PII. S016117120201270X http://ijmms.hindawi.com © Hindawi Publishing Corp.
SEPARABLE FUNCTORS IN CORINGS
J. GÓMEZ-TORRECILLAS
Received 5 April 2001 and in revised form 25 October 2001
We develop some basic functorial techniques for the study of the categories of comodules over corings. In particular, we prove that the induction functor stemming from every mor-phism of corings has a left adjoint, called ad-induction functor. This construction general-izes the known adjunctions for the categories of Doi-Hopf modules and entwined modules. The separability of the induction and ad-induction functors are characterized, extending earlier results for coalgebra and ring homomorphisms, as well as for entwining structures.
2000 Mathematics Subject Classification: 16W30.
1. Introduction. The notion of separable functor was introduced by N˘ast˘asescu et al. [12], where some applications for group-graded rings were done. This notion fits satisfactorily to the classical notion of separable algebra over a commutative ring. Every separable functor between abelian categories encodes a Maschke’s theorem, which explains the interest concentrated in this notion within the module-theoretical developments in recent years. Thus, separable functors have been investigated in the framework of coalgebras [8], graded homomorphisms of rings [9,10], Doi-Koppinen modules [6,7], or finally, entwined modules [4,5]. These situations are generaliza-tions of the original study of the separability for the induction and restriction of scalars functors associated to a ring homomorphism done in [12]. It turns out that all the aforementioned categories of modules are instances of comodule categories over suitable corings [3]. In fact, the separability of some fundamental functors relating the category of comodules over a coring and the underlying category of modules has been studied in [3]. Thus, we can expect that the characterizations obtained in [4] of the separability of the induction functor associated to an admissible morphism of en-twining structures and its adjoint generalize to the corresponding functors stemming from a homomorphism of corings. This is done in this paper.
To state and prove the separability theorems, a basic theory of functors between categories of comodules has been developed in this paper, making the arguments independent from the Sweedler’ssigma-notation.The plan here is to use purely cate-gorical methods which could be easily adapted to more general developments of the theory. These methods had been sketched in [1,2] in the framework of coalgebras over commutative rings and are expounded in Sections2,3, and4. InSection 5, a notion of homomorphism of corings is given, which leads to a pair of adjoint functors (the induction functor and its adjoint, called here ad-induction functor). The morphisms of entwining structures [4] are instances of homomorphisms of corings in our setting. Finally, the separability of these functors is characterized.
termK-linear category and functor, forKa commutative ring (see, e.g., [15, Section I.O.2]). There are, however, some minor differences: the notationX∈Ꮽfor a category Ꮽmeans thatXis an object ofᏭ, and the identity morphism attached to any objectX
is represented by the object itself. The notationᏹKstands for the category of all unital
K-modules. The fact thatGis a right adjoint to some functorFis denoted byFG. For the notion of separable functor, the reader is referred to [12]. Finally, letf ,g:X→Y
be a pair of morphisms of right modules over a ringR, and letk:K→Xbe its equalizer (i.e., the kernel off−g). We will say that a leftR-moduleZ preserves the equalizer of(f ,g)ifk⊗RZ:K⊗RZ→X⊗RZ is the equalizer of the pair(f⊗RZ,g⊗RZ). Of course, every flat moduleRZpreserves all equalizers.
2. Bicomodules and the cotensor product functor. First recall from [17] the notion of coring. The concepts of comodule and bicomodule over a coring are generalizations of the corresponding notions for coalgebras. We briefly state some basic properties of the cotensor product of bicomodules. Similar associativity properties were stud-ied in [11] in the framework of coseparable corings and in [2] for coalgebras over commutative rings.
Throughout,A,A,A,...denote associative and unitary algebras over a commuta-tive ringK.
2.1. Corings. AnA-coringis a three-tuple(C,∆C,C)consisting of anA-bimodule Cand twoA-bimodule maps
∆C:C →C⊗AC, C:C →A, (2.1)
such that the diagrams
C ∆C
∆C
C⊗AC
C⊗A∆−C
C⊗AC
∆C⊗AC
C⊗AC⊗AC,
C ∆C
C⊗AC
C⊗A−C
C⊗AA
C ∆C
C⊗AC
C⊗AC
A⊗AC
(2.2)
commute.
From now on,C,C,C,...will denote corings overA,A,A,...,respectively.
2.2. Comodules. A right C-comodule is a pair (M,ρM) consisting of a right A -moduleMand anA-linear mapρM:M→M⊗ACsuch that the diagrams
M ρM
ρM
M⊗AC
M⊗A∆−C
M⊗AC
ρM⊗AC
M⊗AC⊗AC
M ρM
M⊗AC
M⊗A−C
M⊗AA
commute. LeftC-comodules are similarly defined; we use the notationλM for their structure maps. Amorphismof rightC-comodules(M,ρM)and(N,ρN)is anA-linear mapf:M→Nsuch that the following diagram is commutative:
M f
ρM
N
ρN
M⊗AC
f⊗AC
N⊗AC.
(2.4)
The K-module of all right C-comodule morphisms from M to N is denoted by HomC(M,N). TheK-linear category of all rightC-comodules is denoted byᏹC. When C=A, with the trivial coring structure, the categoryᏹAis just the category of all right
A-modules, which is “traditionally” denoted byᏹA.
Coproducts and cokernels in ᏹC do exist and can be already computed in ᏹA. Therefore,ᏹChas arbitrary inductive limits. IfACis a flat module, thenᏹCis easily proved to be an abelian category.
2.3. Bicomodules. LetρM :M→M⊗ACbe a comodule structure over anA−A -bimoduleM, and assume thatρM isA-linear. For any rightA-moduleX, the right
A-linear mapX⊗AρM:X⊗AM→X⊗AM⊗ACmakesX⊗AMa rightC-comodule. This leads to an additive functor−⊗AM:ᏹA→ᏹC. WhenA=AandM=C, the functor −⊗ACis left adjoint to the forgetful functorUA:ᏹC→ᏹA(see [11, Proposition 3.1] and [3, Lemma 3.1]). Now assume that theA−A-bimoduleMis also a leftC-comodule with structure mapλM:M→C⊗AM. It is clear thatρM:M→M⊗ACis a morphism of leftC-comodules if and only ifλM:M→C⊗AMis a morphism of rightC-comodules. In this case, we say thatM is aC−C-bicomodule. The C−C-bicomodules are the objects of aK-linear categoryCᏹCwhose morphisms are thoseA−A-bimodule maps, which are morphisms ofC-comodules and ofC-comodules. Some particular cases are now of interest. For instance, whenC=A, the objects of the categoryAᏹCare the
A−A-bimodules with a rightC-comodule structureρM :M→M⊗ACwhich is A -linear.
2.4. The cotensor product. LetM∈CᏹC
andN∈CᏹC
. We considerM⊗ANand
M⊗AC⊗ANasC−C-bicomodules with structure maps
M⊗AN M⊗AρN→M⊗AN⊗AC, M⊗AN λM⊗AM→C⊗AM⊗AN, (2.5)
M⊗AC⊗AN M⊗AC⊗AρN→M⊗AC⊗AN⊗AC, (2.6)
M⊗AC⊗AN λM⊗AC⊗AN→C⊗AM⊗AC⊗AN. (2.7)
The map
M⊗AN ρM⊗AN−M⊗AλN→M⊗AC⊗AN (2.8)
is then aC−C-bicomodule map. LetMCNdenote the kernel of (2.8). IfC
A and
Proposition2.1. Assume that for everyM∈CᏹC andN∈CᏹC, CA andAC
preserve the equalizer of(ρM⊗AN,M⊗AλN). There is an additive bifunctor
−C−:CᏹC×CᏹC →CᏹC. (2.9)
In particular, the cotensor product bifunctor (2.9) is defined whenC
A andAC are flat modules or whenCis a coseparableA-coring in the sense of[11].
In the special caseC=AandC=A, we have the bifunctor
−C−:AᏹC×CᏹA →AᏹA (2.10)
and, if we further assume thatA=A=K, we have the bifunctor
−C−:ᏹC×Cᏹ →ᏹK. (2.11)
2.5. Compatibility between tensor and cotensor. LetM∈C
ᏹCand N∈CᏹC
be bicomodules. For any rightA-moduleW, consider the commutative diagram
W⊗AMCN
ψ
W⊗AM⊗AN
W⊗AM⊗AC⊗AN
0 W⊗AMCN W⊗AM⊗AN W⊗AM⊗AC⊗AN
(2.12)
whereψis given by the universal property of the kernel in the second row. This leads to the following.
Lemma 2.2. It follows from (2.12) that WA preserves the equalizer of(ρM⊗AN,
M⊗AλN)if and only ifψ:W⊗A(MCN)(W⊗AM)CN. In particular, ψ is an
isomorphism ifWA is flat.
Next, we prove a basic fact concerning with the associativity of cotensor product.
Proposition 2.3. Let M ∈ CᏹC, N ∈ CᏹC, L ∈ CᏹC. Assume that CA, LA,
L⊗ACA, and AC preserve the equalizer of (ρM⊗AN,M⊗AλN), and thatAC, AN, AC⊗AN, andCApreserve the equalizer of(ρL⊗AM,L⊗AλM). Then there is a canon-ical isomorphism of C−C-bicomodules
Proof. SinceCAandACpreserve the equalizer of(ρM⊗AN,M⊗AλN), we know that it is aC−C-subbicomodule. Analogously,L
CMis aC−C-subbicomodule of L⊗AM. In the commutative diagram
0 LCMCN L⊗AMCN L⊗AC⊗AMCN
0 LCM⊗AN L⊗AM⊗AN L⊗AC⊗AM⊗AN
(2.14)
the second row is exact becauseAN preserves the equalizer of(ρL⊗AM,L⊗AλM). The exactness of the first row is then deduced by using thatAC⊗ANis assumed to preserve the equalizer of(ρL⊗AM,L⊗AλM). Now, consider the commutative diagram with exact rows
0 LCMCN L⊗AMCN L⊗AC⊗AMCN
0 LCMCN
ψ1
L⊗AMCN
ψ2
L⊗AC⊗AMCN
ψ3
.
(2.15)
Lemma 2.2gives the isomorphismsψ2andψ3, which induce the isomorphismψ1.
3. Functors between comodule categories. This section contains technical facts concerning withK-linear functors between categories of comodules over corings. Part of these tools were first developed for coalgebras over commutative rings in [1,2]. Roughly speaking, it is proved in this paper an analogue to Watts theorem, which allows to represent good enough functors as cotensor product functors. Also is in-cluded a result which states that, under mild conditions, a natural transformation gives a bicomodule morphism at any bicomodule. This will be used in the statement and proof of our separability theorems inSection 5.
LetCandDbe corings overK-algebrasAandB, respectively, and consider aK-linear functor
F:ᏹC →ᏹD. (3.1)
3.1. LetT be aK-algebra. For everyM∈TᏹC, consider the homomorphism ofK -algebras
TEndTT →HomCT⊗TM,T⊗TMHomC(M,M)→HomDF(M),F(M) (3.2)
which induces a leftT-module structure overF(M)such thatF(M)becomes aT−D -bicomodule. We have twoK-linear functors
−⊗TF(−),F−⊗T−:ᏹT×TᏹC →ᏹD. (3.3)
We construct a natural transformation
LetΥT ,M be the unique isomorphism ofD-comodules making the following diagram commutative
T⊗TF(M)
ΥT ,M
FT⊗TM
F(M)
(3.5)
To prove that ΥT ,M is natural at T, consider a homomorphism f : T → T of right
T-modules and defineg:F(M)→F(M)byg(x)=f (1)xfor everyx∈F(M). Since
gis just the image under (3.2) off (1)∈T, it follows thatgis a morphism of right D-comodules. Moreover,gmakes the following diagram commutative:
F(M) g F(M)
FT⊗TM
F(f⊗TM)
FT⊗TM
(3.6)
In the diagram
F(M) g F(M)
T⊗TF(M)
f⊗TF(M)
ΥT ,M
T⊗TF(M)
ΥT ,M
FT⊗TM
F(f⊗TM)
FT⊗TM
(3.7)
the commutativity of the front rectangle, which gives thatΥT ,Mis natural, follows from the commutativity of the rest of the diagram. From Mitchell’s theorem [13, Theorem 3.6.5], we obtain a natural transformation
Υ−,M:−⊗TF(M) →F−⊗TM. (3.8)
Moreover, ifF preserves coproducts (resp., direct limits, resp., inductive limits) then
ΥX,M is an isomorphism forXT projective (resp., flat, resp., any rightT-module).
Proposition3.1. If the functor F preserves coproducts, thenΥ−,− :− ⊗TF(−)→
F(−⊗T−)is a natural transformation. Moreover, ifFpreserves direct limits, thenΥX,−is
Proof. BySection 3.1,Υ−,M is natural for everyM∈TᏹC. Thus, we have only to
show thatΥX,−is natural for everyX∈ᏹT. We argue first forX=T. Letf:M→Nbe a homomorphism inTᏹC. From the diagram
F(M) F(f ) F(N)
T⊗TF(M)
T⊗TF(f )
ΥT ,M
T⊗TF(M)
ΥT ,N
FT⊗TM
F(T⊗Tf )
FT⊗TN
(3.9)
we get thatΥA,− is natural. Now, use a free presentationT(Ω)→Xto obtain thatΥX,− is natural for a generalXT. The rest of the statements are easily derived from this.
Lemma3.2. Letη:F1→F2be a natural transformation, whereF1,F2:ᏹC→ᏹDare
K-linear functors which preserve coproducts.
(1)For everyM∈TᏹC,ηM:F1(M)→F2(M)is aT−D-bicomodule homomorphism. (2)GivenX∈ᏹT andM∈TᏹC, the diagram
F1
X⊗TM
ηX⊗T M
F2
X⊗TM
X⊗TF1(M)
X⊗TηM
ΥX,M
X⊗TF2(M)
ΥX,M (3.10)
is commutative.
Proof. We need just to prove that (3.10) commutes forX=T. In this case, the
diagram can be factored out as
F1
T⊗TM
ηT⊗T M
F2
T⊗TM
F1(M) ηM
F2(M).
T⊗TF1(M)
T⊗TηM
ΥT ,M
T⊗TF2(M) ΥT ,M
(3.11)
Lemma3.3. LetT ,SbeK-algebras and assume thatF:ᏹC→ᏹDpreserves
coprod-ucts. GivenX∈ᏹS,Y∈SᏹT, andM∈TᏹC, the following formula holds:
ΥX,Y⊗TM◦
X⊗SΥY ,M
=ΥX⊗SY ,M. (3.12)
Proof. The equality will be first proved forX=S. Consider the diagram
S⊗SFY⊗TM
ΥS,Y⊗T M
FY⊗TM
S⊗SY⊗TF(M) S⊗SΥY ,M
ΥS⊗S Y ,M
Y
⊗TF(M).
ΥY ,M
ΥY ,M
FS⊗SY⊗TM FY⊗TM
(3.13)
The back rectangle is commutative by definition ofΥS,Y⊗TM, while the other two par-allelograms are commutative becauseΥ−,− is natural. Therefore, the right triangle is commutative. The equality is now easily extended forX=S(Ω) and, by using a free presentationS(Ω)→X→0, for anyX.
3.2. Natural transformations and bicomodule morphisms. LetM∈C
ᏹC be a bi-comodule. A functorF:ᏹC→ᏹDis said to beM-compatibleifΥ
C,MandΥC⊗AC,Mare isomorphisms. ByProposition 3.1, the functorF isM-compatible for every bicomod-uleM if either the functorF preserves inductive limits orC
Ais flat andF preserves direct limits. In case thatF isM-compatible, defineλF(M)as the uniqueA-linear map making the following diagram commutative
F(M) λF(M)
F(λM)
C⊗AF(M)
ΥC,M
FC⊗AM.
(3.14)
Proposition3.4. LetF be an M-compatible functor which preserves coproducts.
TheA-linear mapλF(M)is a leftC-comodule structure onF(M)such thatF(M)
be-comes aC−D-bicomodule. Moreover, givenF1,F2:ᏹC →ᏹD M-compatible functors and a natural transformationη:F1→F2, the mapηM :F1(M)→F2(M)is a C−D -bicomodule homomorphism.
Proof. In order to prove that the coaction λF(M) is coassociative, consider the
F(M)
λF(M)
λF(M)
C⊗AF(M)
∆C⊗AF(M)
ΥC,M C⊗AF(M) C⊗
AλF(M)
C⊗AF(M)
ΥC,M
C⊗AC⊗AF(M)
C⊗AΥC,M
ΥC⊗AC,M C⊗AFC⊗AM
ΥC,C⊗AM
F(M)
F(λM)
F(λM)
FC⊗AM
F(∆C⊗AM)
FC⊗AM F(C⊗AλM) FC⊗AC⊗
AM
.
(3.15)
We want to see that the top side is commutative. SinceF is assumed to be M -com-patible, we have just to prove that the mentioned side is commutative after composing with the isomorphismΥC⊗AC,M. This is deduced by usingLemma 3.3, in conjunction with the naturality ofΥ−,Mand the very definition ofλF(M). The counitary property is deduced from the commutative diagram
A⊗AF(M) ΥA,M FA⊗AM
F(M)
λF(M) F(λM)
C⊗AF(M)
C⊗AF(M)
ΥC,M
FC⊗AM.
F(C⊗AM) (3.16)
To prove the second statement, consider the diagram
F1(M)
ηM
λF1(M)
F1(λM)
F2(M)
F2(λM)
λF2(M)
F1
C⊗AM ηC⊗AM
ΥC,M
F2
C⊗AM.
ΥC,M C⊗AF
1(M)
C⊗AηM
C⊗AF 2(M)
(3.17)
Theorem 3.5. Assume thatCA is flat. If F :ᏹC →ᏹD is exact and preserves
di-rect limits (e.g., ifF is an equivalence of categories), thenF is naturally isomorphic to
−CF(C).
Proof. LetρN →N⊗AC be a rightC-comodule. We have the following diagram
with exact rows
0 NCF(C)
N⊗AF(C)
ρN⊗AF(C)−N⊗AλF(C)
ΥN,C
N⊗AC⊗AF(C)
ΥN⊗AA,C
0 F(N) FN⊗AC
F(ρN⊗AC−N⊗A∆C)
FN⊗AC⊗AC (3.18)
where the desired isomorphism is given by the universal property of the kernel.
4. Co-hom functors. This section contains a quick study of the left adjoint to a cotensor product functor, if it does exist. The presentation is inspired from the one given in [18, Subsection 1.8] for coalgebras over a field.
LetC,Dbe corings overK-algebrasAandB, respectively.
Definition 4.1. A bicomodule N∈CᏹD is said to bequasi-finite as a rightD
-comodule if the functor−⊗AN:ᏹA→ᏹDhas a left adjoint hD(N,−):ᏹD→ᏹA. This functor is called theco-homfunctor associated toN.
The natural isomorphism which gives the adjunction is denoted by
ΦX,Y : HomAhD(N,X),Y →HomDX,Y⊗AN, (4.1)
forY∈ᏹA,X∈ᏹD.
4.1. Letθ: id→hD(N,−)⊗AN be the unit of the adjunction (4.1). Therefore, the isomorphismΦX,Y is given by the assignmentf(f⊗AN)θX. In particular, the map
X θX→hD(N,X)⊗AN id⊗AλN→hD(N,X)⊗AC⊗AN (4.2)
determines anA-linear map
ρhD(N,X): hD(N,X) →hD(N,X)⊗AC (4.3)
4.2. The following is a basic tool in our investigation.
Proposition4.2. LetNbe aC−D-bicomodule. Assume thatBDpreserves the
equal-izer of(ρY⊗AN,Y⊗AλN)for every rightC-comoduleY (e.g.,BDis flat orCis a cosep-arableA-coring in the sense of[11]).
(1)IfNis quasi-finite as a rightD-comodule, then the natural isomorphism (4.1) re-stricts to an isomorphismHomC(hD(N,X),Y )HomD(X,YCN). Therefore,hD(N,−) is left adjoint to−CN.
(2)Conversely, if−CN:ᏹC→ᏹDhas a left adjoint, thenNis quasi-finite as a right C-comodule.
Proof. (1) We need to prove that iff∈HomA(hD(N,X),Y ), thenfisC-colinear if
and only if the image of(f⊗AN)θXis included inYCN. But these are straightforward computations in view of the definition of the contensor product.
(2) Since the inclusionCCN⊆C⊗ANsplits-off, we get fromSection 2.5the natural isomorphism(−⊗AC)CN −⊗A(CCN) −⊗AN. Now, the functor−⊗ACis right adjoint [3, Lemma 3.1] to the forgetful functor UA: ᏹC →ᏹA and, by hypothesis,
−CNis right adjoint to the functor hD(N,−):ᏹD→ᏹC. This implies that−⊗ANis right adjoint toUAhD(N,−), as desired.
Example4.3. LetN∈CᏹB. ThenNBis quasi-finite if and only if−⊗AN:ᏹA→ᏹB
has a left adjoint, that is, if and only ifANis finitely generated and projective. In such a case, the left adjoint is− ⊗BHomA(N,A):ᏹB→ᏹC. Notice that takingB=K, we obtain a canonical structure of rightC-comodule onN∗=HomA(N,A)for every left
C-comoduleNsuch thatNis finitely generated and projective as a leftA-module.
Example4.4. Given anA-coringCand aK-algebra homomorphismρ:A→B, we
can consider the functor− ⊗AC:ᏹB→ᏹC, which is already the composite of the restriction of scalars functor(−)ρ:ᏹB→ᏹAand the functor−⊗AC:ᏹA→ᏹC. Since these functors have both left adjoints, given, respectively, by the induction functor −⊗AB:ᏹA→ᏹBand the underlying functorUA:ᏹC→ᏹA, we get that the composite functor− ⊗AB:ᏹC→ᏹB is a left adjoint to− ⊗AC:ᏹB →ᏹC. Clearly, − ⊗AC
−⊗B(B⊗AC)and, thus,B⊗ACbecomes a quasi-finite comodule.
5. Separable homomorphisms of corings. We propose a notion of homomorphism of corings which generalizes both the concept of morphism of entwining structures [4] and the coring maps originally considered in [17]. An induction functor is constructed, which is shown to have a right adjoint, called ad-induction functor. The separability of these two functors is characterized in terms which generalize both the previous results on rings [12] and on coalgebras [8]. Our approach rests on the fundamental characterization of the separability of adjoint functors given in [10,14].
Consider anA-coringCand aB-coringD, whereAandBareK-algebras.
Definition5.1. Acoring homomorphism is a pair(ϕ,ρ), whereρ:A→B is a
C ∆C
ϕ
C⊗AC
ϕ⊗Aϕ
D⊗AD
ωD,D
D ∆D D⊗BD
C C
ϕ
A
ρ
D D B
(5.1)
whereωD,D:D⊗AD→D⊗BDis the canonical map induced byρ:A→B.
Throughout this section, we consider a coring homomorphism(ϕ,ρ):C→D. We define the induction and ad-induction functors connecting the categories of comod-ulesᏹCandᏹD.
5.1. We start with some unavoidable technical work. For everyB-bimoduleX, de-note byσX:B⊗AX→X⊗BBtheB-bimodule morphism given byb⊗Axbx⊗B1. GivenB-bimodulesX,Y, a straightforward computation shows that the diagram
B⊗AX⊗AY
B⊗AωX,Y σX⊗AY
X⊗BB⊗AY
X⊗BσY
B⊗AX⊗BY
σX⊗B Y
X⊗BY⊗BB
(5.2)
commutes, whereωX,Y:X⊗AY→X⊗BY is the obvious map. We have as well that, for every homomorphism ofB-bimodulesf:X→Y, the following diagram is commuta-tive:
B⊗AX
B⊗Af
σX
B⊗AY
σY
X⊗BB
f⊗BB
Y⊗BB
(5.3)
5.2. The induction functor. LetλM:M→C⊗AMbe a leftC-comodule. DefineλM:
M→D⊗AMas the composite map
M
λM
λM
D⊗AM
C⊗AM
ϕ⊗AM (5.4)
andλB⊗AM:B⊗AM→D⊗BB⊗AMas
B⊗AM
λB⊗AM
B⊗AλM
D⊗BB⊗AM
B⊗AD⊗AM
Proposition5.2. The homomorphism of leftB-modulesλB⊗AMendowsB⊗AMwith
a structure of leftD-comodule. This gives a functorB⊗A−:Cᏹ→Dᏹ.
Proof. Consider the diagram
M
λM
λM
λM
λM
(2) (3)
D⊗AM
D⊗AλM
D⊗AλM
C⊗AM
ϕ⊗AM
C⊗AλM
D⊗AC⊗AM
D⊗Aϕ⊗AM C⊗AM
ϕ⊗AM
∆C⊗AM
D⊗AD⊗AM
ωD,D⊗AM
C⊗AC⊗AM ϕ⊗Aϕ⊗AM
(1)
D⊗AM ∆
D⊗AM D⊗BD⊗AM.
(5.6)
The pentagon labeled as (1) commutes sinceϕis a homomorphism of corings. The four-edged diagram (2) is commutative sinceMis a left comodule. The commutation of the quadrilateral (3) follows easily from the displayed decomposition ofD⊗AλM. Therefore, the pentagon with bold arrows commutes. Now consider the diagram
B⊗AM
λB⊗AM
B⊗AλM
B⊗AλM
λB⊗AM (4)
D⊗BB⊗AM
D⊗BB⊗AλM
D⊗BλB⊗AM
B⊗AD⊗AM
σD⊗AM
B⊗AD⊗AλM (5)
B⊗AD⊗AM
σD⊗AM
B⊗A∆D⊗AM
B⊗AD⊗AD⊗AM
B⊗AωD,D⊗AM
σD⊗AD⊗AM
D⊗BB⊗AD⊗AM
D⊗BσD⊗AM
B⊗AD⊗BD⊗AM
σD⊗BD⊗AM (6)
D⊗BB⊗AM
∆D⊗BB⊗AM (7)
D⊗BD⊗BB⊗AM
(5.7)
commutes, which gives the pseudo-coassocitative property for the coactionλB⊗AM. To check the counitary property, letιM:M→A⊗AMbe the canonical isomorphism. We get from the commutative diagram
M λM
ιM λM
D⊗AM
D⊗AM C⊗AM
C⊗AM
ϕ⊗AM
A⊗AM ρ⊗AM B⊗AM
(5.8)
that the diagram
B⊗AM
λB⊗AM
B⊗AλM
ιB⊗AM B⊗AιM
D⊗BB⊗AM
D⊗BB⊗AM
B⊗AD⊗AM
σD⊗AM
B⊗AD⊗AM
B⊗AA⊗AM B
⊗Aρ⊗AM B⊗AB⊗AM σB⊗AM B⊗BB⊗AM
(5.9)
commutes, whereιB⊗AM :B⊗AM→B⊗BB⊗AM denotes the canonical isomorphism. Therefore,λB⊗AM:B⊗AM→D⊗BB⊗AMis a leftD-comodule structure map. In order to show that the assignmentMB⊗AM is functorial, we will prove thatB⊗Af is a homomorphism of leftD-comodules for every morphismf:M→NinCᏹ. So, we have to show that the outer rectangle in the following diagram is commutative
B⊗AM
B⊗Af
B⊗AλM
λB⊗AM
B⊗AN
B⊗AλN
λB⊗AN
B⊗AD⊗AM
σD⊗AM
B⊗AD⊗Af
B⊗AD⊗AN
σD⊗AN
D⊗BB⊗AM
D⊗BB⊗Af D⊗BB⊗M. (5.10)
5.3. Proposition 5.2also implies by symmetry that for every rightC-comoduleρM:
M→M⊗AC, the rightB-moduleM⊗ABis endowed with a rightD-comodule structure
ρM⊗AB:M⊗AB→M⊗AB⊗BD given by ρM⊗AB =(M⊗AδD)(ρM⊗AB), whereρM =
(M⊗Aϕ)ρM andδD:D⊗AB→B⊗BDis the obvious map. We can already state the following.
Proposition5.3. The assignmentMM⊗ABestablishes a functor−⊗AB:ᏹC→
ᏹD.
5.4. The ad-induction functor. Consider the leftD-comodule structure
λB⊗AC:B⊗AC →D⊗BB⊗AC (5.11)
defined onB⊗ACinSection 5.2.
We have, as well, a canonical structure of rightC-comodule
B⊗A∆C:B⊗AC →B⊗AC⊗AC. (5.12)
Proposition5.4. TheB−A-bimoduleB⊗ACis aD−C-bicomodule, which is
quasi-finite as a rightC-comodule. Therefore, if ACpreserves the equalizer of(ρY⊗BB⊗A C,Y⊗BλB⊗AC)for every rightD-comoduleY, then the functor
−DB⊗AC:ᏹD →ᏹC (5.13)
is right adjoint to
−⊗AB:ᏹC →ᏹD. (5.14)
Proof. Since the comultiplication∆Cis coassociative, we get that the diagram
C λC
∆C
∆C
D⊗AC
D⊗A∆C
C⊗AC
ϕ⊗AC
C⊗A∆C
C⊗AC⊗AC
ϕ⊗AC⊗AC
C⊗AC
∆C⊗AC
λC⊗AC
D⊗AC⊗AC
(5.15)
B⊗AC
λB⊗AC
B⊗AλC
B⊗A∆C
D⊗BB⊗AC
D⊗BB⊗A∆C B⊗AD⊗AC
B⊗AD⊗A∆C
σD⊗AC
B⊗AD⊗AC⊗AC
σD⊗AC⊗AC
B⊗AC⊗AC B⊗AλC⊗AC
λB⊗AC⊗AC
D⊗BB⊗AC⊗AC
(5.16)
Since we know that the rest of inner diagrams are also commutative, we obtain that the outer diagram commutes, too. This proves thatB⊗ACis aD−C-bicomodule.
We will see thatB⊗ACis a quasi-finite rightC-comodule, that is, that−⊗AB:ᏹC→ ᏹBis left adjoint to−⊗
BB⊗AC:ᏹB→ᏹC. Now, these functors fit in the commutative diagrams
ᏹA −⊗AC ᏹC
ᏹB
−⊗BB
−⊗BB⊗AC
ᏹC
−⊗AB UA
ᏹA
−⊗AB
ᏹB
(5.17)
whereUAdenotes the forgetful functor. Since−⊗ABis left adjoint to−⊗BBandUA is left adjoint to− ⊗AC, we get the desired adjunction. The rest of the proposition follows fromProposition 4.2.
Remark5.5. This proposition applies in the case thatAC is flat or whenD is a
coseparableB-coring in the sense of [11].
5.5. The unit. Let ¯∆C:C→C⊗AB⊗BB⊗ACbe the composite map
C ∆C→C⊗
AC→ ιC⊗AB⊗BB⊗AC, (5.18)
whereιmapsc⊗c∈C⊗ACtoc⊗1⊗1⊗c. This map is a homomorphism ofC -bicomodules. The unitθof the adjunction−⊗AB −⊗BB⊗ACat a rightC-comodule
Mis given by the composite map
M ρM→M⊗AC ςM⊗AC→M⊗AB⊗BB⊗AC, (5.19)
whereςM:M→M⊗AB⊗BBmapsm∈Mtom⊗A1⊗B1. We see, in particular, thatθC= ¯
∆C. ByProposition 4.2,θMfactorizes throughout(M⊗AB)D(B⊗AC)and, therefore, it gives the unitθMfor the adjunction−⊗AB −D(B⊗AC)atM. So, the multiplication
∆Cfinally induces a map, ¯
∆C:C →C⊗ABDB⊗AC, (5.20)
which is a homomorphism ofC-bicomodules.
Theorem 5.6. Assume thatCA preserves the equalizer of (ρY⊗BB⊗B⊗AC,Y⊗B
λB⊗AC)for everyY∈ᏹD, and thatXApreserves the equalizer of(ρC⊗AB⊗BB⊗AC,C⊗A
B⊗BλB⊗AC)for everyX∈ᏹC. The functor−⊗AB:ᏹC→ᏹDis separable if and only if there is a homomorphism ifC-bicomodules
ωC:C⊗ABDB⊗AC →C (5.21)
such thatωC∆¯C=C.
Proof. Assume that−⊗ABis separable. By [10, Theorem 4.1] or [14, Theorem 1.2],
the unit of the adjunction
θ: 1ᏹC →−⊗ABDB⊗AD (5.22)
is split-mono, that is, there is a natural transformationω:(−⊗AB)D(B⊗AC)→1ᏹC such thatωθ=1ᏹC. ByLemma 2.2the functor(− ⊗AB)D(B⊗AC)isC-compatible. ByProposition 3.4,ωCis aC-bicomodule map. Obviously,ωC∆¯C=C.
To prove the converse, we need to construct a natural transformationωfrom the bicomodule mapωC. Given a rightC-comoduleM, consider the diagram
M⊗AB⊗BD⊗BB⊗AC
ρM⊗AB⊗BD⊗BB⊗AC
M⊗AC⊗AB⊗BD⊗BB⊗AC
M⊗AB⊗BB⊗AC
ρM⊗AB⊗BB⊗AC ρM⊗AB⊗BB⊗AC M⊗AB⊗BλB⊗AC
M⊗AC⊗AB⊗BB⊗AC M⊗AρC⊗AB⊗BB⊗AC M⊗AC⊗AB⊗BλB⊗AC
M
⊗ABDB⊗AC
κM
M⊗AC⊗ABD(B⊗AC (5.23)
where the vertical are equalizer diagrams (here, we are using the definition of the cotensor product and the fact thatMApreserves the equalizer of(ρC⊗AB⊗BB⊗AC,C⊗A
B⊗BλB⊗AC)). If we prove that the top rectangle commutes, then there is a unique dotted arrowκM making the bottom rectangle commute. The identity
M⊗AC⊗AB⊗BλB⊗AC
ρM⊗AB⊗BB⊗AC
=ρM⊗AB⊗BD⊗BB⊗AC
M⊗AB⊗BλB⊗AC
(5.24) is obvious; so we need just to prove that
M⊗AρC⊗AB⊗BB⊗AC
ρM⊗AB⊗BB⊗AC
=ρM⊗AB⊗BB⊗BD⊗BB⊗AC
ρM⊗AB⊗BB⊗AC
, (5.25)
which is equivalent to
M⊗AρC⊗AB
ρM⊗AB
=ρM⊗AB⊗BD
and this last identity is easy to check. Now, consider the diagram
M
ρM
θM
M⊗AB⊗BB⊗AC
ρM⊗AB⊗BB⊗AC
M
ρM
θM
M⊗ABDB⊗AC
κM
M⊗AC M⊗A∆C¯
M⊗AC⊗ABDB⊗AC
M⊗AC
M⊗A∆¯C
M⊗AC⊗AB⊗BB⊗AC (5.27)
ByLemma 3.2, we have thatM⊗AθM=θM⊗AC. Sinceθis natural, andθM=∆¯C, this implies that the external rectangle commutes. We know that the four trapezia com-mute, whence the internal rectangle commutes as well. DefineωM=(M⊗AC)(M⊗B
ωC)κM, which gives a natural transformationω:(−⊗AB)D(B⊗AC)→1ᏹC. Moreover,
ωMθM=M⊗ACM⊗AωCκMθM
=M⊗ACM⊗AωCM⊗AθMρM
=M⊗AMρM=M.
(5.28)
Therefore,ωθ=1ᏹC and, by [14, Theorem 1.2], the functor−⊗ABis separable.
The counit mapC:C→Ais a homomorphism ofA-corings, where we consider on
Athe canonicalA-coring structure. When applied toC,Theorem 5.6boils down to the following corollary.
Corollary 5.7(see [3, Theorem 3.5]). For an A-coringC, the forgetful functor
UA:ᏹC→ᏹA is separable if and only if there is anA-bimodule mapγ:C⊗AC→A such thatγ∆C=Cand, in Sweedler’s sigma notation,
c(1)γc(2)⊗Ac=γc⊗Ac(1)
c
(2) ∀c,c∈C. (5.29)
Proof. Obviously, the forgetful functor coincides with−⊗AA, so that we get from
Theorem 5.6 a characterization of the separability of this functor in terms of the existence of aC-bicomodule mapωC:C⊗AC→Csuch thatωC∆C=C. Now, notice that the adjointness isomorphism
transfers faithfully the mentioned properties ofωCto the desired properties ofγ= CωC.
5.6. The counit of the adjunction. Let ˆC be the homomorphism ofB-bimodules that makes commute the following diagram
B⊗AC⊗AB
ˆ C
B⊗A−C⊗AB
B
B⊗AA⊗AB
B⊗Aρ⊗AB
B⊗AB⊗AB
m (5.31)
wherem:B⊗AB⊗AB→Bis the obvious multiplication map. Define, for every rightB -moduleY,χY=µY(Y⊗AˆC), whereµY:Y⊗BB→Yis the canonical isomorphism. This natural transformationχgives the counit of the adjunction− ⊗AB − ⊗B(B⊗AC). By Section 4.2, the counit of the adjunction − ⊗AB −D(B⊗AC) is given by the restriction ofχto(−D(B⊗AC))⊗AB. We use the same notation for this counit. Now, define ˆϕas theB-bimodule map completing the diagram
B⊗AC⊗AB ˆ ϕ
B⊗Aϕ⊗AB
D
B⊗AD⊗AB
mD (5.32)
wheremD:B⊗AD⊗AB→Dis the obvious multiplication map given by theB-bimodule structure ofD. We claim that ˆϕ is aD-comodule map. To prove this, we first show that the diagram
DDB⊗AC⊗AB D⊗BB⊗AC⊗AB
χD
B⊗AC⊗AB λB⊗AC⊗AB
ˆ ϕ
D
(5.33)
is commutative. This is done by the following computation, wherem:B⊗BB⊗AB→B denotes the obvious multiplication map
χDλB⊗AC⊗AB
=µDD⊗AˆC
σD⊗AC⊗AB
B⊗AλC⊗AB
=µDD⊗Bm
D⊗BB⊗Aρ⊗AB
D⊗BB⊗AC⊗AD
◦σD⊗AC⊗AB
B⊗Aϕ⊗AC⊗AB
B⊗A∆C⊗AB
=µDD⊗Bm
D⊗BB⊗AD⊗AB
D⊗BB⊗Aϕ⊗AB
◦σD⊗AC⊗AB
B⊗Aϕ⊗AC⊗AB
B⊗A∆C⊗AB
=µDD⊗Bm
D⊗BB⊗AD⊗AB
σD⊗Aϕ⊗AB
◦B⊗Aϕ⊗AC⊗AB
=µDD⊗Bm
D⊗BB⊗AD⊗AB
σD⊗AD⊗AB
◦B⊗Aϕ⊗Aϕ⊗AB
B⊗A∆C⊗AB
=µDD⊗Bm
D⊗BB⊗BD⊗AB
σD⊗BD⊗AB
B⊗AωD,D⊗AB
◦B⊗Aϕ⊗Aϕ⊗AB
B⊗A∆D⊗AB
=µDD⊗Bm
D⊗BB⊗BD⊗AB
σD⊗BD⊗AB
◦B⊗A∆D⊗AB
B⊗Aϕ⊗AB
=µDD⊗Bm
σD⊗BD⊗AB
B⊗A∆D⊗AB
B⊗Aϕ⊗AB
=µDD⊗Bm
σD⊗BB⊗AB
B⊗AD⊗BD⊗AB
◦B⊗A∆D⊗AB
B⊗Aϕ⊗AB
=µDD⊗Bm
σD⊗BB⊗AB
B⊗Aϕ⊗AB
=µDD⊗Bm
σD⊗AB⊗AB
B⊗Aϕ⊗AB
=ϕ.ˆ
(5.34)
We know thatλB⊗AC⊗ABis a homomorphism ofD-bicomodules and that, bySection 3.2,χDisD-bicolinear, too. This proves that ˆϕ:B⊗AC⊗AB→Dis a homomorphism ofD-bicomodules.
We are now in a position to prove our separability theorem for the ad-induction functor.
Theorem5.8. Assume thatABandACpreserve the equalizer of(ρM⊗BB⊗AC,M⊗B
λB⊗AC)for every rightD-comoduleM(e.g.,ABandACare flat orDis a coseparableB -coring in the sense of[11]). The functor−D(B⊗AC):ᏹD→ᏹCis separable if and only if there exists aD-bicomodule homomorphismνˆD:D→B⊗AC⊗ABsuch thatϕˆνˆD=D.
Proof. If−D(B⊗AC)is separable then, by [14, Theorem 1.2], there exists a
nat-ural transformation
ν: 1ᏹD →−DB⊗AC⊗AB (5.35)
such thatχν=1ᏹD. In particular,χDνD=Dand, byProposition 3.4,νDis a bicomodule map (in fact, we easily get that the functor(−D(B⊗AC))⊗ABisD-compatible from the fact thatDD(B⊗AC)is a direct summand ofD⊗BB⊗ACas a leftB-module). The map
λB⊗AC⊗ABgives an isomorphism ofD-bicomodulesB⊗AC⊗AB(DD(B⊗AC))⊗AB, which implies, after (5.33), that
D=χDνD=ϕˆλB⊗AC⊗AB −1
νD. (5.36)
Thus, ˆνD=(λB⊗AC⊗AB)−1νDis the desiredD-bicomodule map.
For the converse, assume that there is aD-bicomodule map ˆνD:D→B⊗AC⊗AB such that ˆϕνˆD=D. For each rightD-comoduleM, we prove that the homomorphism of the rightD-comodules
M ρM→M⊗BD M⊗BˆνD
factorizes throughout(MD(B⊗AC))⊗AB. SinceABpreserves the equalizer of(ρM⊗B
B⊗AC,M⊗BλB⊗AC), we know that
MDB⊗AC⊗ABMDB⊗AC⊗AB. (5.38)
Therefore, we need just to check the equality
ρM⊗BB⊗AC⊗ABM⊗BνˆDρM=M⊗BλB⊗AC⊗AB
M⊗BνˆDρM. (5.39)
This is done by the following computation:
M⊗BλB⊗AC⊗AB
M⊗BνˆDρM
=M⊗BD⊗BνˆDM⊗B∆DρM=M⊗BD⊗BνˆDρM
=ρM⊗BνˆDρM=ρM⊗BB⊗AC⊗ABM⊗BνˆDρM.
(5.40)
Thus, we have proved that the natural transformation given in (5.37) factorizes throughout a natural transformation
νM:M →MDB⊗AC⊗AB. (5.41)
This means that we have a commutative diagram
M ρM
νM
M⊗BD M⊗BνˆD
M⊗BB⊗AC⊗AB χM
MDB⊗AC⊗AB χM M
(5.42)
Finally, we show thatνM splits offχMby means of the following computation:
ρMχMνM=ρMχM
M⊗BνˆD
ρM
=χM⊗BD
ρM⊗BB⊗AC⊗AB
M⊗BνˆD
ρM (χ is natural)
=M⊗BχD
ρM⊗BB⊗AC⊗AB
M⊗BνˆD
ρM (byLemma 3.2)
=M⊗BχD
M⊗BλB⊗AC⊗AB
M⊗BνˆD
ρM (νˆD is bicolinear)
=M⊗Bϕˆ
M⊗BνˆD
ρM=ρM.
(5.43)
SinceρMis a monomorphism, we getχMνM=M. By [14, Theorem 1.2],−D(B⊗AC) is a separable functor.
By applying the stated theorem toC:C→Awe obtain the following corollary.
Corollary 5.9 (see [3, Theorem 3.3]). Let C be an A-coring. Then the functor
−⊗ACis separable if and only if there exists an invariante∈C(i.e.,e∈Csatisfying
ae=eafor everya∈A) such thatC(e)=1.
Proof.This follows fromTheorem 5.8taking that theA-bimodule homomorphisms
5.7. Final remarks. Brzezi´nski showed [3, Proposition 2.2] that if(A,C)ψis an en-twining structure overK, thenA⊗KCcan be endowed with anA-coring structure in such a way that the category ᏹA⊗KC is isomorphic with the categoryᏹC
A(ψ)of en-twined(A,C)ψ-modules. It turns out that every morphism of entwining structures
(f ,g):(A,C)ψ→(B,D)γ in the sense of [4] gives a coring morphism (f⊗Kg,f ):
A⊗KC→B⊗KD. Some straightforward computations show that the statements of the separability theorem [4, Theorem 3.4] correspond to our Theorems5.6and5.8. In fact, we can give the notions of totally integrable and totally cointegrable morphism of entwining structures to the framework of morphisms of corings(φ,ρ):C→Dby requiring the existence of the splitting bicomodule mapsωCand ˆνD, respectively. In the first case, the existence ofωC can be transferred, if desired, to the existence of certain bimodule map with extra properties by means of the adjointness isomorphism
HomCC⊗ABDB⊗AC,CHomAC⊗ABDB⊗AC,A. (5.44)
Acknowledgment. I would like to thank Tomasz Brzezi´nski for some helpful
comments.
References
[1] K. Al-Takhman,Äquivalenzen zwischen komodul-kategorien von koalgebren über ringen, Ph.D. thesis, Universität Düsseldorf, 1999.
[2] ,Equivalences of comodule categories for coalgebras over rings, preprint, 2001. [3] T. Brzezi´nski,The structure of corings: induction functors, Maschke-type theorem, and
Frobenius and Galois-type properties, to appear in Algebras and Repr. Theory. [4] ,Coalgebra-Galois extensions from the extension theory point of view, Hopf
Alge-bras and Quantum Groups (S. Caenepeel and F. V. Oystaeyen, eds.), Lecture Notes Pure Appl. Math., vol. 209, Dekker, New York, 2000, pp. 47–68.
[5] T. Brzezi´nski, S. Caenepeel, G. Militaru, and S. Zhu,Frobenius and Maschke type theorems for Doi-Hopf modules and entwined modules revisited: a unified approach, Ring Theory and Algebraic Geometry (A. Granja, J. H. Alonso, and A. Verschoren, eds.), Lecture Notes Pure Appl. Math., vol. 221, Dekker, New York, 2001, pp. 1–31. [6] S. Caenepeel, B. Ion, G. Militaru, and S. Zhu,Separable functors for the category of
Doi-Hopf modules. II, Hopf Algebras and Quantum Groups (S. Caenepeel and F. Van Oys-taeyen, eds.), Lecture Notes Pure Appl. Math., vol. 209, Dekker, New York, 2000, pp. 69–103.
[7] S. Caenepeel, G. Militaru, B. Ion, and S. Zhu,Separable functors for the category of Doi-Hopf modules, applications, Adv. Math.145(1999), 239–290.
[8] F. Castaño, J. Gómez-Torrecillas, and C. N˘ast˘asescu,Separable functors in coalgebras. Applications, Tsukuba J. Math.21(1997), 329–344.
[9] ,Separable functors in graded rings, J. Pure Appl. Algebra127(1998), 219–230. [10] A. del Río,Categorical methods in graded ring theory, Publ. Mat.36(1992), 489–531. [11] F. Guzmán,Cointegrations, relative cohomology for comodules, and coseparable corings,
J. Algebra126(1989), 211–224.
[12] C. N˘ast˘asescu, M. van den Bergh, and F. van Oystaeyen,Separable functors applied to graded rings, J. Algebra123(1989), 397–413.
[13] N. Popescu,Abelian Categories with Applications to Rings and Modules, Academic Press, London, 1973.
[14] M. D. Rafael,Separable functors revisited, Comm. Algebra18(1990), 1445–1459. [15] R. N. Saavedra, Catégories Tannakiennes, Lecture Notes in Mathematics, vol. 265,
Springer-Verlag, Berlin, 1972.
[17] M. Sweedler,The predual theorem to the Jacobson-Bourbaki theorem, Trans. Amer. Math. Soc.213(1975), 391–406.
[18] M. Takeuchi,Morita theorems for categories of comodules, J. Fac. Sci. Univ. Tokyo Sect. IA Math.24(1977), 629–644.
J. Gómez-Torrecillas: Departamento de Álgebra, Universidad de Granada, E18071 Granada, Spain