Note that f ∗ g is right A-linear, since g commutes with all elements of A by Proposition 8.14.One can easily verify that (240) defines a left ∗C∗-action.

(1b) We give only the explicit form of the action and leave other verifications to the reader.

Take any f ∈ C∗HomA(N,C), g ∈ ∗C∗ and n ∈ N , then we define

(241) (f ∗ g)(n) = f(n) · g = f(n)(1)g(f(n)(2)).


(2a) Analogously to the adjunction of (FC,GC), the forgetful functor CMA → AMA has a

right adjoint −⊗AC : AMA → CMA. Consequently, for any M ∈ CMA, we have an isomorphismCHomA(M,C) ∼= AHomA(M,A) that is natural in M . Moreover, the action defined in (241)can be restricted to a right ∗C∗-module structure on CHomA(M,C) ∼= AHomA(M,A). For anyf ∈ AHomA(M,A), m ∈M and g ∈ ∗C∗, one defines explicitly

(f ∗ g)(m) = g(m[−1]f(m[0])) = g(m[−1])f(m[0]).

(2b) Since every left C-comodule is also a left ∗C-module (see Section 3.3), by part (1a), we

find that C∗HomA(M, ∗C) ∈M∗C∗ .Suppose now that C is R-locally projective as a right A-module. Then the image of any

f ∈ C∗HomA(M, ∗C) lies within the rational part RRat(C∗). Indeed, for any g ∈ C∗, g ∗ f(m) =f(g · m) = f(g(m[−1])m[0]) = g(m[−1])f(m[0]), so f(m) ∈ RRat(C∗). We can conclude that

C∗HomA(M,C∗) ∼= C∗HomA(M,RRat(C∗)) ∼= CHomA(M,RRat(C∗)).

The observations made in Lemma 8.16 lead to the introduction of the following contravariantfunctors

J : C∗MA →M∗C∗ , J (M) = C∗HomA(M,C∗);K : C∗MA →M∗C∗ , K(M) = C∗HomA(M,C);J ′ : CMA →M∗C∗ , J ′(M) = C∗HomA(M,C∗) ∼= C∗HomA(M,RRat(C∗))

∼= CHomA(M,RRat(C∗));

K′ : CMA →M∗C∗ , K′(M) = AHomA(M,A) ∼= CHomA(M,C).(242)

(The alternative descriptions of J ′ in terms of the R-rational part of C∗ is only considered if C isR-locally projective as a right A-module.) Out of these functors we can construct the k-modules

V6 = Nat(K′,J ′) and V7 = Nat(K,J );

W6 = Nat(J ′,K′) and W7 = Nat(J ,K).

Lemma 8.17. Let N be an C∗-A bimodule. Then

(1) NA ∼= AHomA(A,N) ∼= C∗HomA(C∗, N) ∈ ∗C∗M;(2) C∗HomA(C, N) ∈ ∗C∗M.

Proof. (1) Both isomorphisms follow from Lemma 8.2. We define a left ∗C∗-action on

C∗HomA(C∗, N) with the following formula

(f ∗ ϕ)(g) = ϕ(g ∗ f)

for all ϕ ∈ C∗HomA(C∗, N), f ∈ ∗C∗ and g ∈ C∗.(2) For ϕ ∈ C∗HomA(C, N), f ∈ ∗C∗ and c ∈ C we define

(f ∗ ϕ)(c) = ϕ(c(1)f(c(2))).

One can easily verify this turns C∗HomA(C, N) into a left ∗C∗-module.

Using Lemma 8.17 we can construct the covariant functors

J : C∗MA → ∗C∗M, J (N) = NA ∼= AHomA(A,N) ∼= C∗HomA(C∗, N);

K : C∗MA → ∗C∗M; K(N) = C∗HomA(C, N)(243)

and the k-modules

V8 = Nat(J , K) and W8 = Nat(K, J ).

Let X be any category, F : X → Set a covariant functor and X ∈ X . Recall that by theYoneda Lemma (see e.g. [25, Theorem 1.3.3]) Nat(Hom(X,−), F ) ∼= F (X). Similarly for anycontravariant functor G : X → Set, we have Nat(Hom(−, X), G) ∼= G(X). Of course the YonedaLemma can be applied to the particular case where F = Hom(X,−) and G = Hom(−, X). Inthose cases, Nat(F, F ) and Nat(G,G) can be completed with a semigroup structure, coming fromthe composition of natural transformations. The following Lemma compares these structures withthe semigroup structure of Hom(X,X) (under composition). This result might be well-known,but since we could not find any reference, we include the proof.


Lemma 8.18. Let X be any category and X ∈ X , then we have the following isomorphismsof semigroups

Nat(Hom(X,−),Hom(X,−))op ∼= Hom(X,X) ∼= Nat(Hom(−, X),Hom(−, X)).

Proof. Consider the Yoneda bijection Λ : Nat(Hom(X,−),Hom(X,−)) → Hom(X,X);Λ(α) = αX(X). Let us compute Λ(αβ) = (αβ)X(X) = αX βX(X). Consider the morphismβX(X) : X → X and apply the naturality of the functor Hom(X,−) to this morphism, we obtainαX βX(X) = βX(X) αX(X).

Similarly, starting from the bijection V : Nat(Hom(−, X),Hom(−, X)) → Hom(X,X);V(α) = αX(X), we find V(α β) = (α β)X(X) = αX βX(X). The functor property ofthe contravariant functor Hom(−, X) implies αX βX(X) = αX(X) βX(X) and we find theneeded semigroup morphism.

Proposition 8.19. Let C be an A-coring. Then we have isomorphisms of k-modules

(1) V ′3∼= V7

∼= V8;(2) W ′

3∼= W7

∼= W8;

(3) Nat(J ,J ) ∼= Nat(J , J )op ∼= C∗EndA(C∗) ∼= X;

(4) Nat(K,K) ∼= Nat(K, K)op ∼= C∗EndA(C);(5) Nat(K′,K′) ∼= C∗EndA(C).

If C is R-locally projective as a left A-module, then V6∼= V ′3 , W6

∼= AHomC(C,RatR(∗C)),Nat(J ′,J ′) ∼= AEndC(RatR(∗C)) and Nat(K′,K′) ∼= Nat(K,K).

Proof. All isomorphisms are immediate consequences of the Yoneda Lemma and Lemma8.18.

8.3.4. The coproduct functor. Quasi-Frobenius type properties can not be described by thefunctors F and G alone, we have to incorporate a new functor in our theory (compare also with[75]).

Consider the following coproduct-functor

S : MA →MA, S(M) = M (I),

where I is an arbitrary fixed index set.Applying our previous results, we will give a full description of the sets

Nat(SF ,SF), Nat(MA,SFG), Nat(GSF ,MC).

To improve the readability of the next theorems, let us recall the construction of coproductsin MC. Take (M,ρM ) ∈MC; then (M,ρM )(I) = (M (I), ρ), where the coaction ρ is given by thefollowing composition

(244) ρ : S(M)S(ρM ) // S(M ⊗A C)

∼= // S(M)⊗A C

where we used that the tensor product commutes with coproducts.

Lemma 8.20. Let C be an A-coring. Then we have the following isomorphisms of k-modules

AHomC(C,S(C)) ∼= CHomC(C,S(C⊗A C)) ∼= CHomA(C,S(C)).

Proof. Take γ ∈ AHomC(C,S(C)). Then we define

θ : C∆ // C⊗A C

C⊗Aγ // C⊗A S(C) ∼= S(C⊗A C).

Conversely, given θ ∈ CHomC(C,S(C⊗A C)), define

γ : Cθ // S(C⊗A C) ∼= C⊗A S(C)

ε⊗AC // S(C).

The second isomorphism is constructed in the same way.


Proposition 8.21. There exist maps

Nat(SF ,SF)∼= // CEndA(C(I))

υ //

∼=

C∗EndA(C(I))

Nat(GS,GS)op ∼= //AEndC(C(I))op

υ′ //AEnd∗C(C(I))op

where υ (resp. υ′) is an isomorphism as well if C is locally projective as a left (resp. right)A-module.

Proof. Take α ∈ Nat(SF ,SF). Then we find, by definition, that αC ∈ EndA(S(C)). Takenow N ∈ MA. For any n ∈ N , we can consider the right C-colinear map fn : C → N ⊗A C,fn(c) = n⊗A c. The naturality of α and the commutativity of the tensor product and coproductimply the commutativity of the following diagram.

N ⊗A S(C)∼= // S(N ⊗A C)

αN⊗AC // S(N ⊗A C)∼= // N ⊗A S(C)

S(C) αC

//fn⊗AS(C)

ggOOOOOOOOOOOOS(fn)

OO

S(C)fn⊗AS(C)

77ooooooooooooS(fn)

OO

This implies that αN⊗AC is determined by αC up to isomorphism, as expressed in the followingdiagram.

S(N ⊗A C)αN⊗AC //

∼=

S(N ⊗A C)

N ⊗A S(C)N⊗AαC

// N ⊗A S(C)

∼=

OO

It follows now easily from the naturality of α that αC is left C-colinear, and thus αC ∈ CEndA(C).Moreover, α is completely determined by its value in C. Take any M ∈ MC and consider thefollowing diagram.

S(M)αM //

S(ρM )

S(M)

S(ρM )

S(M ⊗A C)

∼=

αM⊗AC // S(M ⊗A C)

∼=

M ⊗A S(C)M⊗AαC

// M ⊗A S(C) ∼=// S(M)⊗A C

S(M)⊗Aε

jj

The upper quadrangle commutes by the naturality of α, applied on the C-colinear morphismρM : M → M ⊗A C, the lower quadrangle commutes by the previous observations and thecommutativity of the triangle is exactly the counit condition on the comodule S(M). This way wefind an isomorphism Nat(SF ,SF) ∼= CEndA(C(I)). The second horizontal isomorphism is provedin the same way. The vertical isomorphism is a consequence of Lemma 8.1 and Lemma 8.20:

AHomC(C(I),C(I)) ∼= (AHomC(C,C(I)))I ∼= (CHomA(C,C(I)))I ∼= CHomA(C(I),C(I)).

We leave it to the reader that the constructed isomorphisms are algebra morphisms. Themorphisms υ and υ′ follow from the relations between left C-comodules and left C∗-modules (seeSection 3.3).

Lemma 8.22. Let C be an A-coring, B → A a ring morphism and I any index set.

(i) BVI := BHom∗C(C, (∗C)I) ∼= BHom∗C(C(I), ∗C)ξ∼= C∗HomB(C(I),C∗) ∼= C∗HomB(C, (C∗)I);

(ii) for all M ∈ CMC,

CHomC(M,C) ∼= θ ∈ AHomA(M,A) | x[−1]θ(x[0]) = θ(x[0])x[1], for all x ∈M;


(iii) there exist morphisms CHomC(S(C⊗AC),C)ξ1→ AVI ∼= V I

3 , where ξ1 becomes an isomorphismif C is locally projective as a left A-module;

(iv) BHom∗C((∗C)(I),C)∼=C∗HomB(C∗,CI)∼=(CI)B ∼= (CB)I ;(v) BWI := C∗HomB(C∗,C(I)) ∼= BHom∗C(∗C,C(I)) ∼= (C(I))B ∼= (CB)(I).

Proof. (i) The first and last isomorphism are an immediate consequence of Lemma 8.1, thesecond isomorphism is induced by the isomorphism of Proposition 8.10.(ii) Take any γ ∈ CHomC(M,C) and define θ = ε γ. Clearly, θ ∈ AHomA(M,A). Moreover, bythe bi-colinearity of γ we find for all x ∈M ,

x[1] ⊗A γ(x[0]) = γ(x)(1) ⊗A γ(x)(2) = γ(x[0])⊗A x[1].

If we apply C⊗A ε to the first equation, ε⊗A C to the second equation we obtain x[−1]θ(x[0]) =θ(x[0])x[1] = γ(x). Conversely, starting from θ ∈ AHomA(M,A), such that x[−1]θ(x[0]) =θ(x[0])x[1] for all x ∈M , we define γ(x) = x[−1]θ(x[0]).(iii) Denote by ιC⊗AC

` : C⊗AC → S(C⊗AC) and ιC` : C → S(C) the canonical injections. Considerthe following diagrams.

S(C⊗A C) ν // C ∗C

C⊗A C

ιC⊗AC

`

OO

ν`

66nnnnnnnnnnnnnnnC

ϕ`

88qqqqqqqqqqqqqq

ιC`

// S(C)

ϕ

OO

We find that every morphism ν ∈ CHomC(S(C⊗AC),C) is completely determined by the morphismsν` ∈ CHomC(C⊗A C,C) = V2. Similarly, any ϕ ∈ V A

I is completely determined by ϕ` ∈ V3. Theneeded morphism and isomorphism is now a consequence of Proposition 8.12 and Lemma 8.1.(iv) The first isomorphism is a consequence of Lemma 8.1 the second one is a consequence ofLemma 8.2. The last isomorphism is trivial.(v) The second (and first) isomorphism follows from Lemma 8.2, the last one is trivial.

Proposition 8.23. Let C be an A-coring and I any index set, than the following isomorphismshold.

(i) Nat(MA,SFG)∼=(CA)(I);(ii) Nat(GSF ,MC) ∼= CHomC(S(C⊗A C),C).

Proof. (i) Take any ζ ∈ Nat(MA,SFG). Then ζA ∈ HomA(A,C(I)) by definition, andfrom the naturality of ζ we obtain that ζA is left A-linear. Applying the same techniques as in theproof of Proposition 8.21, we find that ζ is completely determined by ζA, and thus we obtain anisomorphism Nat(MA,SFG)∼=AHomA(A,C(I)) ∼= (CA)(I).(ii) The proof is completely similar to part (i). Any ν ∈ Nat(GSF) is completely determined by

νC : SFG(C) = S(C ⊗A C) → C, by definition νC is right C-colinear and the left C-colinearityfollows from the naturality of ν, i.e. νC ∈ CHomC(S(C⊗A C),C).

We give a generalization of Proposition 8.9, the proof is completely similar.

Proposition 8.24. There exist isomorphisms of k-modules

(i) Nat(GSF , 11MC) ∼= Nat(SF ,H);(ii) Nat(11MA

,SFG) ∼= Nat(H,SF).

Proof. (i) The isomorphism follows directly form the adjunction between G and H if we

apply (16).(ii). Take any α ∈ Nat(H,SF), then we define α′ ∈ Nat(11MA

,SFG) as

α′N = αGN λN .

Conversely, for any β ∈ Nat(11MA,SFG) we define β′ ∈ Nat(H,SF) by

β′M = SFκM βHM .

8.4. CHARACTERIZATIONS OF CO-FROBENIUS AND QUASI-CO-FROBENIUS CORINGS 223

If we compute α′′, we find

α′′M = SFκM αGHM λH= αM HκM λH = αM

where we used the naturality of α in the second equality and (18) on the adjunction (G,H) in thethird equality. Similarly, we find

β′′M = SFκGN βHGN λN= SFκGN SFGλN βN= SF(κGN GλN ) βN = βN .

Here we used the naturality of β in the second equality and (17) in the fourth equality.

Consider the functors

Ks : C∗MA →M∗C∗ , Ks(M) = C∗HomA(M,S(C));

Ks : C∗MA → ∗C∗M, Ks(N) = C∗HomA(S(C), N).(245)

As a consequence of the Yoneda Lemma, we immediately obtain the following

Proposition 8.25. With notation as introduced before, the following isomorphisms hold:

(i) Nat(Ks,Ks) ∼= Nat(Ks, Ks)op ∼= C∗EndA(S(C));(ii) Nat(Ks,J ) ∼= Nat(J , Ks) ∼= C∗HomA(S(C),C∗);(iii) Nat(Ks, J ) ∼= Nat(J ,Ks) ∼= C∗HomA(C∗,S(C)).

8.4. Characterizations of co-Frobenius and quasi-co-Frobenius corings

8.4.1. Locally Frobenius corings.

Lemma 8.26. Let C be an A-coring and B → A a ring morphism. And take any j ∈BHom∗C(C, ∗C). Then Im j is a right ideal in ∗C and (Im j)B is a right ideal in (∗C)B.

Proof. Take f ∈ Im j, i.e. f = j(c) for some c ∈ C. Then for any g ∈ ∗C, f ∗ g = j(c) ∗ g =j(c · g) ∈ Im j by the right ∗C-linearity of j.Suppose now that f = j(c) ∈ (Im j)B and g ∈ (∗C)B. We have to check that j(c · g) commuteswith all b ∈ B. We find bj(c · g) = bj(c) ∗ g = j(c)b ∗ g = j(c) ∗ gb = j(c · g)b.

Lemma 8.27. Let C be anA-coring andB → A a ring morphism. Consider j ∈ BHom∗C(C, ∗C).The restriction of j on CB defines a map

j′ : CB → (∗C)B

that is Z(B)-(∗C)B-bilinear, where Z(B) denotes the center of B. Moreover, Im j′ is a right idealin (∗C)B.

Proof. Take c ∈ CB and b ∈ B. Then bj(c) = j(bc) = j(cb) = j(c)b, so j(c) ∈ (∗C)B. Sincej is B-∗C-bilinear and CA ⊂ C is a bimodule with restricted actions of Z(B) ⊂ B and (∗C)B ⊂ ∗C,it is immediately clear that j′ is a Z(B)-(∗C)B bilinear map. The last assertion is proved as thesecond part of Lemma 8.26.

Theorem 8.28. Let C be an A-coring which is locally projective as a left A module. LetB → A be a ring morphism and I any index set. Consider j ∈ AHom∗C(C, (∗C)I), denote for thecorresponding element in C∗HomA(C(I),C∗) and denote ′ : (C(I))B → (C∗)B for the restriction of. Then the following statements are equivalent

(i) for all c1, . . . , cn ∈ C and f ∈ (C∗)B, there exists an element g ∈ Im ′ such that g(ci) = f(ci)(i.e. Im ′ is dense in the finite topology on (C∗)B);

(ii) for all c1, . . . , cn ∈ C, there exists an element e ∈ Im ′ such that e(ci) = ε(ci);(iii) for all c1, . . . , cn ∈ C and f ∈ (C∗)B, there exists an element g ∈ Im ′ such that f · ci = g · ci

(i.e. Im ′ is dense in the C-adic topology on (C∗)B);(iv) for all c1, . . . , cn ∈ C, there exists an element e ∈ Im ′ such that e · ci = ci;(v) there exist B-linear local right inverses for , i.e. for all c1, . . . , cn ∈ C, there exists a ∈

C∗HomB(C∗, (C)(I)) such that ((f))(ci) = f(ci) for all f ∈ C∗;


(vi) for all c1, . . . , cn ∈ C, there exists an element z ∈ (CB)(I) such that ci = z·j(ci) =∑

` z`j`(ci)for all i = 1, . . . , n.

Moreover, if any of these conditions are satisfied, then

(a) there exist B-linear local left inverses for j, i.e. for all c1, . . . , cn ∈ C, there exists a ∈BHom∗C((∗C)I ,C) such that ci = (j(ci))

(b) j is injective;(c) C is Im -locally projective as a right A-module.

Proof. (i) ⇒ (ii) Trivial.

(ii) ⇒ (i) By Lemma 8.27 we know that Im ′ is a left ideal in (C∗)B. The statement follows nowimmediately.(i) ⇒ (iii) Take ci and f as in statement (iii). Then f · ci = f(ci(1))ci(2). By statement (i) we

find an element g ∈ Im ′ such that g(ci(1)) = f(c(i(1))) for all i. Consequently, g · ci = f · ci.(iii) ⇒ (iv) Follows again from the fact that Im ′ is a left ideal in (C∗)B.

(iv) ⇒ (ii) Take ci as in statement (ii). From (iv) we know that we can find an e ∈ Im ′ such

that e · ci = e(ci(2))ci(2) = ci. Apply ε to this last equation, then we find e(ci) = ε(ci).(iv) ⇒ (vi) Consider c1, . . . , cn ∈ C. Then we know from (iv) that there exists an element

e ∈ Im ′ such that e · ci = ci. We can write e = ′(z) =∑′`(z`) for some z = (z`) ∈ (CB)(I).

We will show that this z is the needed one. Recall from Proposition 8.12 that j` is a right C-colinearmorphism from C to Rat∗C(∗C). We find

z · j(ci) =∑`

z` · j`(ci) =∑`

z`(1)j`(ci)(z`(2))

=∑`

j`(ci)[0](z`)j`(ci)[1] =∑`

j`(ci(1))(z`)ci(2)

=∑`

`(z`)(ci(1))ci(2) = e(ci(1))ci(2) = ci,

where we used Lemma 8.11 in the third equation.(vi) ⇒ (v) Take ci ∈ C as in the statement of (v). Choose representatives cji , cki

∈ C such that

∆(ci) =∑

i cki⊗A cji for all i. By (vi) we can find a z = (z`) ∈ (CB)(I) such that for all cki

,

(246) cki= z · (j(cki

)).

Now by Lemma 8.22 we can associate to z an element ∈ C∗HomB(C∗,C(I)), defined as (f) = f ·zfor all f ∈ C∗. We find

((f))(ci) = (f · z)(ci) = (f ∗ (z))(ci) =∑`

(f ∗ `(z`))(ci)

=∑`,i

f(`(z`)(cki)cji) =

∑`,i

f(j`(cki)(z`)cji)

=∑i

f(ε(cki)cji) = f(ci).

Where the one but last equation follows by applying ε on (246).(v) ⇒ (i) For every f ∈ (C∗)B, we have (f) ∈ (CB)(I). Consequently we can choose g = ′((f)).

Suppose now that the conditions (i)− (vi) are satisfied. (vi) ⇒ (a) Follows immediately from

Lemma 8.22. To prove (b), suppose j(c) = 0 for some c ∈ C, then by statement (a) we canfind such that c = (j(c)) = (0) = 0, so j is injective. Finally, we find by (vi) on every setc1, . . . , cn ∈ C an element (z`) ∈ (CB)(I), such that we can compute

ci =∑`

z` · j`(ci) =∑`

z`(1)j`(ci)(z`(2)) =∑`

z`(1)`(z`(2))(ci).

This means that z`(1), `(z`(2)) is a local dual basis for ci, so C is locally projective as a rightA-module.


Remark 8.29. It follows immediately from the proof that, even if C is not necessary lo-cally projective as a left A-module, the first four statements of Theorem 8.28 remain equivalentstatements if we replace Im ′ by any left ideal in (C∗)B.

Definitions 8.30. If C is an A-coring that is locally projective as a left A-module and suchthat the equivalent conditions (i)-(vi) of Theorem 8.28 are satisfied we call C a left B-locallyquasi-Frobenius coring.

If C is left A-locally quasi-Frobenius, we will just say that C is left locally quasi-Frobenius.If C is a B-locally quasi-Frobenius coring such that the index-set I of Theorem 8.28 can be

chosen to have only 1 element, then we say that C is left B-locally Frobenius.

Corollary 8.31. Let C be an A-coring that is left B-locally quasi-Frobenius with Frobeniusmorphism j : C → (∗C)I , denote as in Theorem 8.28 the corresponding morphism : C(I) → C∗

with restricted morphism ′ : (C(I))B → (C∗)B. Then the following statements hold.

(i) Im is a B-ring with left local units, where : C(I) → C∗. Moreover, Im acts with localunits on every left C-comodule;

(ii) Im ′ is a ring with left local units and Im ′ acts with local units on every left C-comodule.

Proof. (i). Clearly Im is a B-ring. Since Im ′ ⊂ Im , the remaining part of the statement

follows by part (ii).(ii). Let (ci) be any element of Im where (ci) ∈ C(I). Then denote by z = (z`) ∈ (CB)(I) the

element satisfying condition (vi) of Theorem 8.28. Write e = ′(z`) = (z`) ∈ Im ′, we claim thate is a left local unit for (ci). Indeed,

(z`) ∗ (ci) = ((z`) · ci

)= (

∑`

`(z`) · ci)

= (∑`

`(z`)(ci(1))ci(2)) = (∑`

j`(ci(1))(z`)ci(2))

= (∑`

j`(ci)[0](z`)j`(ci)[1]) = (∑`

z`(1)j`(ci)(z`(2)))

= (z · j(ci)) = (ci).

Here we used the left C∗-linearity of in the first equality, Lemma 8.11 in the fifth equality andpart (vi) of Theorem 8.28 in the last equality.

Let M be any left C-comodule. The action of f ∈ Im ′ on m ∈ M is given by f · m =f(m[−1])m[0]. That there exists local units for this action is a direct consequence of Theorem 8.28,part (ii).

Theorem 8.32. Let C be an A-coring Then the following statements hold.

(1) If j ∈ AHom∗C(C, (∗C)I) is injective then the restriction j′ : CA → ((∗C)A)I is alsoinjective.

(2) Consider ring morphisms B → B′ → A. If C left B′-locally quasi-Frobenius then C is leftB-locally quasi-Frobenius.

(3) If A is a PF-ring and C is ∗C∗-locally projective then the following statements are equiv-alent

(i) C is left locally quasi-Frobenius;(ii) C is left B-locally quasi-Frobenius for an arbitrary ring morphism B → A;(iii) C is left quasi-co-Frobenius;(iv) The restriction j′ : CA → ((∗C)A)I of the Frobenius map j is injective (i.e. CA is a

torsionless right (∗C)A-module).

Proof. (1) Trivial.

(2) If C is left B′-locally Frobenius, then the B′-linear local left inverses for j from the equivalent

condition (vi) of Theorem 8.28 are clearly also B-linear local left inverses.(3) From part (1) and (2) we know already (iii) ⇒ (iv) and (i) ⇒ (ii). From Theorem 8.28

we know that (ii) ⇒ (iii). So we only have to prove (iv) ⇒ (i). Let us denote = ξ(j) ∈C∗HomA(C(I),C∗) by the isomorphism of Lemma 8.22(i). We will show that Im ′ is dense in


the finite topology on ∗C∗, which is equivalent condition (i) of Theorem 8.28 applied to thesituation B = A. Since C is ∗C∗-locally projective, the canonical map C → (∗C∗)∗ is injective.Moreover A is a PF-ring, so a subset P ⊂ ∗C∗ is dense in the finite topology if and only if theorthogonal complement P⊥ of P is trivial (see [5, Theorem 1.8]). Take any c ∈ (Im ′)⊥. Then′(d`)(c) = j′(c)(d`) = 0 for all d` ∈ C(I). This implies c ∈ ker j′. By the injectivity of j′ we findc = 0, so (Im ′)⊥ = 0 and Im ′ is dense in the finite topology on ∗C∗.

Corollary 8.33. Let C be an A-coring Then the following statements hold.

(1) If j ∈ AHom∗C(C, ∗C) is injective then the restriction j′ : CA → ∗C is also injective.(2) Consider ring morphisms B → B′ → A. If C left B-locally Frobenius then C is left

B′-locally Frobenius.(3) If A is a PF-ring and C is ∗C∗-locally projective then the following statements are equiv-

alent(i) C is left co-Frobenius;(ii) C is left locally Frobenius;(iii) C is left B-locally Frobenius for an arbitrary ring morphism B → A;(iv) The restriction j′ : CA → ∗C of the Frobenius map j is injective.

Proof. This is proved in the same way as Theorem 8.32.

Proposition 8.34. Let C be an A-coring and B → A any ring morphism. If C is left B-locallyquasi-Frobenius, then C∗Rat is an exact functor. In particular, if A is a QF-ring, then then C is aleft semiperfect coring.

Proof. By part (2) of Corollary 8.33, we know that C is also k-locally quasi-Frobenius. Thisimplies by Theorem 8.28 that Im is dense in C∗. Also by Theorem 8.28, we know that C is locallyprojective as a right A-module, so Lemma 8.22(i) implies that Im is contained in C∗Rat(C∗).We can conclude that C∗Rat(C∗) itself is dense in C∗. By [43, Proposition 2.6] the density of

C∗Rat(C∗) is equivalent to the exactness of C∗Rat. Moreover, if A is a QF-ring, this conditionis again equivalent to C being a left semiperfect coring (see [43, Theorem 4.3] or [65, Theorem3.8])

8.4.2. Characterization of Frobenius corings. Considering the objects C and ∗C in thecategory AM∗C and the objects C and C∗ in the category C∗MA, we obtain as in Section 8.1.3the following Morita contexts:

N(C, ∗C) = (AEnd∗C(C),AEnd∗C(∗C),AHom∗C(∗C,C),AHom∗C(C, ∗C), , •);N(C,C∗) = (C∗EndA(C), C∗EndA(C∗), C∗HomA(C∗,C), C∗HomA(C,C∗), , •).

If we consider the contravariant functors J and K, from (242) and the covariant functors J and

K from (243), then we can construct another two Morita contexts

Y(K,J ) = (Nat(K,K),Nat(J ,J ),Nat(J ,K),Nat(K,J ), 4, N);

Y(K, J ) = (Nat(K, K),Nat(J , J ),Nat(J , K),Nat(K, J ), 4, N).

Consider the functors F , G and H as in Section 8.3.1. We can construct the Morita contextthat connects the functors F and H in the category of functors from MC to MA and all naturaltransformations between them.

N(F ,H) = (Nat(F ,F),Nat(H,H),Nat(H,F),Nat(F ,H),,).

Although the functors F and G are not contained in the same category, we can apply the resultsof Section 8.2 to obtain a Morita context (232) connecting the functors F and G.

M(F ,G) = (Nat(G,G),Nat(F ,F)op,Nat(MA,FG),Nat(GF ,MC), ♦, ).

Similarly, we find a Morita context connecting the functors G and H,

M(G,H) = (Nat(H,H),Nat(G,G)op,Nat(MC,GH),Nat(HG,MA), O, H).


Theorem 8.35. Let C be an A-coring. With notation as above, we have the following diagramof morphisms of Morita contexts.

Ntop(F ,H)n

''OOOOOOOOOOOn

wwooooooooooo

M(F ,G)

f

m// Mop(G,H)

Ntop(C,C∗)a //

b

N(C, ∗C)

b

Ytop(K,J ) a // Y(K, J )

Here the upper script ‘op’ indicates the opposite Morita context and ‘t’ denotes the twisted Moritacontext (see Section 8.1.3). For an arbitrary coring C, the morphisms a, a, b, b,m, n, n are iso-morphism of Morita contexts. If C is locally projective as a left A-module, then f becomes anisomorphism of Morita contexts as well.

Proof. The algebra isomorphisms for m, n and n follow immediately from Proposition 8.14and Remark 8.15. The maps that describe the isomorphisms for the connecting bimodules aregiven in equations (237) and (238).The algebra isomorphisms for a, a, b and b follow from Proposition 8.19 in combination with Re-mark 8.15. The isomorphisms for the connecting bimodules of a are given in Proposition 8.10 andProposition 8.13, for a, b and b they follow from Proposition 8.19.The first algebra morphism of f is constructed as follows. We know by Remark 8.15 that Nat(G,G) ∼=CEndA(C)op

. Hence we have an algebra map

f1 : Nat(G,G) ∼= CEndA(C)op → C∗EndA(C)op.

The algebra map f2 : Nat(F ,F)op → C∗EndA(C∗) is given explicitly in Remark 8.15, and thebimodule maps f3 and f4 follow from Proposition 8.13 and Proposition 8.12 respectively. Moreover,we f2 and f3 are always bijective and when C is flat as left A-module, then f1 is an isomorphismby a rationality argument and f4 is an isomorphism by Proposition 8.12.We leave it to the reader to verify that all given (iso)morphisms of algebras and bimodules doindeed form Morita morphisms and that the stated diagrams of Morita morphisms commute.

Corollary 8.36. Let C be an A-coring. There exists a split epimorphism j ∈ C∗HomA(C,C∗)if and only if there exists a split monomorphism ∈ AHom∗C(C, ∗C). If any of these equivalentconditions holds then C finitely generated and projective as a right A-module.

Proof. We will prove a more general version of this corollary in Corollary 8.40

As a corollary we obtain the well-known characterization of Frobenius corings in terms ofFrobenius functors.

Corollary 8.37 (characterization of Frobenius corings). Let C be an A-coring, then thefollowing statements are equivalent;

(i) C ∼= ∗C in AM∗C (i.e. C is a Frobenius coring);(ii) C ∼= C∗ in C∗MA;(iii) there exists a Frobenius system, consisting of a pair (z, θ), with θ ∈ AHomA(C⊗A C, A) and

z ∈ CA such that the following conditions hold:• c(1)θ(c(2) ⊗A d) = θ(c⊗A d(1))d(2), for all c, d ∈ C,• θ(z ⊗A c) = θ(c⊗A z) = εC(c), for all c ∈ C;

(iv) the functors H and F are naturally isomorphic;(v) (G,F) is a pair of adjoint functors, and therefore (F ,G) is a Frobenius pair;(vi) (H,G) is a pair of adjoint functors, and therefore (G,H) is a Frobenius pair;(vii) the functors J and K are naturally isomorphic;

(viii) the functors J and K are naturally isomorphic;


(ix) AC is finitely generated and projective and the functors J ′ and K′ are isomorphic;(x) left hand versions of (iii)− (viii), replacing MC by CM and MA by AM.

Proof. The first statement is true if and only if there exists a pair of invertible elementsin the Morita context N(C, ∗C). From (the left hand version of) Corollary 8.36 we know thatthe isomorphism C ∼= C∗ implies that C is finitely generated and projective as a left and rightA-module. The equivalence of (i), (ii) and (iv)− (viii) follows now immediately from the (anti-)isomorphisms of Morita contexts from Theorem 8.35. Note that (F ,G) is always a pair of adjointfunctors and therefore the adjointness of (G,F) means exactly that (F ,G) is a Frobenius pair. Thesame reasoning holds for the pair (G,H). Since for a coring that is finitely generated and projectiveas a left A-module the categories MC and M∗C are isomorphic, we obtain that the functor J ′,respectively K′, is isomorphic with J , respectively K. The equivalence with (iii) follows fromthe equivalent discriptions of the sets of natural transformations V and W in Section 8.3.2.The equivalence of (i) and (ii) imply the equivalence with the left hand version of the otherstatements.

Remark 8.38. (1) Note that the left-right symmetry of the notion of a Frobenius exten-sion (or a Frobenius coring), is by the previous Corollary a consequence of the isomorphismof Morita contexts between N(C, ∗C) and Ntop(C,C∗). We will see that this isomorphismis missing in Theorem 8.39 if we study the quasi-co-Frobenius property in Section 8.4.3.This (partially) explains why the notion of a quasi-co-Frobenius coring is not left-rightsymmetric.

(2) Considering the functors J ′ and K′ (see (242)), we can construct another Morita contextY(K′,J ′). A pair of invertible elements in this context describes when C ∼= RRat(C∗).This can be in particular of interest when C = C is a coalgebra over a field, since in thatcase we know from [78] C is at the same time left and right co-Frobenius if and only ifC ∼= Rat(C∗).

8.4.3. Quasi-co-Frobenius corings and related functors. Let I be any index set and con-sider the objects C and (∗C)I in the category AM∗C and the objects (C)(I) and C∗ in the category

C∗MA, we obtain in this way the Morita contexts

N((∗C)I ,C) = (AEnd∗C((∗C)I),AEnd∗C(C),AHom∗C(C, (∗C)I),AHom∗C((∗C)I ,C), , •);N(C(I),C∗) = (C∗EndA(C(I)), C∗EndA(C∗), C∗HomA(C∗,C(I)), C∗HomA(C(I),C∗), , •).

Consider again the functors J and J from (242) and (243) and the functors Ks and Ks from(245). We can construct the following Morita contexts.

Y(Ks,J ) = (Nat(Ks,Ks),Nat(J ,J ),Nat(J ,Ks),Nat(Ks,J ), 4, N);

Y(Ks, J ) = (Nat(Ks, Ks),Nat(J , J ),Nat(J , Ks),Nat(Ks, J ), 4, N).

Dually, we can consider functors I = AHom∗C(−, ∗C),Ls = AHom∗C(−,C(I)) : AM∗C → M∗C∗

and I = AHom∗C(∗C,−), Ls = AHom∗C(C(I),−) : AM∗C → ∗C∗M. Out of these functors we

construct Morita contexts Y(Ls, I) and Y(Ls, I).Consider the functors F ,G,H and S from Section 8.3. We immediately obtain the following Moritacontext.

N(SF ,H) = (Nat(SF ,SF),Nat(H,H),Nat(H,SF),Nat(SF ,H),,).

Applying the techniques of Section 8.2, we find a Morita context of type (232) connecting F andGS and a context connecting SF and G.

M(F ,GS) = (Nat(GS,GS),Nat(F ,F)op,Nat(MA,FGS),Nat(GSF ,MC), ♦, );

M(SF ,G) = (Nat(G,G), (Nat(SF ,SF)op,Nat(MA,SFG),Nat(GSF ,MC), ♦, ).

Let us give the explicit form of the connecting maps. Denote α ∈ Nat(GSF ,MC), β ∈Nat(MA,FGS), β′ ∈ Nat(MA,SFG) and γ ∈ Nat(GSF ,MC), N ∈ MA and M ∈ MC


then

(β♦γ)N = γGSN GSβN , (γβ)M = FγM βFM ;(β ′♦α)N = αGN Gβ′N , (αβ′)N = SFαM β′SFM .

The left hand versions of the previous two contexts can be obtained by considering the functorsCF = F ′, CH = H′ : CM → AM, CG = G′ : AM → CM and S ′ : AM → AM. This way, weobtain contexts N(S ′F ′,H′), M(F ′,G′S ′) and M(S ′F ′,G′).

Theorem 8.39. Let C be an A-coring and keep the notation from above.

(i) There exist morphisms of Morita contexts as in the following diagram.

N(S ′F ′,H′)n

wwooooooooooon′

((QQQQQQQQQQQQ

M(F ,GS) m //

f

Mtop(S ′F ′,G′)

N(C(I), ∗C)b

''NNNNNNNNNNNb′

xxqqqqqqqqqqq

Y(Ls, I) c // Y(Ls, I)

For an arbitrary coring C, the morphisms b, b′, c and m, n, n′ are isomorphisms of Moritacontexts, if C is locally projective as left A-module, then f is an isomorphism of Moritacontexts as well.

(ii) There exist morphisms of Morita contexts as in the following diagram.

N(SF ,H)n

wwoooooooooooon′

''PPPPPPPPPPPP

M(F ′,G′S ′) m //

f

Mtop(SF ,G)

N(C(I),C∗)b′

xxpppppppppppb

''OOOOOOOOOOO

Y(Ks,J ) c // Yop(Ks, J )

For an arbitrary coring C, the morphisms b, b′, c and m, n, n′ are isomorphisms of Moritacontexts, if C is locally projective as left A-module, then f is an isomorphism of Moritacontexts as well.

(iii) There exists a anti-morphism of Morita contexts

N(C(I),C∗)a // N(C, (∗C)I).

Proof. (i). The algebra isomorphisms for n, n′ and m follow immediately from Proposi-tion 8.14, Remark 8.15 and Proposition 8.21. The maps that describe the isomorphisms for theconnecting bimodules are given in Proposition 8.24, together with their left-right dual versions.The Morita morphisms b and b′ are obtained by left-right duality out of b and b′ of part (ii). Toconstruct b we can work as follows. The algebra isomorphism b2 follows from Proposition 8.19in combination with Remark 8.15. The algebra isomorphism b1 is given in Proposition 8.25. Theisomorphisms for the connecting bimodules of b3 and b4 are given in Proposition 8.25. The mor-phism b′ is constructed in the same way.


The first algebra morphism of f is constructed as follows. We know by Proposition 8.21 thatNat(GS,GS) ∼= CEndA(C(I))

op. Hence we have an algebra map

f1 : Nat(GS,GS) ∼= CEndA(C(I))op → C∗EndA(C(I))

op.

The algebra map f2 : Nat(F ,F)op → C∗EndA(C∗) is given explicitly in Remark 8.15, and thebimodule maps f3 and f4 follow from Lemma 8.22 and Proposition 8.23 respectively. Moreover,we f2 and f3 are always bijective and when C is locally projective as left A-module, then f1 is anisomorphism by a rationality argument and f4 is an isomorphism by Lemma 8.22.We leave it to the reader to verify that all given (iso)morphisms of algebras and bimodules doindeed form Morita morphisms and that the stated diagrams of Morita morphisms commute.(ii). Follows by left-right duality.

(iii). Consider the element ε ∈ Hom(C(I), A), defined by the following diagram,

C(I)ε // A

C

ι`

OO

ε

77pppppppppppppp

Then we have map C∗EndA(C(I)) → AHomA(C(I), A) defined by composing with ε on the left.Combining Lemma 8.1 and Lemma 8.2 we find that

AHomA(C(I), A) ∼= (AHomA(C, A))I = ∗C∗I ∼= ((∗C)I)A ∼= AHom∗C((∗C)I , (∗C)I).

Composing these maps we obtain a linear map a1 : C∗EndA(C(I)) → AEnd∗C((∗C)I) one can easilycheck that this is an anti-algebra morphism. The algebra map a2 : C∗EndA(C∗) → AEnd∗C(C)op isconstructed as in the proof of Theorem 8.35 part (i). From Lemma 8.1 we obtain an isomorphisma3 : C∗HomA(C(I),C∗) → AHom∗C(C, (∗C)I). The last morphism a4 : C∗HomA(C∗,C(I)) →AHom∗C((∗C)I ,C) is constructed as follows. Denote f(ε) = (z`) for any f ∈ C∗HomA(C∗,C(I)).Then we define a4(f)(f`) =

∑` z` · f`, for (f`) ∈ (∗C)I . The reader can check that the four

morphisms together make up an anti-morphism of Morita contexts.

Corollary 8.40. Consider an A-coring C. There exists a split epimorphism of the formj ∈ C∗HomA(C(I),C∗) if and only if there exists a split monomorphism ∈ AHom∗C(C, (∗C)I),whose left inverse is induced by an element (z`) ∈ (CA)(I). If any of these equivalent conditionsholds then C finitely generated and projective as a right A-module.

Proof. Consider the anti-morphism of Morita contexts a of Theorem 8.39(iii). First note thatthe condition for the left inverse of means exactly that it lies inside the image of a4. Suppose jhas a right inverse . Consider the morphism of Morita contexts a from Theorem 8.39. Then weobtain that a4() is a left inverse for a3(j). For the converse, suppose that has a left inverse ofthe form a4(). We know that a3 is an isomorphism, so we can write = a3(j) for some morphismj ∈ C∗HomA(C(I),C∗). Then we find a4() a3(j) = C = a2(C∗). Since a is an anti-morphismof Morita contexts we find that is a right inverse for j. Finally, denote (z`) ∈ (CA)(I) for therepresentative of the left inverse of . Then we find for all c ∈ C,

c =∑`

z`(1)`(c)(z`(2)) =∑`

z`(1)j`(z`(2))(c),

i.e. z`(1), j`(z`(2)) is a finite dual basis for C as a right A-module.

Theorem 8.41. Suppose that C is an A-coring which is locally projective as a left A-module.Then

(i) The following statements are equivalent(a) C is left locally quasi-Frobenius;(b) there exists a C∗-A bilinear map : C(I) → C∗ such that

T () = ψ | ψ ∈ C∗HomA(C∗,C(I)) ⊂ C∗EndA(C∗) ∼= (∗C∗)op

acts unital on all objects of the generating subcategory CfgpM of CM;


(c) there exists a C∗-A bilinear map : C(I) → C∗ such that T () acts with right local unitson C;

(d) there exists a natural transformation J ′ : G′S ′F ′ → CM such that

R(J ′) = J ′′β′ | β′ ∈ Nat(AM,F ′G′S ′) ⊂ Nat(F ′,F ′)op

acts unital on the generating subcategory CfgpM of CM;

(e) F ′ is a right CfgpM-locally quasi-adjoint for G′;

(f) there exists a natural transformation J : GSF →MC such that

S(J) = β ♦J | β ∈ Nat(MA,SFG) ⊂ Nat(G,G)op

acts unital on the generating subcategory MCfgp of MC;

(g) G is a left MCfgp-locally quasi-adjoint pair for F ;

(h) there exists a natural transformation α ∈ Nat(Ks,J ) such that

α β | β ∈ Nat(J ,Ks) ⊂ Nat(J ,J ) ∼= ∗C∗op

acts with right local units on C;(j) there exists a natural transformation α ∈ Nat(J , Ks) such that

β α | β ∈ Nat(Ks, J ) ⊂ Nat(J , J ) ∼= ∗C∗

acts with left local units on C;(ii) dually, we can characterize right locally quasi-Frobenius corings; in particular C is right locally

quasi-Frobenius if and only if G′ is a left MCfgp-locally quasi-adjoint for F ′ if and only if F is

a right CfgpM-locally quasi-adjoint for G;

(iii) C is left and right locally quasi-Frobenius if and only if (F ,G) is aMCfgp-locally quasi-Frobenius

pair of functors if and only if (F ′,G′) is a CfgpM-locally quasi-Frobenius pair of functors.

Proof. (i). (a) ⇒ (b) Suppose that C is left locally Frobenius. We know by Corollary 8.31

that Im ′, where ′ : (C(I))A → (C∗)A, acts with (left) local units on the objects of CM, andtherefore unital on the objects of C

fgpM (see Theorem 2.34). Since C∗HomA(C∗,C(I)) ∼= (C(I))A

(Lemma 8.22), we can identify Im ′ with T ()op and the statement follows.(b) ⇒ (c). Follows by Theorem 2.34.

(c) ⇒ (a). Follows by Theorem 8.28 and Corollary 8.31 using the same interpretation of T () as

in the proof of part (a) ⇒ (b).(b) ⇔ (d). Since a left locally quasi-Frobenius coring is locally projective as left A-module, this is

in fact an immediate consequence of the isomorphism of Morita contexts f of Theorem 8.39, part(ii). We give however a direct proof.Condition (d) means that for any M ∈ C

fgpM and any left A-module morphism f : F ′M → F ′M ′

with M ′ ∈ CM, we can find a β ∈ Nat(AM,F ′G′S ′) such that

(J ′′β)M ′ = F ′J ′M ′ βF ′M ′ f = f

i.e. the following diagram commutes

(247) F ′Mf

&&NNNNNNNNNNNf

xxppppppppppp

F ′M ′(J ′′β)M′ //

βF′M′ &&NNNNNNNNNNN F ′M ′

F ′G′S ′F ′M ′F ′J ′

M′

88ppppppppppp

where the commutativity of the lower triangle is nothing else than the definition of ′. Since C islocally projective as a left A-module, we find by Theorem 8.39 an isomorphism of Morita contextsf : N(C(I),C∗) → M(F ′,G′S ′). This implies that that the existence of a natural transformation


β is equivalent to the existence of a morphism ψ ∈ C∗HomA(C∗,C(I)). We can translate diagram(247) now into the following diagram

Mf

yyrrrrrrrrrrrf

%%LLLLLLLLLLL

M ′

ψ1 %%KKKKKKKKKK M ′

C(I) ⊗AM ′ψ2

99ssssssssss

where ψ1 and ψ2 are given by

ψ1(m) = ψ(ε)⊗A m(248)

ψ2((c`)⊗A m) = (c`) ·m,(249)

where m ∈M ′ and c` ∈ C(I). So the above diagram commutes if and only if m = (ψ(ε)) ·m forall m ∈ Im f , i.e. if and only if we can find a local unit for all elements of Im f and this local unithas to be of the form ψ(ε). Note that this local unit is exactly an element of Im ′ ∼= T ()op.If condition (b) holds, then we know that there exists such a unit for all left C-comodules that arefinitely generated and projective as a left A-module, so in particular we find a local unit for Im f ,and thus condition (d) holds as well. Conversely, if condition (d) holds, than we find as above alocal unit in Im ′ ∼= T ()op for all modules of the form Im f . Taking M ′ = M and f the identitymap, we obtain a local unit for all M ∈ C

fgpM, i.e. (b) is satisfied as well.

(d) ⇔ (e). Follows directly from the definition.

(c) ⇔ (f). Condition (f) means that for any M ∈ MCfgp and f : M → G(N) = N ⊗A C with

N ∈MA, there exists β ∈ Nat(MA,SFG) such that

(β ♦J)N f = JGN GβN f = f,

or the following diagram commutes.

Mf

xxrrrrrrrrrrrf

&&LLLLLLLLLLL

G(N)(β♦J)N //

GβN %%KKKKKKKKKKG(N)

GSFG(N)JGN

99ssssssssss

Since C is locally projective as a left A-module, we find by Theorem 8.39 an isomorphism betweenthe Morita contexts N(C(I),C∗) and Mtop(SF ,G). Thus, the existence of β as above is equivalentto the existence of an C∗-A bilinear map ψ : C∗ → C(I) such that the following diagram commutes

(250) Mf

vvmmmmmmmmmmmmmmf

((QQQQQQQQQQQQQQ

N ⊗A C

ψ1 ''PPPPPPPPPPPP N ⊗A C

N ⊗A C(I) ⊗A C

ψ2

77nnnnnnnnnnnn

where ψ1 and ψ2 are given by

ψ1(n⊗A c) = n⊗A ψ(ε)⊗A c(251)

ψ2(n⊗ (ci)⊗ c) = n⊗A (ci) · c.(252)

Here we denoted n ∈ N , c ∈ C and (ci) ∈ C(I). Then diagram (250) commutes if and only if∑i ni ⊗ (ψ(ε)) · ci =

∑i ni ⊗A ci for all

∑i ni ⊗A ci ∈ Im f . Suppose that condition (f) holds

and take any c ∈ C. Put N = A and M = cA, the cyclic right A-module generated by c and let

8.4. REFERENCES 233

f : M → A ⊗A C ∼= C be the canonical injection. Then by diagram (250), we obtain a left localunit ψ(ε) on M , i.e. we find left local unit in T ()op for c. This shows that (f) implies (c).Conversely, if condition (c) is satisfied, then we know that we can find a left local unit in T () forany finite number of elements in C. Take any M ∈ MC

fgp and f : M → N ⊗A C. Then Im f isalso finitely generated. Take a finite number of generators for Im f and denote representatives ofthem by ni⊗A ci (to reduce the number of indices, we omit a summation if we denote an elementof N ⊗A C). By (c) we know that we can find a left local unit e = ψ(ε) ∈ T ()op for thegenerators ci, i.e. such that e · ci = ci for all i, for a particular choice of ψ ∈ C∗HomA(C∗,C(I)).If we define ψ1 and ψ2 as in (251), we find that diagram (250) commutes and we obtain indeedthat (c) implies (f).(f) ⇔ (g). Follows directly from the definition.

(c) ⇔ (h) ⇔ (j). Follows from the isomorphisms of Morita contexts b and b′ of Theorem 8.39.

(ii). Follows by left-right duality.

(iii). Is a direct combination of the first two parts.

Remark 8.42. If one takes the index-set I to contain a single element in the previous Theorem,then we obtain a characterization of locally Frobenius corings (and consequently of co-Frobeniuscorings if the base ring is a PF-ring). In particular, we find that an A-coring C is left locallyFrobenius if and only if F ′ is a right C

fgpM-locally adjoint for G′ if and only if G is a left MCfgp-

locally adjoint for F . Moreover C is at the same time left and right locally Frobenius if and onlyif (F ,G) is a MC

fgp-locally Frobenius pair if and only if (F ′,G′) is a CfgpM-locally Frobenius pair.

References

All results in this chapter are deduced from the author’s joint work with M. Iovanov [77].

Chapter 9Applications to Galois Theory

In this final chapter, we study the relation between Galois theory for comodules developed inPart II and the sebarability and Frobenius properties of Chapters 7 and 8.

In Section 9.1 we show that if C is a coseparable coring, every firm R-C Galois comoduleinduces an equivalence of categories between the category of comodules over C and modules overthe endomorphism ring of the Galois comodule. We prove as well a stronger version of the Galoiscomodule structure theorem if C is moreover projective as A-module. Then the surjectivity of thecanonical map implies its bijectivity and hence we obtain an equivalence of categories.

In Section 9.2 we show how Frobenius properties induce more symmetry in the Morita contextsassociated to corings. This makes it possible to construct new Morita contexts which generalizesome well-known results in Hopf algebra theory. We give a partial converse for this construction: ifthe Morita context is strict, then the extra symmetry of the context implies the Frobenius property.

In Section 9.3 we consider, under finiteness conditions, the coring and its dual as a Galoiscomodule. In the finite case, the Galois theory of the dual of the coring appears to be trivial andin the locally finite case we recover the theory of rational modules. The Galois theory of the coringitself is closely related to the quasi co-Frobenius property.

9.1. Separable corings and Galois comodules

From the characterization of separable corings in Chapter 7, it follows that an A-coring C is acoseparable coring if and only if the forgetful functor FC : MC →MA is a separable functor (seeCorollary 7.17). In particular, by Raphael’s theorem (Theorem 7.1), this means that the unit ofthe adjunction (FC,GC) has a left inverse. We know from Corollary 3.9 that for any M ∈MC, theunit %M of this adjunction is given by the comultiplication %M = ρM : M →M ⊗A C. Therefore,C is coseparable if and only if there exists for all M ∈MC a map

ρM : M ⊗A C →M

such that ρM is natural in M and ρM ρM = M . The naturality implies in particular that for anyϕ ∈ EndC(M), ρM (ϕ⊗A C) = ρM . Hence, ρM is R-C bicolinear for any (not necessary unital)ring R whose action on M is defined by a ring morphism R→ EndC(M). Since we know as wellfrom Corollary 7.17 that C is coseparable if and only if the forgetful functor CF : CM → AMis separable, the coseparability of C implies furthermore for all N ∈ CM the existence of an C-Sbicolinear map

λN : C⊗A N → N,

left inverse for the comultiplication λN : N → C⊗AN , where S is any ring whose action on N isinduced by a ring morphism S → CEnd(N).

235

236 CHAPTER 9. APPLICATIONS TO GALOIS THEORY

The following lemma can be found (in case where R is a unital ring) in [117, 2.13], we give asketch of the prove for sake of completeness.

Lemma 9.1. Let R be a firm ring and C an A-coring. Take M ∈ MC, N ∈ CMR andP ∈ RM, the natural map

f : (M ⊗C N)⊗R P →M ⊗C (N ⊗R P )

is an isomorphism in each of the following situations:

(i) ρM : M →M ⊗A C has a left inverse ρM in MC;(ii) λN : N → C⊗A N has a left inverse λN in CMR.

Proof. Let us show that the equalizer defining the cotensor product

0 // M ⊗C Ni // M ⊗A N

ρM⊗AN //

M⊗AλN// M ⊗A C⊗A N .

is a contractable equalizer. We define maps α : M ⊗AN →M ⊗CN and β, β′ : M ⊗AC⊗AN →M ⊗A N as follows. Under the conditions of (i) we put α = (ρM ⊗A N) (M ⊗A λN ) andβ = ρM ⊗AN , if (ii) is satisfied, then we define α = (M⊗A λN )(ρM ⊗AN) and β′ = M⊗A λN .Then α, β and β′ are right R-linear maps. Moreover, one can easily verify that α i = M ⊗C N ,β (ρM ⊗AN) = M ⊗AN = β′ (M ⊗A λN ) and α i = β (N ⊗A λN ) = β′ (ρM ⊗AN). By[13, Proposition 3.3.2], any functor preserves a contractable equalizer. Therefore, we obtain the

following diagram, applying the functor −⊗R P : MR → Ab,

0 // (M ⊗C N)⊗R P //

f

M ⊗A N ⊗R P //

∼=

// M ⊗A C⊗A N ⊗R P∼=

0 // M ⊗C (N ⊗R P ) // M ⊗A N ⊗R P // // M ⊗A C⊗A N ⊗R P

Since we know that both horizontal rows are equalizers, f is an isomorphism by the universalproperty of the equalizer.

Theorem 9.2. Let C be a coseparable A-coring and R a firm ring that is flat as a left R-module. If Σ a firm R-C Galois comodule and R is an ideal in T = EndC(Σ), then − ⊗R Σ :MR →MC is an equivalence of categories.

Proof. Since Σ ∈ RMC is R-firmly projective as a right A-module, we know by Proposi-tion 3.53 that Σ† = Σ∗ ⊗R R ∈ CMR. From the observations in the beginning of this section, we

know that the coactions ρΣ : Σ → Σ⊗A C and λΣ† : Σ† → C⊗A Σ† have a left inverse in RMC,respectively CMR. Hence it follows by Lemma 9.1 that for all N ∈ MR and all M ∈ MC, thefollowing isomorphisms hold

(N ⊗R Σ)⊗C Σ† ∼= N ⊗R (Σ⊗C Σ†)

(M ⊗C Σ†)⊗R Σ ∼= M ⊗C (Σ† ⊗R Σ)

Applying Theorem 4.22 and Theorem 4.25, we obtain that both − ⊗R Σ and its right adjointHomC(Σ,−)⊗R R ' −⊗C Σ† are fully faithful.

Lemma 9.3. Let C be an A-coring, R a firm ring that is faithfully flat as a left R-moduleand Σ ∈ RMC that is R-firmly projective as right A-module. If R is a left ideal in T = EndC(Σ)and (Σ† ⊗R Σ) ⊗C Σ ∼= Σ† ⊗R (Σ ⊗C Σ), then Σ is a firm R-C Galois comodule if and only ifcan : Σ† ⊗R Σ → C is a split epimorphism in MC.

Proof. Consider the following exact sequence in MC

(253) 0 // K // Σ† ⊗R Σcan // C

where K is the kernel of can. If can is split epi in MC, then this means that the above sequenceis split. If we apply the functor −⊗C Σ† : MC →MR to this sequence we obtain

0 // K ⊗C Σ† // (Σ† ⊗R Σ)⊗C Σ†can⊗CΣ† // C⊗C Σ† ∼= Σ†

9.2. FROBENIUS PROPERTIES AND MORITA THEORY FOR COMODULES 237

as − ⊗C Σ† ' HomC(Σ,−) ⊗R R is left exact, having a left adjoint − ⊗R Σ : MR → MC.Furthermore, we have

(Σ† ⊗R Σ)⊗C Σ† ∼= Σ† ⊗R (Σ⊗C Σ)∼= Σ† ⊗R EndC(Σ)⊗R R∼= Σ† ⊗R R ∼= Σ†

where we applied Theorem 4.13 to obtain the second isomorphism and Lemma 2.14 to obtain thethird isomorphism. Therefore, K ⊗C Σ† ∼= HomC(Σ,K) ⊗R R = 0 and since R is faithfully flatas left R-module, HomC(Σ,K) = 0. Since the (253) is split, K is a direct summand of Σ† ⊗R Σ.We know that R generates MR (see Lemma 2.9), hence Σ generates Σ†⊗RΣ and K. Combiningour results, we find that K = 0 and can is bijective.

Corollary 9.4. Let C be an A-coring, R a firm ring that is faithfully flat as a left R-moduleand Σ ∈ RMC that is R-firmly projective as right A-module. Suppose further that R is a leftideal in T = EndC(Σ) and C is a coseparable coring that is projective as right A-module. If canis surjective, then −⊗R Σ : MR →MC is an equivalence of categories.

Proof. By Theorem 9.2 and Lemma 9.3 we only have to show that can is split epi in MC.Since C is projective as right A-module, can is split epi in MA and because of the coseparabilityof C this implies that can is also split epi in MC.

9.2. Frobenius properties and Morita theory for comodules

9.2.1. Frobenius corings. Let C be an A-coring, Σ a right C-comodule and consider theMorita context

M(Σ) = (T, ∗C,Σ, Q, O, H),

constructed in (143). In the next Proposition we examine this context in the situation where C isa Frobenius coring.

Proposition 9.5. Let C be a Frobenius A-coring, Σ a right C-comodule and M(Σ) theMorita context associated to Σ as introduced in (143). Then there exists an isomorphism ofA-T bimodules J : Σ∗ → Q. The Morita context M(Σ) is isomorphic to the Morita contextM(Σ) = (T, ∗C,Σ,Σ∗, µ, τ), where the left ∗C-action on Σ∗ and the maps µ and τ are givenexplicitly by

(g · f)(u) = θ(z(1)g(z(2))⊗A f(u[0])u[1]

),

µ : Σ∗ ⊗T Σ → ∗C, µ(f ⊗T u)(c) = θ(c⊗A f(u[0])u[1]),

τ : Σ⊗∗C Σ∗ → T, τ(u⊗∗C f)(v) = u[0]θ(u[1] ⊗A f(v[0])v[1]),

where f ∈ Σ∗, g ∈ ∗C, u, v ∈ Σ, c ∈ C and (z,θ) is a Frobenius system for C.Conversely, if C satisfies the equivalent conditions of Theorem 5.19, and if Σ∗ and Q are

isomorphic as A-T bimodules, then C is a Frobenius coring.

Proof. Applying the functor G = HomC(Σ,−) to the Frobenius map j ∈ AHom∗C(C, ∗C),we obtain the following isomorphism in MT

HomC(Σ, j) : HomC(Σ,C) → HomC(Σ, ∗C).

Now HomC(Σ,C) ∼= HomA(Σ, A) = Σ∗ and Hom∗C(Σ, ∗C) ∼= CHom(C,Σ∗) = Q (see Lemma 5.8and Remark 5.9). Hence we obtain an isomorphism of right T -modules J : Σ∗ → Q =HomC(Σ, ∗C). A straightforward computation shows that J is given by the formula

J(f)(u) = j(f(u[0])u[1]),

for all f ∈ Σ∗ and u ∈ Σ. Let us show that J is left A-linear.

J(af)(u) = j((af)(u[0])u[1]) = j(a(f(u[0])u[1])) = aj(f(u[0])u[1]) = aJ(f)(u),


where we used the A-linearity of Σ∗ and j. The left ∗C-module structure on Q can be transferredto Σ∗:

(g · f)(u) = J−1(g · J(f))(u) = ε j−1((g · J(f))(u))= ε j−1(g ∗ J(f)(u)) = ε j−1(g ∗ j(f(u[0])u[1]))

= ε(z · (g ∗ j(f(u[0])u[1]))) = ε(z(1) · (j(f(u[0])u[1])(z(2)g(z(2)))))

= ε(z(1)θ(z(2)g(z(3))⊗A f(u[0])u[1]))

= ε(z(1))θ(z(2)g(z(3))⊗A f(u[0])u[1])

= θ(z(1)g(z(2))⊗A f(u[0])u[1]).

Now take the Morita context M(Σ) from (143). Using the isomorphisms J and

β : Hom∗C(Σ, ∗C) → CHom(C,Σ∗) = Q, β(q)(u)(c) = q(c)(u),

we find the connecting maps of the Morita context:

µ(f ⊗T u)(c) = β(J(f))(c)(u) = J(f)(u)(c)= j(f(u[0])u[1])(c) = θ(c⊗A f(u[0])u[1]).

Remark that µ(f ⊗T u) = J(f)(u).

τ(u⊗∗C f)(v) = u[0](β(J(f))(u[1])(v)) = u[0](J(f)(v)(u[1]))

= u[0]j(f(v[0])v[1])(u[1]) = u[0]θ(u[1] ⊗A f(v[0])v[1]).

Conversely, assume that C satisfies the equivalent conditions of Theorem 5.19, and let J : Σ∗ → Qbe an isomorphism of A-T bimodules. Then we find that J ⊗T Σ : Σ∗ ⊗T Σ → Q ⊗T Σ,H : Q ⊗T Σ → ∗C and can : Σ∗ ⊗T Σ → C are A-∗C bimodule isomorphisms. Then j =H (J ⊗T Σ) can−1 is an A-∗C bimodule isomorphism between C and ∗C, and we find that ∗C isFrobenius.

Recall that the ring extension R/A is called Frobenius if there exists a Frobenius system (e, ν)consisting of an A-bimodule map ν : R→ A and e = e1 ⊗A e2 ∈ R⊗A R (summation implicitlyunderstood) such that

(254) re1 ⊗A e2 = e1 ⊗A e2rfor all r ∈ R, and

(255) ν(e1)e2 = e1ν(e2) = 1.

The element e is called a Casimir element. As in the coring case, a ring extension is Frobenius ifand only if to the restrictions of scalars MR →MA is a Frobenius functor, which means that itsleft adjoint HomA(R,−) and right adjoint −⊗AR are isomorphic functors MA →MR (see [46,Sections 3.1 and 3.2]). This is furthermore equivalent to R being finitely generated and projectiveas right A-module and R and HomA(R,A) being isomorphic as R-A bimodules.

The following theorem shows that Frobenius corings and Frobenius ring extensions are into aone-to-one correspondence.

Theorem 9.6. Let R/A be a Frobenius ring extension with Frobenius system (e, ν). PutC = (R,∆, ν), where we define ∆ : R → R ⊗A R, ∆(r) = re = er. Then C is an A-coring.Moreover, C∗ ∼= R as rings and C is a Frobenius coring.

Conversely, if C is a Frobenius coring, then i : A→ ∗C, i(a) = εa is a Frobenius ring extension.

Proof. From the definition of ∆ and (254) it is clear that ∆ is A-bilinear. Let us check that∆ is coassociative

(C⊗A ∆) ∆(r) = (C⊗A ∆)(re1 ⊗A e2) = re1 ⊗A e2e1 ⊗A e2

(∆⊗A C) ∆(r) = (∆⊗A C)(re1 ⊗A e2) = re1e1 ⊗A e2 ⊗A e2 = re1 ⊗A e2e1 ⊗A e2

here we made use of (254) in the last equality and we denoted e = e1 ⊗A e2 = e1 ⊗A e2. Thecounit property follows immediately from (255).

Consider C∗ = AHom(R,A) which is an A-ring with structure maps

(af)(r) = af(r) (fa)(r) = f(ar) (f ∗ g)(r) = f(g(re1)e2)


for all a ∈ A, f, g ∈ ∗C and r ∈ R. We define maps

α : HomA(R,A) → R α(f) = f(e1)e2

β : R→ HomA(R,A) β(r)(s) = ν(rs)

where we denote f ∈ HomA(R,A), and r, s ∈ R. Take any f, g ∈ HomA(R,A), then we find

α(f ∗ g) = (f ∗ g)(e1)e2 = f(g(e1)e2e1)e2

α(f) · α(g) = (f(e1)e2)(g(e1)e2) = f(e1)e2g(e1)e2 = f(g(e1)e2e1)e2

where we applied (254) in the last equation with r = g(e1)e2. Therefore α is a ring morphism.Let us now check that α and β are inverses.

α β(r) = ν(re1)e2 = ν(e1)e2r = r

β α(f)(r) = ν(f(e1)e2r) = f(e1)ν(e2r)= f(e1ν(e2r)) = f(re1ν(e2)) = f(r)

Here we applied (254), (255) and the linearity of f and ν. As R/A is Frobenius, we knowfurthermore that R ∼= HomA(R,A) as R-A bimodules. Combined with the above ring morphism,we find that C ∼= C∗ as C∗-A bimodule and C is a Frobenius coring.

Conversely, if C is a Frobenius coring, then we know that C is finitely generated and projective asa left A-module, and C ∼= ∗C in AM∗C. If we put R = ∗C, then we find HomA(R,A) = (∗C)∗ ∼= Cas R-A bimodules. Hence A → R is a Frobenius ring extension by one of the characterizationsgiven before this theorem.

Theorem 9.7. Let i : A→ R be a morphism of rings, and χ : R→ A a grouplike character.Consider the Morita context (B,R,A,Q, τ, µ) from Proposition 5.25. If R/A is Frobenius, withFrobenius system (e, ν), then A is an R-B bimodule, with left R-action

r · a = ν(raχ(e1)e2)

and A ∼= Q as R-B bimodules. Moreover the Morita context of Proposition 5.25 is isomorphic tothe Morita context

(B,R,A,A, τ, µ)

with connecting maps

µ : A⊗B A→ R : µ(a⊗B a′) = aχ(e1)e2a′;τ : A⊗R A→ B : τ(a⊗R a′) = χ(aa′χ(e1)e2).

Conversely, if the Morita context (B,R,A,Q, τ, µ) is strict and A ∼= Q as R-B bimodule, thenR/A is a Frobenius ring extension.

Proof. This follows from Theorem 9.6 and Proposition 9.5.

Let H be a bialgebra and A a left H-module algebra then (A,H, ρ) is a factorization structureif we define

ρ : H ⊗A→ A⊗H, ρ(h⊗ a) = h(1) · b⊗ h(2).

Furthermore εH : H → k is an algebra morphism. Hence, we can apply the results of Section 6.5.8and obtain we a grouplike character

X : R = A#ρH → A,X(a#h) = εH(h)a

and therefore a Morita context (B,A#ρH,A,Q, µ, τ), where

B = b ∈ A | h · b = ε(h)b, for all h ∈ H,Q =

∑i

ai#hi ∈ A#ρH |∑i

h(1) · ai#h(2)hi = ε(h)∑i

ai#hi, for all h ∈ H

The left A#ρH action on A is given by

(a#h)b = a(h · b).


Finally, the connecting maps are given by

µ : A⊗B Q→ A#H, µ(a⊗B (∑i

ai#hi)) =∑i

aai#hi;

τ : Q⊗ρ A→ B, τ(∑i

ai#hi)⊗ρ a) =∑i

aihi · a.

In the particular situation where H is a finite dimensional Hopf algebra over a field k, thereexists another Morita context connecting B and A#H, due to Cohen, Fischman and Montgomery(see [53]). The construction can be generalized to the case where H is a Frobenius Hopf algebraover a commutative ring k (see [52]). This Morita context can be described as follows. Take afree generator t of the space of left integrals in H, and let λ be the distinguished grouplike elementin H∗. Then we have that ht = ε(h)t and th = λ(h)t for all h ∈ H; λ is an algebra map, and Ais a B-A#H bimodule, the right A#H-action is given by

a(b#h) = λ(h(2))S(h(1)) · (ab)

and we have a Morita context

(B,A#H,A,A, τ , µ)

with

τ : A⊗ρ A→ B, τ(a⊗ρ b) = t · (ab);

µ : A⊗B A→ R, µ(a⊗B b) = a(t(1) · b)#t(2).

We refer to [53] for the details. We will now show that this Morita context can be obtained usingProposition 5.25 and Theorem 9.7.If H is Frobenius, then there exists a left integral ϕ in H∗ such that 〈ϕ, t〉 = 1. The element ϕ isa free generator of the space of left integrals in H∗, and (t(2) ⊗ S(t(1)), ϕ) is a Frobenius systemfor H/k (see for example [46, Theorem 31]). This means that

(256) ht(2) ⊗ S(t(1)) = t(2) ⊗ S(t(1))h and 〈ϕ, t(2)〉S(t(1)) = t(2)〈ϕ, S(t(1))〉 = 1

for all h ∈ H.

Proposition 9.8. Let H be a Frobenius Hopf algebra, let t and ϕ be as above, and take aleft H-module algebra A. Then A#H/A is Frobenius, with Frobenius system(

e = (1#t(2))⊗A (1#S(t(1))), ν = IA#ϕ)

Proof. For all a ∈ A and h ∈ H, we have((1#t(2))⊗A (1#S(t(1))

)(a#h) = (1#t(3))⊗A

(S(t(2)) · a#S(t(1))h

)=

((t(3)S(t(2))) · a#t(4)

)⊗A

(1#S(t(1))h

)= (a#t(2))⊗A

(1#S(t(1))h

)= (a#h)

((1#t(2))⊗A (1#S(t(1))

).

It is obvious that ν is left A-linear. It is also right A-linear since

ν((1#h)a) = ν(h(1)a#h(2)) = 〈ϕ, h(2)〉h(1)a = 〈ϕ, h〉a.

Finally, using (256), we find that

ν(1#t(2))(1#S(t(1))) = 1#S(〈ϕ, t(2)〉t(1)) = 1#S(〈ϕ, t〉1) = 1#1

and (1#t(2))ν(1#S(t(1))) = 1#t(2)〈ϕ, S(t(1))〉 = 1#1.

Corollary 9.9. As in Proposition 9.8, let H be a Frobenius Hopf algebra, and A a leftH-module algebra. Then A and Q are isomorphic as A-A#H bimodules and the Morita contextsfrom Proposition 5.25 and [53],[52] are isomorphic.


Proof. The fact that A and Q are isomorphic follows immediately from Theorem 9.7 andProposition 9.8. The connecting isomorphisms are α : A → Q, α(a) = t(1) · a#t(2) and

α−1 = IA#ϕ|Q. Let us check that the right A#H-action on A transported from the one on Qcoincides with the A#H-action from [53]:

a(b#h) = ν(α(a)(b#h)) = ν((

(t(1) · a)#t(2))(b#h)

)= ν

((t(1) · a)(t(2) · b)#t(3)h

)= 〈ϕ, t(2)h〉t(1) · (ab)

= 〈ϕ, t(2)h(3)〉(t(1)h(2)S(h(1))) · (ab) = 〈ϕ, th(2)〉S(h(1)) · (ab)= 〈λ, h(2)〉〈ϕ, t〉S(h(1)) · (ab) = 〈λ, h(2)〉S(h(1)) · (ab)

as needed.

9.2.2. Co-Frobenius corings. Recall from Chapter 8 that a coring C is called left co-Frobeniusif C ∈ AM is locally projective, and if there exists an injective map j ∈ AHom∗C(C, ∗C). Thefollowing result should be viewed as a generalization of [15, Theorem 2.10].

Proposition 9.10. Let C be an A-coring, take Σ ∈ MC and consider the Morita contextM(Σ) of (143) associated to Σ. If C is left co-Frobenius, then there exists a monomorphism ofA-T bimodules J : Σ∗ → Q.Conversely, if the equivalent conditions of Theorem 5.22 are satisfied, and if there exists a monomor-phism of A-T bimodules J : Σ∗ → Q, then C is left co-Frobenius.

Proof. We construct J in the same way as in Proposition 9.5: J(f)(u) = j(f(u[0])u[1]).Let us show that J is injective. If J(f(u)) = 0, then j(f(u[0])u[1]) = 0, for all u ∈ Σ. Sincej is injective, this implies that f(u[0])u[1] = 0, hence 0 = ε(f(u[0])u[1]) = f(u[0])ε(u[1]) =f(u[0]ε(u[1])) = f(u), for all u ∈ Σ, and f = 0. Conversely, if J : Σ∗ → Q is a monomorphismof A-T bimodules, then we have a morphism of A-∗C bimodules J ⊗T Σ : Σ∗ ⊗T Σ → Q⊗T Σ,which is injective since Σ ∈ TM is flat. Theorem 5.22 also tells us that can : Σ∗ ⊗T Σ ∼= C andH : Q⊗T Σ → Rat(∗C) are isomorphisms. Then j = H (J ⊗T Σ) can−1 : C → Rat(∗C) ⊂ ∗Cis a monomorphism of A-∗C bimodules.

Before we state our next results, we need the following Lemma.

Lemma 9.11. Let A be a commutative ring and C an A-coalgebra. If Rat(C∗) is dense in thefinite topology on C∗, then C∗Hom(Rat(C∗),M) = C∗Hom(C∗,M) for every M ∈ C

fgpM.

Proof. Let E(M) be the injective envelope of M ∈ CM. Then by [36, 9.5] E(M) is alsoinjective as a left C∗-module and we can extend any left C∗-linear χ : Rat(C∗) →M ⊂ E(M) toχ : C∗ → E(M).

Since Rat(C∗) is dense, it has a left local units on M , so we can take e ∈ Rat(C∗) such thate ·m = m with m = χ(εC). We find e · χ(εC) = χ(e#εC) = χ(e) ∈ M . Furthermore, for anyf ∈ C∗, χ(f) = χ(f#εC) = f · χ(εC) ∈M , so χ ∈ C∗Hom(C∗,M).

Finally, χ is unique: suppose that there exists a ξ ∈ C∗Hom(C∗,M) which has also the propertythat ξ(f) = χ(f) for all f ∈ Rat(C∗), and take a local unit e ∈ Rat(C∗) for (ξ − χ)(εC). Then

(ξ − χ)(εC) = e · (ξ − χ)(εC) = (ξ − χ)(e) = 0.

Consequently

(ξ − χ)(f) = (ξ − χ)(f#εC) = f · (ξ − χ)(εC) = 0,

finishing the proof.

Proposition 9.12. Let A be a commutative ring, and C an A-coalgebra. Take Σ ∈MC andconsider the Morita context M(Σ) of (143) associated to Σ. If A is a commutative PF ring and Cis right and left co-Frobenius, then there exists a monomorphism of A-B bimodules J : Σ∗ → Qand an epimorphism of A-B bimodules J ′ : Σ∗ → Q. Conversely, if A is a commutative PF ring,and if the equivalent conditions of Theorem 5.22 are satisfied, then the existence of J and J ′ asimplies that C is right and left co-Frobenius.


Proof. The monomorphism J is constructed as in Proposition 9.10.Since A is a PF ring and therefore injective in MA, C is injective in MC. By [36, 9.5], C is alsoinjective as a C∗-module, so the injective right co-Frobenius morphism j′ : C → Rat(C∗) splits,and C is a direct summand of Rat(C∗) as a C∗-module. We obtain an epimorphism

(257) C∗Hom(Rat(C∗),Σ∗) → C∗Hom(C,Σ∗).

C is right co-Frobenius, so it follows from Corollary 8.33 that C is right locally Frobenius, henceRat(C∗) is dense in the finite topology. From Lemma 9.11, it follows that C∗Hom(Rat(C∗),Σ∗) =Σ∗.To prove the converse, we proceed as in Proposition 9.10. The existence of the monomorphism Jimplies that C is right co-Frobenius. Using the fact that can and H are isomorphisms, we find aB-A bimodule epimorphism

′ = H (J ′ ⊗B Σ) can−1 : C → Σ∗ ⊗B Σ → Q⊗B Σ → Rat(C∗)

Since Rat(∗C) is dense, C is left locally Frobenius. So we find by Corollary 8.33 that C is also leftco-Frobenius.

9.3. The coring as a Galois comodule

9.3.1. The coring as a finite Galois comodule. Let C be an A-coring which is finitelygenerated and projective as a right A-module. We can consider C as a right C-comodule, andtherefore, by Corollary 3.55, C∗ is a left C-comodule. Corollary 3.55 tells moreover that we canassociate to any finite comodule two pairs of adjoint functors (GC,HC) and (CG, CH). If we takeΣ = C and B = T = EndC(C) ∼= C∗, then we obtain the following pairs of adjoint functors.

F : MC∗ →MC, F (N) = N ⊗C∗ C,(258)

G : MC →MC∗ , G(M) = HomC(C,M) ∼= M ⊗C C∗,(259)

and

F ′ : C∗M→ CM, F ′(N) = C∗ ⊗C∗ N,

G′ : CM→ C∗M, G′(M) = CHom(C∗,M) ∼= C⊗C M.

Since C is finitely generated and projective as a right A-module, the categories C∗M and CM areisomorphic (see Section 3.3). The isomorphism and its inverse are precisely given by the functorsF ′ and G′.The associated comatrix coring is D = C∗ ⊗C∗ C and the canonical map

can : D → C, can(f ⊗C∗ c) = f(c(1))c(2)

is the canonical isomorphism. We also have two Morita contexts. The first context is M∗(Σ) fromRemark 5.7 (2), with M = C. We find

C′ = (C∗,C∗, Q = EndC(C) ∼= C∗,C∗, 4, N),

with 4 = N the canonical isomorphism C∗ ⊗C∗ C∗ → C∗. This Morita context is the trivial oneconnecting C∗ to itself.The second context is M(Σ) of (143) with Σ = C. Since C is finitely generated and projective asa right A-module, this context is isomorphic to the the context ∗M(C∗) from Remark 5.7 (3) byProposition 5.10. This leads us to

(260) C = (C∗, ∗C,C, Q, τ, µ),

where Q = CHom(C,C∗) ∼= HomC(C, ∗C) = Q and

τ : C⊗∗C Q→ C∗, τ(c⊗ q) = q(c);

µ : Q⊗C∗ C → ∗C, µ(q ⊗ c) = q(c).

We now want to investigate when (F,G) is a pair of inverse equivalences. In the situation where Cis also finitely generated and projective as a left A-module, the answer is given by Theorem 5.19.We obtain the following result.

9.3. THE CORING AS A GALOIS COMODULE 243

Corollary 9.13. Let C be an A-coring which is finitely generated and projective as a leftand right A-module. Then the following assertions are equivalent.

(i) C ∈ C∗M is faithfully flat;(ii) C ∈ C∗M is a progenerator;(iii) the Morita context (260) is strict;(iv) (F,G) from (258-259) is a pair of inverse equivalences.(v) there exist epimorphisms j1 ∈ C∗Hom(Cn,C∗) and j2 ∈ Hom∗C(Cm, ∗C) for positive integers

n and m (in particular, C is left and right k-locally quasi Frobenius)

If A is a PF ring, this is furthermore equivalent to the existence of monomorphisms j1 ∈∗CHom(C, (C∗)m) and j2 ∈ Hom∗C(C, (∗C)n).

Proof. The equivalence of the first four statements follows directly from Theorem 5.19. Letus prove (iv) ⇔ (v). If τ is surjective, then consider an element

∑ni=1 zi ⊗ qi ∈ C ⊗∗C Q which

maps to εC under τ . Then we can define a left C∗-linear map

j1 : Cn → C∗, j1(ci) =n∑i=1

qi(ci),

where (ci)ni=1 ∈ Cn. Furthermore, j1(zi) = εC, hence j1 is surjective. In the same way we constructthe surjective morphism j2 out of µ.

Conversely, given a left C∗-linear surjective map j1 : Cn → C∗, we construct an inverse for τ asfollows. By the universal property of the direct sum, j1 is completely determined by n left C∗ linearmaps ji1 : C → C∗, satisfying the property that j1(ci) =

∑i ji1(ci) for all (ci)ni=1 ∈ Cn. Clearly,

ji1 ∈ Q for all i. By the surjectivity of j1 there exists an element zi ∈ Cn such that j1(zi) = ε.Then we define τ(

∑i zi⊗A ji1) =

∑i ji1(zi) = j1(zi) = εC, hence τ is surjective. In the same way,

the surjectivity of µ follows from the surjectivity of τ .The last statement is a consequence of Theorem 8.32.

Corollary 9.14. Let C be a Frobenius A-coring. Then the adjoint pair (F,G) from (258-259) is a pair of inverse equivalences.

We will now give other sufficient conditions for (F,G) to be a pair of inverse equivalences.

Theorem 9.15. Let C be an A-coring which is finitely generated and projective as a rightA-module. Then C ∈ C∗M is flat and and C∗ ∈ CM is coflat if and only if the adjoint pair (F,G)from (258-259) is a pair of inverse equivalences.

Proof. This is a consequence of Theorem 4.27 (i) ⇔ (x).

Consider again the case Σ = C ∈MCfgp in Corollary 3.55, but this time we take B = A instead

of B = T . The map ` : B = A → T = EndC(C) = C∗, is now the usual ring homomorphismi : A→ C∗, given by i(a)(c) = aεC(c). We have the two following pairs of adjoint functors (F,G)and (F ′, G′).

F : MA →MC, F (N) = N ⊗A C

G : MC →MA, G(M) = M ⊗C C∗

and

F ′ : AM→ CM, F ′(N) = C∗ ⊗A NG′ : CM→ AM, G′(M) = C⊗C M ∼= M

Remark that G′ is the forgetful functor. We know (see e.g. Section 8.3.1) that the forgetfulfunctor G′ has a right adjoint H ′ = C⊗A−. Furthermore, the functor F also has a left adjoint H,which is also a forgetful functor. Remember that the coring C is a Frobenius coring if and only ifthe forgetful functor MC →MA is a Frobenius functor, This is equivalent to the forgetful functorG′ being Frobenius, see Corollary 8.37. Using the adjoint pairs (F,G) and (F ′, G′), we can statemore equivalent conditions:

Proposition 9.16. Let C be an A-coring which is finitely generated and projective as a rightA-module. With notation as above, the following assertions are equivalent:


(i) C is a Frobenius coring;(ii) there is a natural isomorphism of functors G = −⊗C C∗ ' H;(iii) there is a natural isomorphism of functors F ′ = C∗ ⊗A − ' H ′ = C⊗A −;(iv) G = −⊗C C∗ is a left adjoint of F = −⊗A C;(v) the forgetful functor G′ is a left adjoint of F ′ = C∗ ⊗A −.

9.3.2. The coring as an infinite Galois comodule: rationality properties. Consider asbefore the A-coring C which is locally projective as a left A-module. Recall from Section 3.3the construction of the category of rational modules RM∗C and the equivalence of categoriesMC ' RM∗C of Theorem 3.35. If moreover Rat(∗C) is dense in the finite topology on ∗C,then Rat(∗C) is a ring with right local units, in particular a firm ring, and by Lemma 5.20, theequivalence of categories mentioned above can be reduced further to

MC 'MRat(∗C)

The aim of this section is to show that this equivalence can be obtained by use of Galois comodules.

Theorem 9.17. Consider ∗C⊗AC as elementary algebra. Then the right coaction on Rat(∗C),given by

ρ : Rat(∗C) → Rat(∗C)⊗A C, f 7→ f[0] ⊗ f[1],

is a homomorphism of rings.

Proof. The multiplication on Rat(∗C) is induced by the multiplication on ∗C. The multipli-cation on Rat(∗C)⊗A C is defined as usual by

(f ⊗A c) · (g ⊗A d) = f ⊗A g(c)d,for all f, g ∈ Rat(∗C) and c, d ∈ C. With the same notation, we can compute

(261)ρ(f ∗ g) = (f ∗ g)[0] ⊗A (f ∗ g)[1]

= f ∗ g[0] ⊗A g[1]where we used Proposition 3.38 (iii) in the second equation. On the other hand,

(262)ρ(f) · ρ(g) = (f[0] ⊗A f[1]) · (g[0] ⊗A g[1])

= f[0]g[0](f[1])⊗A g[1]Furthermore, f ∗ g[0](c) = g[0](c(1)f(c2)) and

(f[0]g[0](f[1]))(c) = f[0](c)g[0](f[1]) = g[0](f[0](c)f[1]).

Finally, using Lemma 8.11, we find that the right hand sides of (261) and (262) are equal andthus ρ is indeed a homomorphism of rings.

From now on, we will restrict ourselves to the situation where Rat(∗C) is dense in the finitetopology on ∗C. In particular Rat(∗C) is a ring with right local units.

Now we can apply the results of Section 3.4.1 and construct a comatrix coring

C⊗Rat(∗C) Rat(∗C).

The comultiplication and counit are given respectively by

∆ : C⊗R Rat(∗C) → C⊗R Rat(∗C)⊗A C⊗R Rat(∗C)c⊗R f 7→ c⊗R f[0] ⊗A f[1] ⊗R e

ε : C⊗R Rat(∗C) → Ac⊗R f 7→ f(c)

where e is a right local unit for f and we denote R = Rat(∗C) from now on, if we concern onlyits ring structure.

The canonical map is given by the following map

can : C⊗R Rat(∗C) → C, can(c⊗R f) = f[0](c)f[1] (= c(1)f(c(2))),

where the second version of the formula is obtained from Lemma 8.11. The canonical map has aninverse that reads

can−1(c) = c⊗R e,

9.3. THE CORING AS A GALOIS COMODULE 245

where e ∈ R is a local unit for c (i.e. c · e = c). Remark that this isomorphism is nothing else thefirmness isomorphism C ∼= C⊗R R.

We will now apply the results of Section 4.2. The Galois comodule Rat(∗C) induces a pair ofadjoint functors given by

(263) MR

−⊗RRat(∗C) //MC

HomC(Rat(∗C),−)⊗RR

oo

The general theory makes use of the module Σ† = Σ∗ ⊗R R. In our situation, we have thefollowing Lemma, which is, in fact, an application of Proposition 3.53.

Lemma 9.18. With notation and conventions as before, the following holds,

Rat(∗C)† = HomA(Rat(∗C), A)⊗R R ∼= C.

Proof. Take ϕ⊗R r ∈ HomA(Rat(∗C), A)⊗RR then we define α(ϕ⊗R r) = ϕ(r[0])r[1], andconversely we define for all c ∈ C, β(c) = ψc ⊗R e where e ∈ R is a right local unit for c and ψcis defined by ψc(f) = f(c) for all f ∈ Rat(∗C). Then we can check

α β(c) = e[0](c)e[1]= c(1)e(c(2))= c · e = c

where we used Lemma 8.11 in the second equality. Let us as well verify the inverse composition.

(264) β α(ϕ⊗R r) = ψϕ(r[0])r[1] ⊗R e

where e is a right local unit for ϕ(r[0])r[1] and thus also for r. Let us first compute

ψϕ(r[0])r[1](f) = f(ϕ(r[0])r[1])= ϕ(r[0])f(r[1]) = ϕ(r[0]f(r[1]))= ϕ(r ∗ f) = (ϕ · r)(f)

We can now go on with (264),

β α(ϕ⊗R r) = ϕ · r ⊗R e= ϕ⊗R r · e = ϕ⊗R r.

Since the existence of local units in R implies that RR is flat, by Theorem 4.13 and the previouslemma,

HomC(Rat(∗C),−)⊗R R ' −⊗C C,

however, let us give the isomorphisms of this equivalence for completeness sake, they are verysimilar to the ones in the proof of Lemma 9.18. Take M ∈MC then

α : HomC(Rat(∗C),M)⊗R R→M ⊗C C ∼= M,

is an isomorphism where for all ϕ ⊗R r ∈ HomC(Rat(∗C),M) ⊗R R, m ∈ M , α(ϕ ⊗ r) = ϕ(r)and α−1(m) = ψm ⊗ e, where ψm(f) = m · f for all f ∈ R and e is a right local unit for m.

Theorem 9.19. Let C be an A-coring that is locally projective as a left A-module. If Rat(∗C)is dense with respect to the finite topology on ∗C, then the following statements hold true,

(i) (−⊗R R,⊗CC) is a pair of inverse equivalences between the categories MRat(∗C) and MC;(ii) Rat(∗C) is a Galois comodule and faithfully flat as a left Rat(∗C)-module;(iii) Rat(∗C) is a projective generator in MC.

Proof. The first statement follows from the previous observations. The other statementsfollow now from Theorem 4.27.

It follows now that the pair of adjoint functors induced by the Galois comodule Rat(∗C) is canbe written as

MR

−⊗RR //MC

−⊗CC

oo


Since M⊗RR ∼= M for every firm right R-module M and N⊗C C ∼= N for every right C-comoduleN , it is clear that these functors constitute the same equivalence of categories as mentioned inthe beginning of this section.

9.3.3. The coring as an infinite Galois comodule: quasi co-Frobenius corings. Let Abe an A-coring that is locally projective as a left A-module and suppose that Rat(∗C) is dense inthe finite topology on ∗C. As we know from the previous section, we can succesfully apply Galoistheory on the right C-comodule Rat(∗C) and the pair of adjoint functors

MR

−⊗RRat(∗C) //MC

HomC(Rat(∗C),−)⊗RR

oo

is an equivalence of categories (where R = Rat(∗C)). However, by Proposition 3.53 we canalso study the Galois properties of the finite dual of Rat(∗C). We know from Lemma 9.18 thatRat(∗C)† ∼= C and thus we find by Corollary 3.54 another pair of adjoint functors

(265) RMC⊗R− // CM

R⊗RCHom(C,−)

oo

We want to examine when this pair of adjoint functors is an equivalence of categories. The comatrixcoring is given by C⊗R R⊗R CHom(C,C) ∼= C⊗R ∗C, hence we can calculate the canonical mapas follows

can : C⊗R ∗C ∼= C, can(c⊗R f) = c(1)f(c(2)) = c · fSince R = Rat(∗C) is a two-sided ideal in ∗C (see Section 3.3), we conclude that the canonicalmap is an isomorphism, thus C is a firm C-R Galois comodule.

Applying Theorem 4.27, we obtain immediately the following

Theorem 9.20. (C ⊗R −, R ⊗R CHom(C,−)) is a pair of inverse equivalences between thecategories CM and RM if and only if C is faithfully flat as a right R-module.

The study of the Galois comodule C is also related to the Morita context similar to the one of(260). However, we can now restrict ∗C to Rat(∗C). If C is locally projective as a right A-module,we obtain the following Morita context

(266) (Rat(C∗),Rat(∗C),C, Q, τ, µ)

where Q = C∗Hom(C,C∗) ∼= CHom(C,Rat(C∗)) ∼= Hom∗C(C, ∗C) ∼= HomC(C,Rat(∗C)) = Q and

τ : C⊗∗C Q→ C∗, τ(c⊗ q) = q(c);

µ : Q⊗C∗ C → ∗C, µ(q ⊗ c) = q(c).

Let us denote R = Rat(∗C) and S = Rat(C∗). Since CM and SM are isomorphic, the Moritacontext (266) is strict if and only if the pair of adjoint functors (265) is an equivalence of categories.Therefore, we obtain our final result.

Theorem 9.21. Let C be an A-coring that is locally projective as a left and right A-module,such that R = Rat(∗C) and S = Rat(C∗) are dense respectively in the finite topology on ∗C andC∗. Then the following statements are equivalent:

(i) the adjoint functors of (265) induce an equivalence between CM and RM;(ii) −⊗S C is an equivalence between the categories MS and MC;(iii) the Morita context (266) is strict;(iv) C is a left and right k-locally quasi Frobenius coring;

If A is a PF-ring, then the above statements are furthermore equivalent to the fact that there existinjective maps j1 ∈ C∗Hom(C, (C∗)I) and j2 ∈ Hom∗C(C, (∗C)J) for arbitrary index sets I and J .

Proof. The equivalence between (i) and (iii) follows from our previous observations. Theequivalence with (ii) follows from the left-right duality in statement (iv). The equivalence between(iv) and the last statement follows from Theorem 8.32. So we only have to prove (iii) ⇔ (iv).Suppose first that C is left k-locally quasi Frobenius. Then there exists a left C∗-linear morphismj : C(I) → C∗, such that Im j is dense in the finite topology on C∗. Then Im j acts with left local

9.3. REFERENCES 247

units on all left C-comodules, so in particular on Rat(C∗). Take any f ∈ Rat(C∗) and let e ∈ Im j

be a left local unit for f , i.e. f = e · f . Since j is surjective, we can find an element (z`) ∈ C(I),where ` ∈ I and only a finite number of z` differ from zero, such that j(z`) = e. By Lemma 8.1, j iscompletely determined by a set of left C∗-linear morphisms j` : C → C∗, such that j(c`) =

∑` j`(c`)

for all (c`) ∈ C(I). Remark that j` ∈ Q. We find that τ(z`⊗∗C j`) =∑

` j`(z`) = j(z`) = e. Sinceτ is C∗ bilinear, we conclude that τ(z` ⊗∗C j`f) = e · f = f . Therefore, τ is surjective. In thesame way, we find that µ is surjective if C is right k-locally quasi Frobenius.

Conversely, suppose that the Morita context is strict. Since Rat(C∗) is dense in the finitetopology on C∗, it acts with local units on C. Consider a complete set of local units E = e` | ` ∈ Lin Rat(∗C) for C. Since τ is surjective, we find for every e an element

∑iezie ⊗∗C qie such that

τ(∑

iezie ⊗∗C qie) = e. Denote by I the set of all indices ie. Then we define a map

j : C(I) → C∗, j(ck) =∑k∈I

qk(ck).

By construction the image of j contains E, hence C is left k-locally quasi Frobenius by Theo-rem 8.28. The proof of the fact that C is right k-locally quasi Frobenius is similar.

References

The results of Section 9.1, Section 9.2 and Section 9.3.1 are an extended and updated versionof some results that appeared in the joint work with S. Caenepeel and E. De Groot [42]. ForSection 9.1 we used the easier approach of [117] (there for finite Galois comodules). The connectionwith the Cohen-Fischman-Montgomery context in Section 9.2 appeared already in the paper withS. Caenepeel and S. Wang [47]. Section 9.3.2 is deduced from the joint work with J. Gomez-Torrecillas [73]. Section 9.3.3 contains new results.

Appendix ANederlandse Samenvatting

Zever,... Gezever !...– Frankie Loosveld, Eilandbewoner

1.1. Ringen en coringen

Kijken we even terug naar de zin op de eerste pagina van dit book : “Van zodra er bestaanis, is er ook mede-bestaan (co-bestaan)”. Deze uitspraak is, binnen de tak van de Wiskunde diecategorietheorie heet, een evidentie. Ze betekent dat elke constructie kan worden gedualiseerd.Eenvoudig gezegd betekent dualiseren het omkeren van pijlen. Het is dit principe dat aan de basisligt van het bestaan van co-algebra’s en meer algemeen coringen, het belangrijkste onderwerp vandeze thesis.

Een algebra, of meer algemeen een ring, is een algebraısche structuur, die ruwweg gezegd eenverzameling is, waarvan we de elementen kunnen vermenigvuldigen, i.e. uit twee gegeven getallenuit een algebra kunnen we een nieuw getal maken. Deze bewerking moet uiteraard aan een aantaleigenschappen voldoen. We eisen bijvoorbeeld dat de vermenigvuldiging associatief is en dat ereen eenheidselement 1 bestaat:

(r · s) · t = r · (s · t), 1 · r = r = r · 1 voor alle r, s, t ∈ R,

waar R onze algebra of ring is. Stellen we de vermenigvuldiging voor door een afbeelding µ :R ⊗ R → R, en de eenheid door een afbeelding η : Z → R, dan kunnen we dit schematisch alsvolgt voorstellen.

R⊗R⊗R11⊗µ //

µ⊗11

R⊗R

µ

R⊗R µ

// R

R∼= //

∼=

R⊗ Z

11⊗η

Z⊗Rη⊗11

// R⊗R

µ

hhQQQQQQQQQQQQQQQ

Wiskundig ingewijden zullen opmerken dat het gebruik van het symbool ⊗ (het tensorproduct)impliceert dat de vermenigvuldiging distributief is ten opzichte van de optelling. Een moduul overeen ring R kan begrepen worden als een meetkundige ruimte, die gecoordinatiseerd wordt aande hand van R. De ‘verzameling’ van al deze modulen (wiskundig-technisch gezien is dit geenverzameling meer, maar een klasse) samen met hun onderlinge verbanden, wordt de categorie vanalle modulen genoemd en genoteerd met MR. Deze categorie bevat alle informatie over de ringR. Gekende voorbeelden van ringen zijn de verzamelingen van de gehele getallen Z, de rationalegetallen Q en de reele getallen R. Verder zijn R en Q modulen over Z en R is een moduul over Q.

249

250 APPENDIX: NEDERLANDSE SAMENVATTING

Door de definities van een algebra en een ring te dualiseren, komen we tot de definitie vaneen co-algebra en meer algemeen een coring. Ruwweg gezegd, is een coring een verzameling metde mogelijkheid om elementen te covermenigvuldigen, wat intuıtief kan begrepen worden als “vaneen getal twee getalletjes maken”, hoewel het eigenlijk om een som van koppels van getallen gaat,maar dit is eerder een wiskundig-technische bijkomstigheid. Associativiteit wordt vervangen doorco-associativiteit, een eenheid door een co-eenheid. Schematisch ziet de definitie er als volgt uit.We noemen C een A-coring, indien er afbeeldingen ∆ : C → C ⊗A C en ε : C → A bestaan, dievoldoen aan eigenschappen zoals uitgedrukt in de onderstaande diagrammen.

C∆ //

∆

C⊗A C

11⊗A∆

C∼= //

∆ ((RRRRRRRRRRRRRRRR

∼=

C⊗A A

C⊗A C∆⊗A11

// C⊗A C⊗A C A⊗A C C⊗A Cε⊗A11

oo

11⊗Aε

OO

Co-algebra’s zijn een eenvoudige coringen, met een commutatieve basisring. Men kan ook co-modulen over een coring definieren en de klasse van alle comodulen, samen met de onderlingeverbanden heet de categorie van alle comodulen en wordt genoteerd met MC. Zoals in het gevalvan ringen zit in MC alle eigenschappen van de coring C vervat.

Coringen en comodulen werden geıntroduceerd door Sweedler in 1975 (zie [109]). Omdater echter zo goed als geen niet-triviale voorbeelden beschikbaar waren, bleef de interesse voorcoringen uit en werden deze structuren min of meer vergeten. Deze situatie veranderde drastischop het einde van de vorige eeuw, toen Takeuchi opmerkte dat coringen geconstrueerd kunnenworden uit zogenaamde verstrengelende structuren. Deze verstrengelende structuren bestaan uiteen algebra A, een co-algebra C en een afbeelding ψ : C ⊗ A → A ⊗ C, die op een natuurlijkemanier de algebraısche eigenschappen van A met de co-algebraısche eigenschappen van C opmengt(of verstrengelt). De reden waarom het verband met verstrengelende structuren het onderzoeknaar coringen opnieuw aantrekkelijk maakte, is dat verstrengelende structuren en de bijhorendeverstrengelde modulen een helder kader bieden om de vele soorten van Hopf modulen te bestuderendie waren geconstrueerd tijdens de laatste decenia van de twintigste eeuw, zoals Yetter-Drinfeldmodulen, Long dimodulen, gegradeerde modulen en Doi-Hopf modulen. Voor al deze structurenbestaat tot op vandaag een grote interesse en ze zijn van belang in vele deelgebieden van dealgebra, meetkunde en wiskundige natuurkunde. Dit verband tussen coringen en verstrengelendestructuren werd voor het eerst uitvoerig besproken in [28].

Eenvoudig gezegd is een Hopf-algebra een algebra die tegelijkertijd een co-algebra is, voorzienvan compatibiliteitscondities tussen beide structuren. Een belangrijk onderwerp binnen de studievan de Hopf algebra’s, is de zogenaamde Hopf-Galois theorie. Deze theorie veralgemeent deklassieke Galois theorie voor eindige velduitbreidingen en handelt over equivalenties tussen cate-gorieen van Hopf modulen over een Hopf algebra en categorieen van gewone modulen over eenalgebra. Dit is interessant omdat we op deze manier de theorie van Hopf modulen kunnen reducerentot een theorie over gewone modulen. Deze laatste hebben doorgaans een eenvoudigere structuuren zijn dus vaak beter gekend. Een fundamenteel resultaat binnen deze Hopf-Galois theorie werdgeformuleerd door Schneider in [104]. Een eenvoudige versie van deze structuurstelling zegt dater een equivalentie tussen een categorie van Hopf modulen en een categorie van modulen bestaat,indien een bepaalde afbeelding, de kanonieke afbeelding genaamd, bijectief (i.e. er komt overalprecies 1 pijl toe, de afbeelding is omkeerbaar) is en er aan een technische voorwaarde is voldaandie we ‘getrouwe platheid’ noemen. In een sterkere vorm zegt deze structuurstelling dat, onderenkele bijkomende voorwaarden, de surjectiviteit (i.e. er komt overal minstens 1 pijl toe) van decanonische afbeelding voldoende is om de vernoemde equivalentie van categorieen te bekomen.

Aangezien coringen gebruikt kunnen worden om Hopf algebra’s in een ruimer kader te bestud-eren, is het niet verwonderlijk dat ook de Hopf-Galois theorie werd veralgemeend naar de taal vancoringen en comodulen. Reeds in [28] staat een versie van de (eenvoudige) structuurstelling, voorcoringen met een zogenaamd groepachtig element. Dit werd veralgemeend in [63] tot de theorievan eindige Galois comodulen. Enkele aspecten van deze eindige theorie onderzochten we in eengezamenlijk werk met Caenepeel en De Groot [42]. Het belangrijkste onderwerp van deze thesisis het ontwikkelen van een theorie van oneindige Galois comodulen.

1.3. SEPARABILITEITSEIGENSCHAPPEN EN FROBENIUS CORINGEN 251

1.2. Comatrix coringen en Galois comodulen

De sleutel tot het ontwikkelen van een Galois theorie voor comodulen is de constructie van eenbepaald soort coringen, gekend als comatrix coringen. Dit zijn de duale constructies van matrixringen. De eindigheidsconditie leek bij de constructie van de eerste comatrix coringen uit [63]essentieel te zijn. Indien we deze constructies wilden veralgemenen, hadden we dus nood aannieuwe technieken. In samenwerking met Gomez-Torrecillas bleek het mogelijk om technieken diewe ontwikkeld hadden in [113] te combineren deze van El Kaoutit en Gomez-Torrecillas uit [64],om de gewenste oneindige comatrix coringen te bekomen. Een speciaal type van deze coringen, diebeschreven worden aan de hand van colimieten, werd ontwikkeld in samenwerking met Caenepeelen De Groot, [41]. Voor de technische details verwijzen we naar Hoofdstukken 2 en 3.

Deze comatrix coringen staan ons toe om een Galois theorie voor coringen te ontwikkelen.Startend van een gegeven coring en een comoduul, kan men een comatrix coring construeren diemet de gegeven coring verbonden is door een kanonieke afbeelding. We konden de (eenvoudige)structuurstelling van Schneider veralgemenen naar deze situatie en bekomen aldus

Stelling A.1. Zij C een A-coring die plat is als links A-moduul en R een stevige ring. Defunctor −⊗RΣ : MR →MC is een equivalentie van categorieen als en slechts als Σ getrouw platis als links A-moduul, can een isomorfisme is en er een ringmorfisme i : R→ Σ⊗RΣ∗ bestaat datvan Σ een stevig links R-moduul maakt.

Er bestonden nog andere versies van Galois theorie voor comodulen. In deze thesis besprekenwe deze verschillende theorieen en onderzoeken we hun verbanden. Dit werk is een combinatie vande het onderzoek uit [42], [41], [73], [112] en enkele nieuwe resultaten.

We ontwikkelen ook een Galois theorie binnen het kader van de bicategorieen. Op deze manierkunnen we Galois theorie voor coringen unificeren met andere theorieen, zoals de comonadiciteit vanfunctoren en de theorie van C-ringen. Dit gedeelte van het onderzoek is nog in opbouw en gebeurtin samenwerking met Gomez-Torrecillas [73] Voor meer details verwijzen we naar Hoofdstuk 4.

In Hoofdstuk 5 bespreken we enkele speciale gevallen van de Galois theorie voor comodulen.We kunnen aan een coring een duale ring associeren. Indien de coring aan een eindigheidsvoor-waarde voldoet, is de categorie van comodulen over de coring isomorf aan de categorie van modulenover deze duale ring. Op deze manier reduceert Galois theorie zich van een theorie die verbandentussen een categorie van comodulen en een categorie van modulen onderzoekt naar een theoriedie verbanden onderzoekt tussen twee categorieen van modulen, i.e. Morita theorie. We veral-gemenen op deze manier opnieuw enkele welbekende resultaten uit de theorie van Hopf algebra’s.We ontwikkelden deze theorie in samenwerking met Caenepeel en Wang voor coringen met eengroepachtig element [47], [48], met Caenepeel en De Groot voor (eindige) comodulen [42], in eeniets algemenere vorm met Bohm [24]. In de versie die voorkomt in Hoofdstuk 5, worden enkeleresultaten nog verder veralgemeend.

Door de Morita theorie van Hoofdstuk 5 verder uit te breiden naar een Morita theorie voorcoring extensies, wordt het mogelijk om in Hoofdstuk 6 een theorie van gekliefde bicomodulen teontwikkelen. Een gekliefd bicomoduul induceert altijd een equivalentie van categorieen zoals inStelling A.1, zelfs indien het bicomoduul niet getrouw plat is. Deze theorie unificeert alle vroegergekende theorieen over gekliefde uitbreidingen van ringen, een sterk ontwikkelde tak binnen deHopf-Galois theorie. We ontwikkelden dit deel van de thesis in samenwerking met Bohm [24], enenkele speciale gevallen in vroeger werk met Caenepeel en Wang [47].

1.3. Separabiliteitseigenschappen en Frobenius coringen

In het laatste deel van de thesis bespreken we separabele en Frobenius functoren. Functorenworden gebruikt om verbanden tussen verschillende categorieen te onderzoeken. Eigenschappenzoals Frobenius en separabiliteit voor functoren hebben interessante implicaties voor de Galoistheorie die we hiervoor hebben ontwikkeld.

In Hoofdstuk 7 bespreken we eerst de separabiliteitseigenschap. In een van de zeldzame artike-len over coringen die dateren van voor de coring-revolutie in 1999, bespreekt Guzman separabiliteitvoor coringen, en karakteriseert coseparabele coringen aan de hand van cohomologische eigenschap-pen [76]. Een verwant resultaat is dat van Nakajima [94], die, ook aan de hand van cohomologie,

252 APPENDIX: NEDERLANDSE SAMENVATTING

separabileit voor co-algebra-extensies onderzoekt. Hoewel beide theorieen duidelijk verwant zijn,zijn ze niet uit elkaar af te leiden. Door over te schakelen op een algemener kader, namelijk danvan zogenaamde cotripels (dit zijn structuren die nauw verwant zijn aan coringen, maar ‘leven’ ineen andere categorie), was het mogelijk om beide theorieen te unificeren. Dit werk gebeurde insamenwerking met El Kaoutit [67].

Hoofdstuk 8 handelt over Frobenius eigenschappen. Frobenius, co-Frobenius en quasi co-Frobenius eigenschappen voor co-algebra’s, Hopfalgebra’s, ring uitbreidingen en bimodulen zijnde laatste tientallen jaren intens bestudeerd. Sinds de hernieuwde interesse voor coringen is ookduidelijk geworden dat de Frobenius eigenschappen het eenvoudigst binnen het kader van coringenkunnen worden behandeld. Frobenius coringen zijn reeds enige tijd goed begrepen. Zo bestaater een interessante karakterisatie van Frobenius coringen aan de hand van zogenaamde Frobeniusfunctoren, dit zijn functoren die tegelijkertijd links en rechts toegevoegd zijn [46]. Co-Frobeniusen quasi co-Frobenius coringen zijn echter veel minder goed begrepen. Zo bleef de categorischeinterpretatie van deze eigenschap lange tijd mysterieus. In Hoofdstuk 8 geven we deze karakter-isatie. Het voordeel van onze aanpak is dat we de vroegere resultaten over Frobenius coringenbeter kunnen interpreteren. Dit onderzoek werd uitgevoerd in samenwerking met Iovanov [77]

In het laatste Hoofdstuk bespreken we de invloed van de Frobenius eigenschap en separabiliteitvan coringen op hun Galois theorie. Zo kunnen we dankzij separabiliteit de sterke vorm van Schnei-ders structuurstelling voor oneindige Galois comodulen aantonen. Frobenius coringen inducereneen grotere symmetrie in de Morita contexten van Hoofdstuk 5. Verder blijkt het mogelijk omquasi co-Frobenius coringen te karakteriseren aan de hand van Galois theorie, waarbij de coringzelf de rol van Galois comoduul vervult.

Bibliography

[1] G. D. Abrams, Morita equivalence for rings with local units, Comm. Algebra, 11 (8), (1983) 801–837.[2] J. Y. Abuhlail, A note on coseparable coalgebras, preprint, 2006.[3] J. Y. Abuhlail, Rational modules for corings, Comm. Algebra, 31 (12), (2003) 5793–5840.[4] J. Y. Abuhlail, Morita contexts for corings and equivalences, in: Hopf algebras in noncommutative geometry

and physics, volume 239 of Lecture Notes in Pure and Appl. Math., pp. 1–19, Dekker, New York, 2005.[5] J. Y. Abuhlail, On the linear weak topology and dual pairings over rings, Topology Appl., 149 (1-3), (2005)

161–175.[6] J. N. Alonso Alvarez, J. M. Fernandez Vilaboa, R. Gonzalez Rodrıguez and A. B. Rodrıguez Raposo, Weak

C-cleft extensions, weak entwining structures and weak Hopf algebras, J. Algebra, 284 (2), (2005) 679–704.

[7] P. N. Anh and L. Marki, Morita equivalence for rings without identity, Tsukuba J. Math., 11 (1), (1987) 1–16.[8] A. Ardizzoni, Separable functors and formal smoothness, 2004, to appear in J. K-theory, URL http://arXiv.

org/abs/math/0407095.[9] A. Ardizzoni, C. Menini and D. Stefan, Hochschild cohomology and smoothness in monoidal categories, J.

Pure Appl. Algebra, 208, (2007) 297–330.[10] M. Auslander and O. Goldman, The Brauer group of a commutative ring, Trans. Amer. Math. Soc., 97, (1960)

367–409.[11] G. Azumaya, Some characterizations of regular modules, Publ. Mat., 34 (2), (1990) 241–248.[12] M. Barr and G. S. Rinehart, Cohomology as the derived functor of derivations, Trans. Amer. Math. Soc., 122,

(1966) 416–426.[13] M. Barr and C. Wells, Toposes, triples and theories, Repr. Theory Appl. Categ., (12), (2005) x+288 pp.

(electronic), corrected reprint of the 1985 original [MR0771116].[14] H. Bass, Algebraic K-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968.[15] M. Beattie, S. Dascalescu and S. Raianu, Galois extensions for co-Frobenius Hopf algebras, J. Algebra, 198 (1),

(1997) 164–183.[16] J. Benabou, Introduction to bicategories, in: Reports of the Midwest Category Seminar, pp. 1–77, Springer,

Berlin, 1967.[17] G. Bohm, Galois theory for Hopf algebroids, Ann. Univ. Ferrara Sez. VII Sci. Mat., 51, (2005) 233–262.[18] G. Bohm, Integral theory for Hopf algebroids, Algebr. Represent. Theory , 8 (4), (2005) 563–599.[19] G. Bohm, Internal bialgebroids, entwining structures and corings, in: Algebraic structures and their represen-

tations, volume 376 of Contemp. Math., pp. 207–226, Amer. Math. Soc., Providence, RI, 2005.[20] G. Bohm and T. Brzezinski, Strong connections and the relative Chern-Galois character for corings, Int. Math.

Res. Not., (42), (2005) 2579–2625.[21] G. Bohm and T. Brzezinski, Cleft extensions of Hopf algebroids, Appl. Categ. Structures, 14 (5-6), (2006)

431–469.[22] G. Bohm and K. Szlachanyi, Hopf algebroids with bijective antipodes: axioms, integrals, and duals, J. Algebra,

274 (2), (2004) 708–750.[23] G. Bohm and J. Vercruysse, Morita theory of comodules, submitted, URL http://arXiv.org/abs/0710.

1017.[24] G. Bohm and J. Vercruysse, Morita theory for coring extensions and cleft bicomodules, Adv. Math., 209 (2),

(2007) 611–648.[25] F. Borceux, Handbook of categorical algebra. 1: Basic category theory, volume 50 of Encyclopedia of Mathe-

matics and its Applications, Cambridge University Press, Cambridge, 1994.[26] F. Borceux, Handbook of categorical algebra. 2: Categories and structures, volume 51 of Encyclopedia of

Mathematics and its Applications, Cambridge University Press, Cambridge, 1994.[27] T. Brzezinski, On modules associated to coalgebra Galois extensions, J. Algebra, 215 (1), (1999) 290–317.[28] T. Brzezinski, The structure of corings: induction functors, Maschke-type theorem, and Frobenius and Galois-

type properties, Algebr. Represent. Theory , 5 (4), (2002) 389–410.[29] T. Brzezinski, Galois comodules, J. Algebra, 290 (2), (2005) 503–537.[30] T. Brzezinski, A note on coring extensions, Ann. Univ. Ferrara Sez. VII Sci. Mat., 51, (2005) 15–27.[31] T. Brzezinski, L. El Kaoutit and J. Gomez-Torrecillas, The bicategories of corings, J. Pure Appl. Algebra,

205 (3), (2006) 510–541.[32] T. Brzezinski and J. Gomez-Torrecillas, On comatrix corings and bimodules, K-Theory , 29 (2), (2003) 101–

115.

253

http://arXiv.org/abs/math/0407095


http://arXiv.org/abs/0710.1017

http://arXiv.org/abs/0710.1017

254 BIBLIOGRAPHY

[33] T. Brzezinski, L. Kadison and R. Wisbauer, On coseparable and biseparable corings, in: Hopf algebras innoncommutative geometry and physics, volume 239 of Lecture Notes in Pure and Appl. Math., pp. 71–87,Dekker, New York, 2005.

[34] T. Brzezinski and S. Majid, Coalgebra bundles, Comm. Math. Phys., 191 (2), (1998) 467–492.[35] T. Brzezinski and R. B. Turner, The Galois theory of matrix C-rings, Appl. Categ. Structures, 14 (5-6), (2006)

471–501.[36] T. Brzezinski and R. Wisbauer, Corings and comodules, volume 309 of London Mathematical Society Lecture

Note Series, Cambridge University Press, Cambridge, 2003.[37] S. Caenepeel, Brauer groups, Hopf algebras and Galois theory, volume 4 of K-Monographs in Mathematics,

Kluwer Academic Publishers, Dordrecht, 1998.[38] S. Caenepeel, Galois corings from the descent theory point of view, in: Galois theory, Hopf algebras, and

semiabelian categories, volume 43 of Fields Inst. Commun., pp. 163–186, Amer. Math. Soc., Providence, RI,2004.

[39] S. Caenepeel and E. De Groot, Modules over weak entwining structures, in: New trends in Hopf algebra theory(La Falda, 1999), volume 267 of Contemp. Math., pp. 31–54, Amer. Math. Soc., Providence, RI, 2000.

[40] S. Caenepeel and E. De Groot, Galois corings applied to partial Galois theory, in: Proceedings of the Interna-tional Conference on Mathematics and its Applications (ICMA 2004), pp. 117–134, Kuwait Univ. Dep. Math.Comput. Sci., Kuwait, 2005.

[41] S. Caenepeel, E. De Groot and J. Vercruysse, Constructing infinite comatrix corings from colimits, Appl. Categ.Structures, 14 (5-6), (2006) 539–565.

[42] S. Caenepeel, E. De Groot and J. Vercruysse, Galois theory for comatrix corings: descent theory, Morita theory,Frobenius and separability properties, Trans. Amer. Math. Soc., 359 (1), (2007) 185–226 (electronic).

[43] S. Caenepeel and M. Iovanov, Comodules over semiperfect corings, in: Proceedings of the InternationalConference on Mathematics and its Applications (ICMA 2004), pp. 135–160, Kuwait Univ. Dep. Math. Comput.Sci., Kuwait, 2005.

[44] S. Caenepeel and K. Janssen, Partial entwining structures, 2006, to appear in Comm. Algebra, URL http:

//arxiv.org/abs/math/0610524.[45] S. Caenepeel and G. Militaru, Maschke functors, semisimple functors and separable functors of the second

kind: applications, J. Pure Appl. Algebra, 178 (2), (2003) 131–157.[46] S. Caenepeel, G. Militaru and S. Zhu, Frobenius and separable functors for generalized module categories and

nonlinear equations, volume 1787 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2002.[47] S. Caenepeel, J. Vercruysse and S. Wang, Morita theory for corings and cleft entwining structures, J. Algebra,

276 (1), (2004) 210–235.[48] S. Caenepeel, J. Vercruysse and S. Wang, Rationality properties for Morita contexts associated to corings,

in: Hopf algebras in noncommutative geometry and physics, volume 239 of Lecture Notes in Pure and Appl.Math., pp. 113–136, Dekker, New York, 2005.

[49] F. Castano Iglesias, J. Gomez Torrecillas and C. Nastasescu, Frobenius functors: applications, Comm. Algebra,27 (10), (1999) 4879–4900.

[50] S. U. Chase, D. K. Harrison and A. Rosenberg, Galois theory and Galois cohomology of commutative rings,Mem. Amer. Math. Soc. No., 52, (1965) 15–33.

[51] S. U. Chase and M. E. Sweedler, Hopf algebras and Galois theory, Lecture Notes in Mathematics, Vol. 97,Springer-Verlag, Berlin, 1969.

[52] M. Cohen and D. Fischman, Semisimple extensions and elements of trace 1, J. Algebra, 149 (2), (1992)419–437.

[53] M. Cohen, D. Fischman and S. Montgomery, Hopf Galois extensions, smash products, and Morita equivalence,J. Algebra, 133 (2), (1990) 351–372.

[54] C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Pure and AppliedMathematics, Vol. XI, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962.

[55] S. Dascalescu, C. Nastasescu and S. Raianu, Hopf algebras: An introduction, volume 235 of Monographs andTextbooks in Pure and Applied Mathematics, Marcel Dekker Inc., New York, 2001.

[56] Y. Doi, Homological coalgebra, J. Math. Soc. Japan, 33 (1), (1981) 31–50.[57] Y. Doi, Generalized smash products and Morita contexts for arbitrary Hopf algebras, in: Advances in Hopf

algebras (Chicago, IL, 1992), volume 158 of Lecture Notes in Pure and Appl. Math., pp. 39–53, Dekker, NewYork, 1994.

[58] Y. Doi and M. Takeuchi, Hopf-Galois extensions of algebras, the Miyashita-Ulbrich action, and Azumayaalgebras, J. Algebra, 121 (2), (1989) 488–516.

[59] M. Dokuchaev and R. Exel, Associativity of crossed products by partial actions, enveloping actions and partialrepresentations, Trans. Amer. Math. Soc., 357 (5), (2005) 1931–1952 (electronic).

[60] M. Dokuchaev, M. Ferrero and A. Paques, Partial actions and Galois theory, J. Pure Appl. Algebra, 208 (1),(2007) 77–87.

[61] S. Eilenberg and J. C. Moore, Adjoint functors and triples, Illinois J. Math., 9, (1965) 381–398.[62] S. Eilenberg and J. C. Moore, Foundations of relative homological algebra, Mem. Amer. Math. Soc. No., 55,

(1965) 39.[63] L. El Kaoutit and J. Gomez-Torrecillas, Comatrix corings: Galois corings, descent theory, and a structure

theorem for cosemisimple corings, Math. Z., 244 (4), (2003) 887–906.

http://arxiv.org/abs/math/0610524


BIBLIOGRAPHY 255

[64] L. El Kaoutit and J. Gomez-Torrecillas, Infinite comatrix corings, Int. Math. Res. Not., (39), (2004) 2017–2037.[65] L. El Kaoutit and J. Gomez-Torrecillas, Morita duality for corings over quasi-Frobenius rings, in: Hopf algebras

in noncommutative geometry and physics, volume 239 of Lecture Notes in Pure and Appl. Math., pp. 137–153,Dekker, New York, 2005.

[66] L. El Kaoutit, J. Gomez-Torrecillas and F. J. Lobillo, Semisimple corings, Algebra Colloq., 11 (4), (2004)427–442.

[67] L. El Kaoutit and J. Vercruysse, Cohomology for bicomodules. Separable and Maschke functors, 2006, toappear in J. K-Theory, URL http://arxiv.org/abs/math/0608195.

[68] G. S. Garfinkel, Universally torsionless and trace modules, Trans. Amer. Math. Soc., 215, (1976) 119–144.[69] J. Gomez-Torrecillas, Separable functors in corings, Int. J. Math. Math. Sci., 30 (4), (2002) 203–225.[70] J. Gomez Torrecillas, Comonads and Galois corings, Appl. Categ. Structures, 14 (5-6), (2006) 579–598.[71] J. Gomez Torrecillas and C. Nastasescu, Quasi-co-Frobenius coalgebras, J. Algebra, 174 (3), (1995) 909–923.[72] J. Gomez Torrecillas and J. Vercruysse, Galois theory in bicategories, submitted.[73] J. Gomez Torrecillas and J. Vercruysse, Comatrix corings and Galois comodules over firm rings, Algebr.

Represent. Theory , 10 (3), (2007) 271–306.[74] F. Grandjean and E. M. Vitale, Morita equivalence for regular algebras, Cahiers Topologie Geom. Differentielle

Categ., 39 (2), (1998) 137–153.[75] G. Guo, Quasi-Frobenius corings and quasi-Frobenius extensions, Comm. Algebra, 34 (6), (2006) 2269–2280.[76] F. Guzman, Cointegrations, relative cohomology for comodules, and coseparable corings, J. Algebra, 126 (1),

(1989) 211–224.[77] M. Iovanov and J. Vercruysse, Co-Frobenius corings and adjoint functors, 2006, to appear in J. Pure Appl.

Algebra, URL http://arxiv.org/abs/math/0610853.[78] M. C. Iovanov, Co-Frobenius coalgebras, J. Algebra, 303 (1), (2006) 146–153.[79] N. Jacobson, Basic algebra. I, W. H. Freeman and Co., San Francisco, Calif., 1974.[80] D. W. Jonah, Cohomology of coalgebras, Memoirs of the American Mathematical Society, No. 82, American

Mathematical Society, Providence, R.I., 1968.[81] C. Kassel, Quantum groups, volume 155 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1995.[82] M. Kleiner, Induced modules and comodules and representations of BOCSs and DGCs, in: Representations of

algebras (Puebla, 1980), volume 903 of Lecture Notes in Math., pp. 168–185, Springer, Berlin, 1981.[83] M. Kleiner, The dual ring to a coring with a grouplike, Proc. Amer. Math. Soc., 91 (4), (1984) 540–542.[84] M. Kleiner, Integrations and cohomology theory, J. Pure Appl. Algebra, 38 (1), (1985) 71–86.[85] H. F. Kreimer and M. Takeuchi, Hopf algebras and Galois extensions of an algebra, Indiana Univ. Math. J.,

30 (5), (1981) 675–692.[86] S. Lack and R. Street, The formal theory of monads. II, J. Pure Appl. Algebra, 175 (1-3), (2002) 243–265,

special volume celebrating the 70th birthday of Professor Max Kelly.[87] T. Leinster, Basic bicategories, URL http://arXiv.org/abs/math/9810017.[88] S. Mac Lane, Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics, Springer-

Verlag, New York, 1998, second edition.[89] S. Mac Lane and R. Pare, Coherence for bicategories and indexed categories, J. Pure Appl. Algebra, 37 (1),

(1985) 59–80.[90] L. Marın, Morita equivalence based on contexts for various categories of modules over associative rings, J.

Pure Appl. Algebra, 133 (1-2), (1998) 219–232, ring theory (Miskolc, 1996).[91] A. Masuoka, Corings and invertible bimodules, Tsukuba J. Math., 13 (2), (1989) 353–362.[92] S. Montgomery, Hopf algebras and their actions on rings, volume 82 of CBMS Regional Conference Series in

Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1993.[93] B. J. Muller, The quotient category of a Morita context, J. Algebra, 28, (1974) 389–407.[94] A. Nakajima, Coseparable coalgebras and coextensions of coderivations, Math. J. Okayama Univ., 22 (2),

(1980) 145–149.[95] C. Nastasescu, M. Van den Bergh and F. Van Oystaeyen, Separable functors applied to graded rings, J.

Algebra, 123 (2), (1989) 397–413.[96] M. O’uchi, Coring structures associated with multiplicative unitary operators on Hilbert C∗-modules, Far East

J. Math. Sci. (FJMS), 11 (2), (2003) 121–136.[97] M. O’uchi, Coring structures on a Hilbert C∗-module of compact operators, Far East J. Math. Sci. (FJMS),

15 (2), (2004) 193–201.[98] B. Pareigis, Non-additive ring and module theory. II. C-categories, C-functors and C-morphisms, Publ. Math.

Debrecen, 24 (3-4), (1977) 351–361.[99] D. Quillen, Module theory over non-unital rings, notes, 1997.

[100] M. D. Rafael, Separable functors revisited, Comm. Algebra, 18 (5), (1990) 1445–1459.[101] A. V. Rojter, Matrix problems and representations of BOCSs, in: Representation theory, I (Proc. Workshop,

Carleton Univ., Ottawa, Ont., 1979), volume 831 of Lecture Notes in Math., pp. 288–324, Springer, Berlin,1980.

[102] P. Schauenburg, Bialgebras over noncommutative rings and a structure theorem for Hopf bimodules, Appl.Categ. Structures, 6 (2), (1998) 193–222.

[103] P. Schauenburg, Doi-Koppinen Hopf modules versus entwined modules, New York J. Math., 6, (2000) 325–329(electronic).




256 BIBLIOGRAPHY

[104] H.-J. Schneider, Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math., 72 (1-2), (1990)167–195, hopf algebras.

[105] Z. Skoda, Distributive laws for actions of monoidal categories, 2004, URL http://arXiv.org/abs/math/

0406310.[106] B. Stenstrom, Rings of quotients, Springer-Verlag, New York, 1975, die Grundlehren der Mathematischen

Wissenschaften, Band 217, An introduction to methods of ring theory.[107] R. Street, The formal theory of monads, J. Pure Appl. Algebra, 2 (2), (1972) 149–168.[108] R. Street, Categorical structures, in: Handbook of algebra, Vol. 1, pp. 529–577, North-Holland, Amsterdam,

1996.[109] M. Sweedler, The predual theorem to the Jacobson-Bourbaki theorem, Trans. Amer. Math. Soc., 213, (1975)

391–406.[110] M. Takeuchi, as referred to in MR 200c 16047, by A. Masuoka.[111] H. Tominaga, On s-unital rings, Math. J. Okayama Univ., 18 (2), (1975/76) 117–134.[112] J. Vercruysse, Equivalences between categories of modules and categories of comodules, 2006, to appear in

Acta Math. Sin. (Engl. Ser.), URL http://arxiv.org/abs/math/0604423.[113] J. Vercruysse, Local units versus local projectivity dualisations: corings with local structure maps, Comm.

Algebra, 34 (6), (2006) 2079–2103.[114] R. Wisbauer, Foundations of module and ring theory, volume 3 of Algebra, Logic and Applications, Gordon

and Breach Science Publishers, Philadelphia, PA, 1991, german edition, a handbook for study and research.[115] R. Wisbauer, On the category of comodules over corings, in: Mathematics & mathematics education (Beth-

lehem, 2000), pp. 325–336, World Sci. Publ., River Edge, NJ, 2002.[116] R. Wisbauer, From Galois field extensions to Galois comodules, in: Advances in ring theory, pp. 263–281,

World Sci. Publ., Hackensack, NJ, 2005.[117] R. Wisbauer, On Galois comodules, Comm. Algebra, 34 (7), (2006) 2683–2711.[118] M. Zarouali-Darkaoui, Comatrix coring generalized and equivalences of categories of comodules, 2005, URL

arXiv.org/abs/math/0505534.[119] B. Zimmermann-Huisgen, Pure submodules of direct products of free modules, Math. Ann., 224 (3), (1976)

233–245.




arXiv.org/abs/math/0505534

Index

Michel, we zitten met ne ring!– Frankie Loosveld, eilandbewoner

2-category, 10α-condition, 58E -injective object, 194η-multiplication, 70I -exact sequence, 194ε-comultiplication, 65

adjoint pair, 22algebra, 17

bi-equivalence, 12bicategory, 9

enriched, 31bicomodule, 20

cleft, 178weak, 178

bicomodules, 21, 195bimodule, 17bocs, 1

canonical cotriple morphism, 120canonical map, 120Casimir element, 101, 238category

concrete, 31Hom-Class, 9Hom-Set, 9monoidal, 15small category, 9strict monoidal, 15

cleft extensionby a coalgebra, 182by a Hopf algebra, 182by a Hopf algebroid, 187by a partial group action, 184

coalgebra, 20, 21separable, 209

cocone, 35coderivation, 198, 206, 209coderivations

inner, 198coideal, 78cointegration, 198, 207, 209

inner, 199universal, 200

colimit, 35comatrix coring, 97

finite, 98infinite, 98

comatrix coring context, 54

comodule, 20, 73(C, A)-injective, 116cofirm, 70coflat, 74comonadic-Galois, 116, 146firm Galois, 120Galois, 146

comodule algebra, 85comonad, 19comonad morphism, 23complete set

of idempotent local units, 48of local units, 47

compositionhorizontal, 10vertical, 9

contractable equalizer, 117coring, 21, 73

– extension, 83co-Frobenius, 212coseparable, 40, 206Dorroh coring, 78Frobenius, 212lax, 84quasi co-Frobenius, 212

cotensor product, 28cotriple, 20

lax, 84cotriple functor, 21

directed system, 36Doi-Hopf structure, 85Doi-Koppinen structure, 85Dorroh overring, 38dual basis, 57

idempotent, 59dual pair, 52

firm, 53

Eilenberg-Watts theoremfor comodules, 117over firm rings, 41

entwined module, 83weak, 183

entwining structure, 83cleft, 180, 186completely factorizable, 86factorizable, 86lax, 84weak, 183

257

258 INDEX

cleft, 183equalizer condition, 139

factorization structure, 85cleft, 188

finite topology, 52Frobenius system

of a Frobenius coring, 227of a Frobenius ring extension, 238

functorexact, 39Frobenius pair of –s, 212Maschke, 194separable, 193

fundamental theorem for Hopf algebras, 182

Godement product, 10grouplike character, 161grouplike element, 74

Hopf algebra, 182Hopf algebroid, 186

interchange law, 10internal equivalence, 12invertible elements, 22, 146, 212

lax Functor, 11lax natural transformation, 11local property, 11locally faithful, 11

matrix C-ring, 144mixed distributive laws, 80modification, 11module, 17, 18R-rational, 89, 93R-rational part of a, 93faithfully flat, 45firm, 39firmly projective, 53flat, 31strongly locally projective, 59totally faithful, 44weakly locally projective, 57

module coalgebra, 85monad, 17, 18monoid, 17Morita context, 22, 146

anti-morphism of, 146morphism of, 146morphism of –s, 212strict, 146

multiplicative approximation, 51

normal basis property, 179for entwining structures, 181weak, 179

pseudo natural transformation, 11pseudo-functor, 11

reflexive, 148relative injectivity, 194representable coalgebra, 80representation, 18, 65ring, 19

firm, 39

with idempotent local units, 48morphism of, 48

with local units, 46morphism of, 47

ring extensionFrobenius, 238

sequence, 39co-exact, 194cosplit, 194exact, 39

smash product, 85strongly locally projective

in a Grotendieck category, 135module, 59

Sweedler coring, 74

Takeuchi context, 22triad, 18triple, 18

weakly locally projectivein a Grotendieck category, 135module, 57

Galois Theory for Corings and Comoduleshomepages.vub.ac.be/~jvercruy/phd.pdf · 5.4. Structure...

Documents

Transcript of Galois Theory for Corings and Comoduleshomepages.vub.ac.be/~jvercruy/phd.pdf · 5.4. Structure...