Entwined modules over linear categories and Galois extensions
aa r X i v : . [ m a t h . C T ] J un Entwined modules over linear categories and Galois extensions
Mamta Balodi ∗ Abhishek Banerjee † Samarpita Ray ‡ Department of Mathematics, Indian Institute of Science, Bangalore - 560012, India.
Abstract
In this paper, we study modules over quotient spaces of certain categorified fiber bundles. Theseare understood as modules over entwining structures involving a small K -linear category D and a K -coalgebra C . We obtain Frobenius and separability conditions for functors on entwined modules.We also introduce the notion of a C -Galois extension E ⊆ D of categories. Under suitable conditions,we show that entwined modules over a C -Galois extension may be described as modules over thesubcategory E of C -coinvariants of D . MSC(2010) Subject Classification:
Keywords:
Entwining structures, entwined modules, rings with several objects, Frobenius conditions,separability conditions, coalgebra-Galois extensions
The purpose of this paper is to study a theory of modules over quotient spaces of certain categorifiedfiber bundles. Suppose that X is an affine scheme over a field K and let G be an affine algebraic groupscheme with a free action σ : X × G −→ X on X . Let Y be the quotient given by the coequalizer X × G pr / / σ / / X p −→ Y (1.1)If X −→ Y is faithfully flat and the canonical map can : X × G −→ X × Y X is an isomorphism, then X is said to be (see, for instance, [24], [28]) a principal fiber bundle over Y with group G .The algebraic counterpart of (1.1) consists of an algebra A , a Hopf algebra H and a coaction ρ : A −→ A ⊗ H that makes A into a right H -comodule algebra. Let B := A coH = { a ∈ A | ρ ( a ) = a ⊗ H } be thealgebra of coinvariants of A , i.e., B is given by the equalizer B −→ A in / / ρ / / A ⊗ H (1.2)In this case, there is a canonical map can : A ⊗ B A −→ A ⊗ H determined by setting can ( x ⊗ y ) = x · ρ ( y ).If the Hopf algebra H has bijective antipode, B −→ A is a faithfully flat extension and can : A ⊗ B A −→ A ⊗ H is an isomorphism, it was shown by Schneider [28] that modules over B may be recovered as thecategory of “( A, H )-Hopf modules.”We work with a small K -linear category D , a K -coalgebra C and an “entwining structure” ψ consistingof a collection of morphisms ψ = { ψ XY : C ⊗ Hom D ( X, Y ) −→ Hom D ( X, Y ) ⊗ C } ( X,Y ) ∈ Ob ( D ) ∗ Email: [email protected] † Email: [email protected] ‡ Email: [email protected] M ( ψ ) C D of modules overthe entwining structure ( D , C, ψ ) (see Definition 2.2). These may be seen as modules over a “categoricalquotient space” of D with respect to the coalgebra C and the entwining ψ .The notion of a C -Galois extension E ⊆ D of categories is introduced in Section 4. Additionally, a C -Galois extension gives rise to a canonical entwining structure on D . Under certain conditions, we showthat modules over the category E of C -coinvariants of D may be described as modules over the canonicalentwining structure.Entwining structures for algebras were introduced by Brzezi´nski and Majid in [7] and it was realized inBrzezi´nski [3] that entwined modules provide a unifying formalism for studying diverse concepts such asrelative Hopf modules, Doi-Hopf and Yetter-Drinfeld modules as well as coalgebra Galois extensions. Infact, the study of entwining structures for algebras and entwined modules over them is well developedin the literature and we refer the reader, for instance, to [1], [3] [5], [9], [10], [12], [21], [27] for more onthis subject.Our notion of modules over an entwining structure ( D , C, ψ ) builds on the analogy of Mitchell [22] whichsays that a small K -linear category should be seen as a “ K -algebra with several objects.” In particular,the category M ( ψ ) C D also generalizes the “relative ( D , H )-Hopf modules” studied in our previous work in[2], where H is a Hopf algebra and D is an H -comodule category in the sense of Cibils and Solotar [16].In other words, D is a small K -linear category whose morphism spaces are equipped with a coaction of H that is compatible with composition. When D has a single object, it reduces to an ordinary H -comodulealgebra and the relative ( D , H )-Hopf modules reduce to the usual notion of relative Hopf modules (seeTakeuchi [30]).For Doi-Hopf modules, Frobenius and separability conditions were studied extensively in a series ofpapers [13], [14], [15]. Later, Brzezi´nski studied Frobenius and Maschke type theorems for entwinedmodules in [4]. In this paper, we proceed in a manner analogous to the unified approach of Brzezi´nski,Caenepeel, Militaru and Zhu [8] for studying Frobenius and separability conditions for entwined modulesover ( D , C, ψ ).The idea is as follows: the “categorical quotient space” of D with respect to C and ψ may be thoughtof as a subcategory of D and M ( ψ ) C D plays the role of modules over this subcategory. Although this“subcategory” of D need not exist in an explicit sense, we would like to study the properties of thisextension of categories. In particular, we would like to know if it behaves like a separable, split orFrobenius extension of small K -linear categories. For this, we turn to a pair of functors F : M ( ψ ) C D −→ M od - D G : M od - D −→ M ( ψ ) C D Here F is the left adjoint and behaves like an “extension of scalars” whereas its right adjoint G behaveslike a “restriction of scalars.” We recall here (see [8, Theorem 1.2]) that in the classical case of anextension R −→ S of rings inducing the pair of adjoint functors M od - R G ←−−−−−−−−−−−−−−−−−−−−→ F M od - S given byextension and restriction of scalars, we have: R −→ S is split extension ⇔ Left adjoint F : M od - R −→ M od - S is separable R −→ S is separable extension ⇔ Right adjoint G : M od - S −→ M od - R is separable R −→ S is Frobenius extension ⇔ ( F, G ) is Frobenius pair of functorsIt is therefore natural to study criteria for the separability of the functors F and G as well as conditionsfor ( F , G ) to be a Frobenius pair of functors.In this paper, we will always use the following convention: for f ∈ Hom D ( Y, X ) and c ∈ C , we write ψ Y X ( c ⊗ f ) = f ψ ⊗ c ψ ∈ Hom D ( Y, X ) ⊗ C with the summation omitted. We write h : D op ⊗ D −→ V ect K for the canonical D - D -bimodule h ( Y, X ) =
Hom D ( Y, X ). The entwining structure makes h ⊗ C into a D - D -bimodule by setting( h ⊗ C )( Y, X ) :=
Hom D ( Y, X ) ⊗ C (cid:0) ( h ⊗ C )( φ ) (cid:1) ( f ⊗ c ) := φ ′′ f φ ′ ψ ⊗ c ψ for any ( Y, X ) ∈ Ob ( D op ⊗ D ), φ := ( φ ′ , φ ′′ ) ∈ Hom D op ⊗D (cid:0) ( Y, X ) , ( Y ′ , X ′ ) (cid:1) , f ∈ Hom D ( Y, X ) and c ∈ C . We consider a collection θ := { θ X : C ⊗ C −→ End D ( X ) } X ∈ Ob ( D ) of K -linear maps satisfying2he following conditions:( θ X ( c ⊗ d )) ◦ f = f ψψ ◦ θ Y (cid:0) c ψ ⊗ d ψ (cid:1) θ X ( c ⊗ d ) ⊗ d = ( θ X ( c ⊗ d )) ψ ⊗ c ψ for any f ∈ Hom D ( Y, X ). Let V be the K -space consisting of all such θ . Our first result gives conditionsfor the functors F and G to be separable. Theorem A. (see 3.7, 3.8, 3.10 and 3.11) Let D be a small K -linear category, ( C, ∆ C , ε C ) be a K -coalgebra and let ( D , C, ψ ) be a right-right entwining structure.(a) Let V = N at ( G F , M ( ψ ) C D ) be the space of natural transformations from G F to M ( ψ ) C D . Then:(1) There is an isomorphism V ∼ = V of K -vector spaces.(2) The functor F is separable if and only if there exists θ ∈ V such that θ X ◦ ∆ C = ε C · id X ∀ X ∈ Ob ( D ) (b) Let W = N at (1 Mod - D , F G ) be the space of natural transformations from Mod - D to F G . Then:(1) There is an isomorphism of K -vector spaces from W to W = N at ( h, h ⊗ C ) .(2) The functor G is separable if and only if there exists η ∈ W = N at ( h, h ⊗ C ) such that ( id h ⊗ ε C ) η = id h . The next result gives conditions for ( F , G ) to be a Frobenius pair. Theorem B. (see 3.14) Let D be a small K -linear category, ( C, ∆ C , ε C ) be a K -coalgebra and let ( D , C, ψ ) be a right-right entwining structure. Then, ( F , G ) is a Frobenius pair if and only if there exist θ ∈ V and η ∈ W such that the following conditions hold: ε C ( d ) f = X ˆ f ◦ θ X ( c f ⊗ d ) ε C ( d ) f = X ˆ f ψ ◦ θ X ( d ψ ⊗ c f ) for any f ∈ Hom D ( X, Y ) , d ∈ C and η ( X, Y )( f ) = P ˆ f ⊗ c f . More generally, the D - D -bimodule h ⊗ C may be treated as a functor h ⊗ C : D −→ M ( ψ ) C D by setting(see Lemma 2.4) ( h ⊗ C )( Y ) := Hom D ( − , Y ) ⊗ C ( h ⊗ C )( f )( Z )( g ⊗ c ) := f g ⊗ c for f ∈ Hom D ( Y, X ) and g ⊗ c ∈ Hom D ( Z, Y ) ⊗ C . Additionally, let C be a finite dimensional coalgebraand let C ∗ = Hom ( C, K ) be the linear dual of C . Then, we show that there is a functor C ∗ ⊗ h : D −→ M ( ψ ) C D . Theorem C. (see 3.19) Let ( D , C, ψ ) be an entwining structure and let C be a finite dimensionalcoalgebra. Then, the following statements are equivalent:(i) ( F , G ) is a Frobenius pair.(ii) C ∗ ⊗ h and h ⊗ C are isomorphic as functors from D to M ( ψ ) C D . In the final part of this paper, we study coalgebra Galois extensions of categories in a manner analogousto Brzezi´nski [3], Brzezi´nski and Hajac [6] and Caenepeel [11]. For this, we suppose that every morphismspace
Hom D ( X, Y ) carries the structure of a C -comodule ρ XY : Hom D ( X, Y ) −→ Hom D ( X, Y ) ⊗ C , f P f ⊗ f . This allows us to define a category E of C -coinvariants of D (see Definition 4.5). Further,we say that D is a C -Galois extension of E if the canonical map can X : h ⊗ E Hom D ( X, − ) −→ Hom D ( X, − ) ⊗ C is an isomorphism for each X ∈ Ob ( D ) (see Definition 4.7). We show that a C -Galois extension leads toa canonical entwining structure. 3 heorem D. (see 4.9) Let D be a C -Galois extension of E . Then, there exists a unique right-rightentwining structure ( D , C, ψ ) which makes Hom D ( − , Y ) an object in M ( ψ ) C D for every Y ∈ Ob ( D ) withits canonical D -module structure and right C -coactions { ρ XY } X ∈ Ob ( D ) . Conversely, under suitable conditions, an entwining structure ( D , C, ψ ) may be used to express D asa C -Galois extension. In that case, the category M ( ψ ) C D reduces to the category of modules over the C -coinvariants of D . Theorem E. (see 4.12 and 4.21) Let C be a K -coalgebra and D be a small K -linear category suchthat Hom D ( X, Y ) has a right C -comodule structure ρ XY for every X, Y ∈ Ob ( D ) . Let E be the sub-category of C -coinvariants of D . If there exists a convolution invertible collection Φ = { Φ XY : C −→ Hom D ( X, Y ) } X,Y ∈ Ob ( D ) of right C -comodule maps, then the following are equivalent:(i) D is a C -Galois extension of E .(ii) There exists a right-right entwining structure ( D , C, ψ ) such that Hom D ( − , Y ) is an object in M ( ψ ) C D for every Y ∈ Ob ( D ) with its canonical D -module structure and right C -coactions { ρ XY } X ∈ Ob ( D ) .(iii) For any f ∈ Hom D ( X, Y ) , the morphism P f ◦ Φ ′ ZX ( f ) ∈ Hom E ( Z, Y ) for every Z ∈ Ob ( D ) ,where Φ ′ is the convolution inverse of Φ .In this case, the categories M ( ψ ) C D and Mod- E are equivalent. Notations:
Throughout the paper, K is a field, C is a K -coalgebra with comultiplication ∆ C andcounit ε C . We shall use Sweedler’s notation for the coproduct ∆ C ( c ) = c ⊗ c , and for a coaction ρ M : M −→ M ⊗ C , ρ M ( m ) = m ⊗ m with the summation omitted. We denote by C ∗ the linear dualof C . Sometimes when the coaction is clear from context, we will omit the subscript. In this section, we introduce a categorical generalization of entwining structures and entwined modules.We prove that the category of entwined modules is a Grothendieck category. We begin by recalling thedefinition of modules over a category (see, for instance, [23, 29]).
Definition 2.1.
A right module over a small K -linear category D is a K -linear functor D op −→ V ect K ,where V ect K denotes the category of K -vector spaces. Similarly, a left module over D is a K -linearfunctor D −→
V ect K . The category of all right (resp. left) modules over D will be denoted by M od - D (resp. D - M od ). For each X ∈ Ob ( D ), the representable functors h X := Hom D ( − , X ) and X h := Hom D ( X, − ) areexamples of right and left modules over D respectively. Unless otherwise mentioned, by a D -module wewill always mean a right D -module.Let C be a K -coalgebra and let D be a small K -linear category. Suppose that we have a collection of K -linear maps ψ = { ψ XY : C ⊗ Hom D ( X, Y ) −→ Hom D ( X, Y ) ⊗ C } ( X,Y ) ∈ Ob ( D ) We use the notation ψ XY ( c ⊗ f ) = f ψ ⊗ c ψ for c ∈ C and f ∈ Hom D ( X, Y ). We will say that the tuple( D , C, ψ ) is a (right-right) entwining structure if the following conditions hold:( gf ) ψ ⊗ c ψ = g ψ f ψ ⊗ c ψψ (2.1) ε C ( c ψ )( f ψ ) = ε C ( c ) f (2.2) f ψ ⊗ ∆ C ( c ψ ) = f ψψ ⊗ c ψ ⊗ c ψ (2.3) ψ XX ( c ⊗ id X ) = id X ⊗ c (2.4)for each f ∈ Hom D ( X, Y ), g ∈ Hom D ( Y, Z ) and c ∈ C . Throughout this paper, ( D , C, ψ ) will alwaysbe an entwining structure. A morphism between entwining structures ( D ′ , C ′ , ψ ′ ) and ( D , C, ψ ) is apair ( F , σ ) where F : D ′ −→ D is a functor and σ : C ′ −→ C is a counital coalgebra map such that F ( f ′ ψ ′ ) ⊗ σ ( c ′ ψ ′ ) = F ( f ′ ) ψ ⊗ σ ( c ′ ) ψ for any c ′ ⊗ f ′ ∈ C ′ ⊗ Hom D ′ ( X ′ , Y ′ ) where X ′ , Y ′ ∈ Ob ( D ′ ).4 efinition 2.2. Let M be a right D -module with a given right C -comodule structure ρ M ( Y ) : M ( Y ) −→M ( Y ) ⊗ C on M ( Y ) for each Y ∈ Ob ( D ) . Then, M is said to be an entwined module over ( D , C, ψ ) ifthe following compatibility condition holds: ρ M ( Y ) ( M ( f )( m )) = (cid:0) M ( f )( m ) (cid:1) ⊗ (cid:0) M ( f )( m ) (cid:1) = M ( f ψ )( m ) ⊗ m ψ (2.5) for every f ∈ Hom D ( Y, X ) and m ∈ M ( X ) . We denote by M ( ψ ) C D the category whose objects areentwined modules over ( D , C, ψ ) and whose morphisms are given by Hom M ( ψ ) C D ( M , N ) := { η ∈ Hom
Mod - D ( M , N ) | η ( X ) : M ( X ) −→ N ( X ) is C -colinear ∀ X ∈ Ob ( D ) } We now give an important example of entwining structures.
Example 2.3.
Let D be a right co- H -category (see [16] or the description in [2, Definition 2.4] ) and C be a right H -module coalgebra. Then, the triple ( D , C, ψ ) is an entwining structure, where ψ is given by: ψ XY : C ⊗ Hom D ( X, Y ) −→ C ⊗ Hom D ( X, Y ) ⊗ H ∼ = −→ Hom D ( X, Y ) ⊗ C ⊗ H −→ Hom D ( X, Y ) ⊗ C Explicitly, we have ψ XY ( c ⊗ f ) := f ⊗ cf for any f ∈ Hom D ( X, Y ) and c ∈ C . In this case, anentwined module is precisely a right D -module with a given right C -comodule structure on M ( X ) foreach X ∈ Ob ( D ) and satisfying the following compatibility condition (cid:0) M ( f )( m ) (cid:1) ⊗ (cid:0) M ( f )( m ) (cid:1) = M ( f )( m ) ⊗ m f We will refer to these modules as (right-right) Doi-Hopf modules and their category will be denoted by M C D . If D is a right co- H -category with a single object, i.e., an H -comodule algebra, then M C D recoversthe classical notion of Doi-Hopf modules (see [18] ). In the particular case where C = H , the right-rightDoi-Hopf modules have been referred to as relative Hopf modules in [2, § . Lemma 2.4.
Let ( D , C, ψ ) be an entwining structure and let M be a right D -module. Then, we mayobtain an object M ⊗ C ∈ M ( ψ ) C D by setting ( M ⊗ C )( X ) := M ( X ) ⊗ C ( M ⊗ C )( f )( m ⊗ c ) := M ( f ψ )( m ) ⊗ c ψ for X ∈ Ob ( D ) , f ∈ Hom D ( Y, X ) and m ⊗ c ∈ M ( X ) ⊗ C . In fact, this determines a functor from M od - D to M ( ψ ) C D .Proof. The fact that
M ⊗ C is a right D -module follows from (2.1). For each X ∈ Ob ( D ), it may beverified that M ( X ) ⊗ C has a right C -comodule structure given by π r M ( X ) ⊗ C ( m ⊗ c ) := ( id M ( X ) ⊗ ∆ C )( m ⊗ c ) = m ⊗ c ⊗ c (2.6)It remains to check the compatibility condition in (2.5). By definition, we have( M ( f ψ )( m ) ⊗ c ψ ) ⊗ ( M ( f ψ )( m ) ⊗ c ψ ) = M ( f ψ )( m ) ⊗ ( c ψ ) ⊗ ( c ψ ) = M ( f ψψ )( m ) ⊗ c ψ ⊗ c ψ (using (2.3))= ( M ⊗ C )( f ψ )( m ⊗ c ) ⊗ c ψ Lemma 2.5.
Let ( D , C, ψ ) be an entwining structure and N be a right C -comodule. Then, for each X ∈ Ob ( D ) we may obtain an object N ⊗ h X ∈ M ( ψ ) C D by setting ( N ⊗ h X )( Y ) := N ⊗ h X ( Y ) (2.7)( N ⊗ h X )( f )( n ⊗ g ) := n ⊗ gf (2.8) for Y ∈ Ob ( D ) , f ∈ Hom D ( Z, Y ) , n ⊗ g ∈ N ⊗ h X ( Y ) . In fact, this determines a functor from Comod - C to M ( ψ ) C D . roof. By definition, it follows that N ⊗ h X is a right D -module. Further, for each Y ∈ Ob ( D ), we definea K -linear map σ rN ⊗ h X ( Y ) : N ⊗ h X ( Y ) −→ N ⊗ h X ( Y ) ⊗ C as follows σ rN ⊗ h X ( Y ) ( n ⊗ g ) := n ⊗ g ψ ⊗ n ψ (2.9)We now verify that the map defined in (2.9) makes N ⊗ h X ( Y ) a right C -comodule. We have( σ r ⊗ id C ) σ r ( n ⊗ g ) = ( σ r ⊗ id C )( n ⊗ g ψ ⊗ n ψ ) = n ⊗ g ψψ ⊗ n ψ ⊗ n ψ = n ⊗ g ψ ⊗ ∆ C ( n ψ ) (by (2.3))= ( id N ⊗ h X ( Y ) ⊗ ∆ C ) σ r ( n ⊗ g )Moreover, using (2.2) we have( id N ⊗ h X ( Y ) ⊗ ε C ) σ r ( n ⊗ g ) = ( id N ⊗ h X ( Z ) ⊗ ε C )( n ⊗ g ψ ⊗ n ψ )= n ⊗ ε C ( n ψ ) g ψ = n ⊗ ε C ( n ) g = n ⊗ g It remains to verify the condition in (2.5). We have (cid:0) ( N ⊗ h X )( f )( n ⊗ g ) (cid:1) ⊗ (cid:0) ( N ⊗ h X )( f )( n ⊗ g ) (cid:1) = n ⊗ ( gf ) ψ ⊗ n ψ = n ⊗ g ψ f ψ ⊗ n ψψ (by (2.1))= ( N ⊗ h X )( f ψ )( n ⊗ g ψ ) ⊗ n ψψ = ( N ⊗ h X )( f ψ )(( n ⊗ g ) ) ⊗ ( n ⊗ g ) ψ It follows from Lemma 2.4 and Lemma 2.5 that both h Y ⊗ C and C ⊗ h Y are objects in M ( ψ ) C D forevery Y ∈ Ob ( D ). Lemma 2.6.
Let ( D , C, ψ ) be an entwining structure. Then, for each Y ∈ Ob ( D ) , we get a morphism Ψ Y : C ⊗ h Y −→ h Y ⊗ C in M ( ψ ) C D given by Ψ Y ( X ) := ψ XY .Proof. First we verify that Ψ Y is a morphism of right D -modules. For any f ∈ Hom D ( X ′ , X ), g ∈ Hom D ( X, Y ) and c ∈ C , we have(( h Y ⊗ C )( f )) ψ XY ( c ⊗ g ) = ( h Y ⊗ C )( f )( g ψ ⊗ c ψ ) = g ψ f ψ ⊗ c ψψ = ( gf ) ψ ⊗ c ψ = ψ X ′ Y ( c ⊗ gf ) = ψ X ′ Y ( C ⊗ h Y )( f )( c ⊗ g )Next, we will show that Ψ Y ( X ) is C -colinear for every X ∈ Ob ( D ). We have (cid:0) ψ XY ( c ⊗ g ) (cid:1) ⊗ (cid:0) ψ XY ( c ⊗ g ) (cid:1) = g ψ ⊗ ∆ C ( c ψ )= g ψψ ⊗ c ψ ⊗ c ψ (by (2.3))= ( ψ XY ⊗ id )( c ⊗ g ) ⊗ ( c ⊗ g ) (by (2.9))We now recall from [22, §
3] and [23] the notion of a finitely generated module over a category. Given
M ∈
M od - D , we set el ( M ) := ` X ∈ Ob ( D ) M ( X ) to be the collection of all elements of M . Since D is small,we note that el ( M ) is a set. If m ∈ el ( M ) is such that m ∈ M ( X ), we will write | m | = X . Definition 2.7.
Let D be a small preadditive category and let M be a right D -module. For each m ∈ el ( M ) , we consider the corresponding morphism η m : h | m | −→ M . A family of elements { m i ∈ el ( M ) } i ∈ I is said to be a generating set for M if the induced morphism η : M i ∈ I h | m i | −→ M (0 , ..., , id | m i | , , ..., m i is an epimorphism in M od - D . In other words, every element m ∈ el ( M ) may be expressed as a sum m = P i ∈ I M ( f i )( m i ) , where each f i ∈ Hom D ( | m | , | m i | ) and all but finitely many { f i } i ∈ I are zero. emma 2.8. Let ( D , C, ψ ) be an entwining structure and let M be an entwined module. We consideran element m ∈ el ( M ) . Then, there exists a finite dimensional C -subcomodule V m of M ( | m | ) containing m and a morphism η m : V m ⊗ h | m | −→ M in M ( ψ ) C D such that η m ( | m | )( m ⊗ id | m | ) = m .Proof. By [17, Theorem 2.1.7], we know that there exists a finite dimensional C -subcomodule V m ⊆M ( | m | ) such that m ∈ V m . Now, we consider the D -module morphism η m : V m ⊗ h | m | −→ M definedby setting η m ( Y )( v ⊗ f ) := M ( f )( v ) for any Y ∈ Ob ( D ) , f ∈ Hom D ( Y, | m | ) and v ∈ V m . We also have ρ M ( Y ) (cid:0) η m ( Y )( v ⊗ f ) (cid:1) = ρ M ( Y ) (cid:0) M ( f )( v ) (cid:1) = M ( f ψ )( v ) ⊗ v ψ = η m ( Y )( v ⊗ f ψ ) ⊗ v ψ = ( η m ( Y ) ⊗ id C ) (cid:0) ρ V m ⊗ h | m | ( Y ) ( v ⊗ f ) (cid:1) (by (2.9))This shows that η m ( Y ) is C -colinear for each Y ∈ Ob ( D ). Hence, η m is a morphism in M ( ψ ) C D such that η m ( | m | )( m ⊗ id | m | ) = m . Proposition 2.9.
Let ( D , C, ψ ) be an entwining structure. Then, the category M ( ψ ) C D of entwinedmodules is a Grothendieck category.Proof. Given a morphism η : M −→ N in M ( ψ ) C D , let Ker ( η ) and Coker ( η ) be respectively thekernel and cokernel in M od - D . Since Comod - C is an abelian category, we know that Ker ( η )( X ), Coker ( η )( X ) ∈ Comod - C for each X ∈ Ob ( D ). It is easily seen that Ker ( η ) and Coker ( η ) satisfythe compatibility condition in (2.5), i.e., Ker ( η ), Coker ( η ) ∈ M ( ψ ) C D . Since limits and colimits in M ( ψ ) C D are obtained from those in M od - D and Comod - C , it is clear that M ( ψ ) C D is a cocompleteabelian category satisfying (AB5).By Lemma 2.8, there is an epimorphism M m ∈ el ( M ) η m : M m ∈ el ( M ) V m ⊗ h | m | −→ M for any M ∈ M ( ψ ) C D . As such, the collection { V ⊗ h X } , where X ranges over all objects in D and V ranges over all (isomorphism classes of) finite dimensional C -comodules gives a set of generators for M ( ψ ) C D in the sense of [20, Proposition 1.9.1]. Corollary 2.10.
The category M C D of Doi-Hopf modules is a Grothendieck category. Let F : M ( ψ ) C D −→ M od - D be the forgetful functor. The next result shows that the functor F has aright adjoint. Lemma 3.1.
The forgetful functor F : M ( ψ ) C D −→ M od - D has a right adjoint G : M od - D −→ M ( ψ ) C D given by G ( N ) := N ⊗ C for each N ∈
M od - D .Proof. From Lemma 2.4, we know that G ( N ) = N ⊗ C ∈ M ( ψ ) C D for each N ∈
M od - D . We define α : Hom M ( ψ ) C D (cid:0) M , G ( N ) (cid:1) −→ Hom
Mod - D ( F ( M ) , N ) by setting α ( ξ )( X )( m ) := ( id N ( X ) ⊗ ε C )( ξ ( X )( m ))for each ξ : M −→ N ⊗ C in M ( ψ ) C D , X ∈ Ob ( D ) and m ∈ M ( X ).We also define β : Hom
Mod - D ( F ( M ) , N ) −→ Hom M ( ψ ) C D (cid:0) M , G ( N ) (cid:1) by setting β ( η )( X )( m ) := η ( X )( m ) ⊗ m for each η : M −→ N in M od - D , X ∈ Ob ( D ) and m ∈ M ( X ). First we check that α ( ξ ) and β ( η ) aremorphisms in M od - D and M ( ψ ) C D respectively. Using the fact that id N ⊗ ε C : N ⊗ C −→ N and ξ areright D -module morphisms, for any f ∈ Hom D ( Y, X ), we have N ( f ) (cid:0) α ( ξ )( X )( m ) (cid:1) = N ( f ) (cid:16) ( id N ( X ) ⊗ ε C ) (cid:0) ξ ( X )( m ) (cid:1)(cid:17) = ( id N ( Y ) ⊗ ε C )( N ( f ) ⊗ id C ) (cid:0) ξ ( X )( m ) (cid:1) = ( id N ( Y ) ⊗ ε C ) (cid:0) ξ ( Y ) M ( f )( m ) (cid:1) = α ( ξ )( Y )( M ( f )( m ))7e also have( N ⊗ C )( f ) ( β ( η )( X )( m )) = ( N ⊗ C )( f )( η ( X )( m ) ⊗ m )= N ( f ψ ) η ( X )( m ) ⊗ m ψ (by Lemma 2.4)= η ( Y ) M ( f ψ )( m ) ⊗ m ψ = η ( Y ) (cid:0)(cid:0) M ( f )( m ) (cid:1) (cid:1) ⊗ (cid:0) M ( f )( m ) (cid:1) (by (2.5))= β ( η )( Y )( M ( f )( m ))Moreover, it is easy to see that β ( η )( X ) is C -colinear for each X ∈ Ob ( D ). We now verify that α and β are inverses to each other. β (cid:0) α ( ξ ) (cid:1) ( X )( m ) = α ( ξ )( X )( m ) ⊗ m = ( id N ( X ) ⊗ ε C ) (cid:0) ξ ( X )( m ) (cid:1) ⊗ m = ( id N ( X ) ⊗ ε C ⊗ id C )( ξ ( X ) ⊗ id C ) ρ M ( X ) ( m )= ( id N ( X ) ⊗ ε C ⊗ id C ) π r N ( X ) ⊗ C (cid:0) ξ ( X )( m ) (cid:1) ( ξ ( X ) is C -colinear)= ξ ( X )( m ) (by (2.6))Further, we have α (cid:0) β ( η ) (cid:1) ( X )( m ) = η ( X )( m ) ε C ( m ) = η ( X )( m ). This proves the result.We now describe the unit µ : 1 M ( ψ ) C D −→ G F and the counit ν : F G −→ Mod - D of the adjunction inLemma 3.1: µ ( M ) : M −→ M ⊗
C µ ( M )( X )( m ) = m ⊗ m (3.1) ν ( N ) = id N ⊗ ε C : N ⊗ C −→ N ν ( N )( X )( n ⊗ c ) = ε C ( c ) n (3.2)for each M ∈ M ( ψ ) C D , N ∈
M od - D , X ∈ Ob ( D ).We recall that a functor F : A −→ B between arbitrary categories is said to be separable if the naturaltransformation η : Hom A ( − , − ) −→ Hom B ( F ( − ) , F ( − ))induced by F is a split monomorphism (see [25], [26, § Theorem 3.2. [26, Theorem 1.2]
Let F : A −→ B be a functor which has a right adjoint G : B −→ A .Let µ and ν be the unit and counit of this adjunction respectively. Then,(i) F is separable if and only if there exists υ ∈ N at ( GF, A ) such that υ ◦ µ = 1 A , the identity naturaltransformation on A .(ii) G is separable if and only if there exists ζ ∈ N at (1 B , F G ) such that ν ◦ ζ = 1 B , the identity naturaltransformation on B . Let ( D , C, ψ ) be an entwining structure. We now investigate the separability of the forgetful functor F : M ( ψ ) C D −→ M od - D . Since F has a right adjoint G , it follows from Theorem 3.2 that the functor F is separable if and only if there exists a natural transformation υ : G F −→ M ( ψ ) C D such that υ ◦ µ = 1 M ( ψ ) C D , where µ is the unit of the adjunction as explained in (3.1). Throughout Section 3, V := N at ( G F , M ( ψ ) C D ) will denote the K -space of all natural transformations from G F to 1 M ( ψ ) C D .We will shortly give another useful interpretation of V . We start by proving few preparatory resultsrequired for this.We recall from Lemma 2.4 and Lemma 2.5 that both h Y ⊗ C and C ⊗ h Y are objects in M ( ψ ) C D forevery Y ∈ Ob ( D ). We define a functor h ⊗ C : D −→ M ( ψ ) C D as( h ⊗ C )( Y ) := h Y ⊗ C (3.3)( h ⊗ C )( f )( Z )( g ⊗ c ) := f g ⊗ c (3.4)for f ∈ Hom D ( Y, X ) , g ∈ h Y ( Z ) and c ∈ C . Similarly, we may also obtain a functor h ⊗ C ⊗ C : D −→ M ( ψ ) C D . 8 emma 3.3. Let f ∈ Hom D ( Y, X ) . For any υ ∈ V and c, d ∈ C , we have (( id h X ⊗ ε C ) υ ( h X ⊗ C )) ( Y )( f ⊗ c ⊗ d ) = f ◦ (( ε C ⊗ id h Y ) υ ( C ⊗ h Y )) ( Y )( c ⊗ id Y ⊗ d ) (3.5) In particular, we have (( id h X ⊗ ε C ) υ ( h X ⊗ C )) ( X )( id X ⊗ c ⊗ d ) = (( ε C ⊗ id h X ) υ ( C ⊗ h X )) ( X )( c ⊗ id X ⊗ d ) (3.6) Proof.
A morphism f : Y −→ X in D induces morphisms h Y ⊗ C −→ h X ⊗ C and h Y ⊗ C ⊗ C −→ h X ⊗ C ⊗ C in M ( ψ ) C D as explained in (3.4). Since υ : G F −→ M ( ψ ) C D is a natural transformation, itfollows that the following diagram commutes: h Y ( Y ) ⊗ C ⊗ C υ ( h Y ⊗ C )( Y ) (cid:15) (cid:15) f / / h X ( Y ) ⊗ C ⊗ C υ ( h X ⊗ C )( Y ) (cid:15) (cid:15) h Y ( Y ) ⊗ C ( id h Y ⊗ ε C )( Y ) (cid:15) (cid:15) f / / h X ( Y ) ⊗ C ( id h X ⊗ ε C )( Y ) (cid:15) (cid:15) h Y ( Y ) f / / h X ( Y )Thus, we have f ◦ (( id h Y ⊗ ε C ) υ ( h Y ⊗ C )) ( Y )( id Y ⊗ c ⊗ d ) = (( id h X ⊗ ε C ) υ ( h X ⊗ C )) ( Y )( f ⊗ c ⊗ d ) (3.7)We now consider the morphism Ψ Y : C ⊗ h Y −→ h Y ⊗ C in M ( ψ ) C D given by Ψ Y ( X ) := ψ XY asin Lemma 2.6. Then, using the naturality of υ : G F −→ M ( ψ ) C D and (2.2) we have the followingcommutative diagram C ⊗ h Y ( Y ) ⊗ C ψ Y Y ⊗ id C −−−−−−→ h Y ( Y ) ⊗ C ⊗ C υ ( C ⊗ h Y )( Y ) y y υ ( h Y ⊗ C )( Y ) C ⊗ h Y ( Y ) ψ Y Y −−−−→ h Y ( Y ) ⊗ C ( ε C ⊗ id h Y )( Y ) y y ( id h Y ⊗ ε C )( Y ) h Y ( Y ) id h Y ( Y ) −−−−−→ h Y ( Y )Using the fact that ψ Y Y ( c ⊗ id Y ) = id Y ⊗ c , we now have(( id h Y ⊗ ε C ) υ ( h Y ⊗ C )) ( Y )( id Y ⊗ c ⊗ d ) = (( ε C ⊗ id h Y ) υ ( C ⊗ h Y )) ( Y )( c ⊗ id Y ⊗ d ) (3.8)Combining (3.7) and (3.8), we have(( id h X ⊗ ε C ) υ ( h X ⊗ C )) ( Y )( f ⊗ c ⊗ d ) = f ◦ (( ε C ⊗ id h Y ) υ ( C ⊗ h Y )) ( Y )( c ⊗ id Y ⊗ d ) (3.9)By putting Y = X and taking f = id X , the result of (3.6) is clear from (3.9). Lemma 3.4.
For any υ ∈ V and Y ∈ Ob ( D ) , we have υ ( C ⊗ C ⊗ h Y ) = id C ⊗ υ ( C ⊗ h Y ) as a morphismof D -modules.Proof. For each d ∈ C , we define η d : C ⊗ h Y −→ C ⊗ C ⊗ h Y by η d ( X )( c ⊗ g ) := d ⊗ c ⊗ g for each X ∈ Ob ( D ), g ∈ h Y ( X ) and c ∈ C . It may be easily verified that η d is a morphism of right D -modules. We now verify that η d ( X ) : C ⊗ h Y ( X ) −→ C ⊗ C ⊗ h Y ( X ) is right C -colinear. We have σ rC ⊗ C ⊗ h Y ( X ) ( η d ( X )( c ⊗ g )) = σ rC ⊗ C ⊗ h Y ( X ) ( d ⊗ c ⊗ g ) = ( d ⊗ c ) ⊗ g ψ ⊗ ( d ⊗ c ) ψ = d ⊗ c ⊗ g ψ ⊗ c ψ = ( η d ( X ) ⊗ id C )( c ⊗ g ψ ⊗ c ψ )= ( η d ( X ) ⊗ id C ) σ rC ⊗ h Y ( X ) ( c ⊗ g )9hus, η d : C ⊗ h Y −→ C ⊗ C ⊗ h Y is a morphism in M ( ψ ) C D . Therefore, using the naturality of υ , wehave the following commutative diagram: C ⊗ h Y ( X ) ⊗ C υ ( C ⊗ h Y )( X ) −−−−−−−−−→ C ⊗ h Y ( X ) η d ( X ) ⊗ id C y y η d ( X ) C ⊗ C ⊗ h Y ( X ) ⊗ C υ ( C ⊗ C ⊗ h Y )( X ) −−−−−−−−−−−→ C ⊗ C ⊗ h Y ( X )Thus, for any g ∈ Hom D ( X, Y ) and c, c ′ ∈ C , we get υ ( C ⊗ C ⊗ h Y )( X )( d ⊗ c ⊗ g ⊗ c ′ ) = ( υ ( C ⊗ C ⊗ h Y )( η d ⊗ id C )) ( X )( c ⊗ g ⊗ c ′ )= ( η d ◦ υ ( C ⊗ h Y )) ( X )( c ⊗ g ⊗ c ′ )= d ⊗ υ ( C ⊗ h Y )( X )( c ⊗ g ⊗ c ′ )= ( id C ⊗ υ ( C ⊗ h Y )) ( X )( d ⊗ c ⊗ g ⊗ c ′ ) (3.10)The result follows.We now proceed to give another interpretation of V = N at ( G F , M ( ψ ) C D ). We consider a collection θ := { θ X : C ⊗ C −→ End D ( X ) } X ∈ Ob ( D ) of K -linear maps satisfying the following conditions:( θ X ( c ⊗ d )) ◦ f = f ψψ ◦ θ Y (cid:0) c ψ ⊗ d ψ (cid:1) (3.11) θ X ( c ⊗ d ) ⊗ d = ( θ X ( c ⊗ d )) ψ ⊗ c ψ (3.12)for any f ∈ Hom D ( Y, X ). Let V be the K -space consisting of all such θ . Proposition 3.5.
Let υ ∈ V = N at ( G F , M ( ψ ) C D ) . For each X ∈ Ob ( D ) , we define a K -linear map θ X : C ⊗ C −→ End D ( X ) c ⊗ d (cid:0) ( id h X ⊗ ε C ) υ ( h X ⊗ C ) (cid:1) ( X )( id X ⊗ c ⊗ d ) Then, θ := { θ X } X ∈ Ob ( D ) is an element in V . Proof.
Since id h X ⊗ ε C : h X ⊗ C −→ h X is a morphism of right D -modules, we have (cid:0) θ X ( c ⊗ d ) (cid:1) ◦ f = ( id h X ⊗ ε C )( Y ) (cid:16) υ ( h X ⊗ C )( X )( id X ⊗ c ⊗ d ) · f (cid:17) (3.13)for f ∈ Hom D ( Y, X ) and c, d ∈ C . Since υ ( h X ⊗ C ) : h X ⊗ C ⊗ C −→ h X ⊗ C is a morphism of right D -modules, we also have( υ ( h X ⊗ C )( X )( id X ⊗ c ⊗ d )) · f = υ ( h X ⊗ C )( Y ) (cid:0) ( id X ⊗ c ⊗ d ) · f (cid:1) = υ ( h X ⊗ C )( Y ) (cid:0) ( h X ⊗ C ⊗ C )( f )( id X ⊗ c ⊗ d ) (cid:1) = υ ( h X ⊗ C )( Y ) (cid:0) ( h X ⊗ C )( f ψ )( id X ⊗ c ) ⊗ d ψ (cid:1) = υ ( h X ⊗ C )( Y ) (cid:0) h X ( f ψψ )( id X ) ⊗ c ψ ⊗ d ψ (cid:1) = υ ( h X ⊗ C )( Y )( f ψψ ⊗ c ψ ⊗ d ψ ) (3.14)The morphism f ψψ : Y −→ X in D induces morphisms h Y ⊗ C −→ h X ⊗ C and h Y ⊗ C ⊗ C −→ h X ⊗ C ⊗ C in M ( ψ ) C D . Therefore, we have( θ X ( c ⊗ d )) ◦ f = ( id h X ⊗ ε C )( Y ) (cid:16) υ ( h X ⊗ C )( X )( id X ⊗ c ⊗ d ) · f (cid:17) (by (3.13))= (( id h X ⊗ ε C ) υ ( h X ⊗ C )) ( Y )( f ψψ ⊗ c ψ ⊗ d ψ ) (by (3.14))= f ψψ ◦ (( id h Y ⊗ ε C ) υ ( h Y ⊗ C )) ( Y )( id Y ⊗ c ψ ⊗ d ψ ) (by (3.7))= f ψψ ◦ θ Y ( c ψ ⊗ d ψ )This proves (3.11). We now verify that θ satisfies (3.12). Using Lemma 2.5, we know that C ⊗ h Y and C ⊗ C ⊗ h Y belong to M ( ψ ) C D for each Y ∈ Ob ( D ). For each X ∈ Ob ( D ), it may be easily seen that10 ⊗ h Y ( X ) is also a left C -comodule with coaction given by ρ lY ( X ) := ∆ C ⊗ id h Y ( X ) . Moreover, it maybe easily verified that the following diagram commutes: C ⊗ h Y ( X ) ρ lY ( X ) −−−−→ C ⊗ C ⊗ h Y ( X ) σ rC ⊗ h Y ( X ) y y σ rC ⊗ C ⊗ h Y ( X ) C ⊗ h Y ( X ) ⊗ C ρ lY ( X ) ⊗ id −−−−−−−→ C ⊗ C ⊗ h Y ( X ) ⊗ C This shows that ρ lY ( X ) is a morphism of right C -comodules. Further, for any g ∈ Hom D ( X, X ′ ), wehave the following commutative diagram: C ⊗ h Y ( X ′ ) ρ lY ( X ′ ) −−−−−→ C ⊗ C ⊗ h Y ( X ′ ) ( C ⊗ h Y )( g ) y y ( C ⊗ C ⊗ h Y )( g ) C ⊗ h Y ( X ) ρ lY ( X ) −−−−→ C ⊗ C ⊗ h Y ( X )Thus, ρ lY : C ⊗ h Y −→ C ⊗ C ⊗ h Y is a morphism of right D -modules. This shows that ρ lY is a morphismin the category M ( ψ ) C D . Therefore, using the naturality of υ and Lemma 3.4, we have the followingcommutative diagram: C ⊗ h Y ( X ) ⊗ C υ ( C ⊗ h Y )( X ) −−−−−−−−−→ C ⊗ h Y ( X ) ρ lY ( X ) ⊗ id C y y ρ lY ( X ) C ⊗ C ⊗ h Y ( X ) ⊗ C υ ( C ⊗ C ⊗ h Y )( X )= id C ⊗ υ ( C ⊗ h Y )( X ) −−−−−−−−−−−−−−−−−−−−−−−−→ C ⊗ C ⊗ h Y ( X )For any c ⊗ id X ⊗ d ∈ C ⊗ h X ( X ) ⊗ C , we set a i ⊗ f i := υ (cid:0) C ⊗ h X (cid:1) ( X )( c ⊗ id X ⊗ d ). Then, we have ρ lX ( X ) ( a i ⊗ f i ) = a i ⊗ a i ⊗ f i = ( id C ⊗ υ ( C ⊗ h X )( X )) ( c ⊗ c ⊗ id X ⊗ d )Now applying the map id C ⊗ ε C ⊗ id h X to both sides, we get a i ⊗ f i = c ⊗ (cid:0) ( ε C ⊗ id h X ) υ ( C ⊗ h X ) (cid:1) ( X )( c ⊗ id X ⊗ d )= c ⊗ (cid:0) ( id h X ⊗ ε C ) υ ( h X ⊗ C ) (cid:1) ( X )( id X ⊗ c ⊗ d ) (by Lemma 3 . c ⊗ θ X ( c ⊗ d )Therefore, we have ψ (cid:0) a i ⊗ f i (cid:1) = (cid:0) θ X ( c ⊗ d ) (cid:1) ψ ⊗ c ψ (3.15)Since υ ( C ⊗ h Y )( X ) is a morphism of right C -comodules, we also have the following commutativediagram: C ⊗ h Y ( X ) ⊗ C υ ( C ⊗ h Y )( X ) −−−−−−−−−→ C ⊗ h Y ( X ) π rC ⊗ h Y ( X ) ⊗ C y y σ rC ⊗ h Y ( X ) C ⊗ h Y ( X ) ⊗ C ⊗ C υ ( C ⊗ h Y )( X ) ⊗ id C −−−−−−−−−−−−→ C ⊗ h Y ( X ) ⊗ C Thus, we have σ rC ⊗ h X ( X ) (cid:0) a i ⊗ f i (cid:1) = a i ⊗ f iψ ⊗ a i ψ = (cid:0) υ ( C ⊗ h X )( X ) ⊗ id C (cid:1)(cid:0) c ⊗ id X ⊗ d ⊗ d (cid:1) Now, applying the map ε C ⊗ id h X ⊗ id C to both sides, we get ε C ( a i )( f iψ ⊗ a iψ ) = (cid:0) ( ε C ⊗ id h X ) υ ( C ⊗ h X ) (cid:1) ( X ) ( c ⊗ id X ⊗ d ) ⊗ d = (cid:0) ( id h X ⊗ ε C ) υ ( h X ⊗ C ) (cid:1) ( X ) ( id X ⊗ c ⊗ d ) ⊗ d (by Lemma 3 . ψ (cid:0) a i ⊗ f i (cid:1) = θ X ( c ⊗ d ) ⊗ d (3.16)It now follows from (3.15) and (3.16) that θ satisfies (3.12).11 roposition 3.6. Let θ ∈ V . Then, we have an element υ ∈ N at ( G F , M ( ψ ) C D ) defined by υ ( M ) : M ⊗ C −→ M , m ⊗ c
7→ M (cid:0) θ X ( m ⊗ c ) (cid:1) ( m ) for M ∈ Ob (cid:0) M ( ψ ) C D (cid:1) , X ∈ Ob ( D ) , m ∈ M ( X ) and c ∈ C .Proof. We need to verify that υ ( M ) : M ⊗ C −→ M is a morphism in M ( ψ ) C D and that υ is indeeda natural transformation. We first verify that υ ( M ) is a morphism of right D -modules. Let f ∈ Hom D ( Y, X ). Then, we have M ( f ) (cid:0) υ ( M )( X ) (cid:1) ( m ⊗ c ) = M ( f ) M (cid:0) θ X ( m ⊗ c ) (cid:1) ( m )= M (cid:0)(cid:0) θ X ( m ⊗ c ) (cid:1) ◦ f (cid:1) ( m )= M (cid:0) f ψψ ◦ θ Y ( m ψ ⊗ c ψ ) (cid:1) ( m ) (by (3.11))= M (cid:0) θ Y ( m ψ ⊗ c ψ ) (cid:1) M ( f ψψ )( m )= M (cid:16) θ Y (cid:0) ( M ( f ψ )( m )) ⊗ c ψ (cid:1)(cid:17)(cid:0) M ( f ψ )( m ) (cid:1) (by (2.5))= υ ( M )( Y ) (cid:0) M ( f ψ )( m ) ⊗ c ψ (cid:1) = υ ( M )( Y )( M ⊗ C )( f )( m ⊗ c )We now verify that υ ( M )( X ) : M ( X ) ⊗ C −→ M ( X ) is a morphism of right C -comodules for every X ∈ Ob ( D ). For each m ⊗ c ∈ M ( X ) ⊗ C , we have (cid:0) υ ( M )( X ) ⊗ id C (cid:1) π r ( m ⊗ c ) = υ ( M )( X )( m ⊗ c ) ⊗ c = M (cid:0) θ X ( m ⊗ c ) (cid:1) ( m ) ⊗ c = M (cid:16) ( θ X (( m ) ⊗ c )) ψ (cid:17) ( m ) ⊗ ( m ) ψ (by (3.12))= M (cid:16) ( θ X ( m ⊗ c )) ψ (cid:17) ( m ) ⊗ ( m ) ψ = ρ M ( X ) ( M ( θ X ( m ⊗ c )) ( m )) (by (2.5))= ρ M ( X ) (cid:0) υ ( M )( X )( m ⊗ c ) (cid:1) It remains to show that υ : G F −→ M ( ψ ) C D is a natural transformation. Let η : M −→ N be amorphism in M ( ψ ) C D . Then, for every X ∈ Ob ( D ) and m ⊗ c ∈ M ( X ) ⊗ C , we have (cid:0) υ ( N )( η ⊗ id C ) (cid:1) ( X )( m ⊗ c ) = υ ( N )( X ) (cid:0) η ( X )( m ) ⊗ c (cid:1) = N ( θ X (( η ( X )( m )) ⊗ c )) ( η ( X )( m )) = N ( θ X ( m ⊗ c )) η ( X )( m ) (since η ( X ) is C -colinear)= η ( X ) M ( θ X ( m ⊗ c )) ( m )= η ( X ) υ ( M )( X )( m ⊗ c )This proves the result. Proposition 3.7.
The K -spaces V = N at ( G F , M ( ψ ) C D ) and V are isomorphic.Proof. We define α : V −→ V by setting α ( υ ) = θ , where θ is the collection of K -linear maps { θ X : C ⊗ C −→ End D ( X ) } X ∈ Ob ( D ) defined by θ X ( c ⊗ d ) := (( id h X ⊗ ε C ) υ ( h X ⊗ C )) ( X )( id X ⊗ c ⊗ d )for c, d ∈ C . Then, α is a well-defined map by Proposition 3.5. We also define β : V −→ V by setting β ( θ ) = υ , where υ : G F −→ M ( ψ ) C D is defined by υ ( M ) : M ⊗ C −→ M , m ⊗ c
7→ M (cid:0) θ X ( m ⊗ c ) (cid:1) ( m ) (3.17)for M ∈ Ob (cid:0) M ( ψ ) C D (cid:1) , X ∈ Ob ( D ) , m ⊗ c ∈ M ( X ) ⊗ C . By Proposition 3.6, β is well-defined. We willnow verify that α and β are inverses of each other. Let θ ∈ V . Then, for any X, Y ∈ Ob ( D ), we have( αβ ( θ )) X ( c ⊗ d ) = ( id h X ⊗ ε C )( X ) ( β ( θ )( h X ⊗ C )( X )( id X ⊗ c ⊗ d ))= ( id h X ⊗ ε C )( X )( h X ⊗ C ) ( θ X (( id X ⊗ c ) ⊗ d )) ( id X ⊗ c ) = ( id h X ⊗ ε C )( X )( h X ⊗ C ) ( θ X ( c ⊗ d )) ( id X ⊗ c )= ( id h X ⊗ ε C )( X ) (cid:16) h X (cid:16) ( θ X ( c ⊗ d )) ψ (cid:17) ( id X ) ⊗ c ψ (cid:17) (by Lemma 2.4)= ( id h X ⊗ ε C )( X ) ( h X ( θ X ( c ⊗ d )) ( id X ) ⊗ d ) (by (3.12))= ( θ X ( c ⊗ d )) ε C ( d ) = θ X ( c ⊗ d )12his proves that (cid:0) αβ ( θ ) (cid:1) X = θ X for all X ∈ Ob ( D ). Therefore, ( αβ )( θ ) = θ . For any υ ∈ V , we nowverify that ( βα )( υ ) = υ . We set θ = α ( υ ). Then, by definition we have( βα )( υ )( M )( X )( m ⊗ c ) = (( β ( θ )) ( M )) ( X )( m ⊗ c )= M (cid:0) θ X ( m ⊗ c ) (cid:1) ( m )= M (cid:0) ( id h X ⊗ ε C )( X ) υ ( h X ⊗ C )( X )( id X ⊗ m ⊗ c ) (cid:1) ( m ) (3.18)For any m ′ ∈ M ( X ), it may be easily verified that η m ′ : h X −→ M defined by η m ′ ( Y )( f ) := M ( f )( m ′ )for each f ∈ Hom D ( Y, X ) is a morphism in
M od - D . By Lemma 2.4, this induces the morphism η m ′ ⊗ id C : h X ⊗ C −→ M ⊗ C in M ( ψ ) C D defined by ( η m ′ ⊗ id C )( Y )( f ⊗ c ) := M ( f )( m ′ ) ⊗ c for f ∈ Hom D ( Y, X )and c ∈ C . Since υ is a natural transformation, it follows easily that the following diagram commutes h X ( X ) ⊗ C ⊗ C η m ′ ( X ) ⊗ id C ⊗ id C −−−−−−−−−−−−→ M ( X ) ⊗ C ⊗ C υ ( h X ⊗ C )( X ) y y υ ( M⊗ C )( X ) h X ( X ) ⊗ C η m ′ ( X ) ⊗ id C −−−−−−−−→ M ( X ) ⊗ C ( id h X ⊗ ε C )( X ) y y ( id M ⊗ ε C )( X ) h X ( X ) η m ′ ( X ) −−−−−→ M ( X )In particular, we have M ((( id h X ⊗ ε C ) υ ( h X ⊗ C )) ( X )( id X ⊗ m ⊗ c )) ( m ) = (( id M ⊗ ε C ) υ ( M ⊗ C )) ( X )( m ⊗ m ⊗ c ) (3.19)The comodule structure on entwined modules determines a morphism in M ( ψ ) C D as follows. We define˜ ρ : M −→ M ⊗ C given by ˜ ρ ( X ) := ρ M ( X ) : M ( X ) −→ M ( X ) ⊗ C for any M ∈ Ob ( M ( ψ ) C D ) and X ∈ Ob ( D ). We first verify that ˜ ρ is a morphism of right D -modules. Forany f ∈ Hom D ( Y, X ) and m ∈ M ( X ), we have( M ⊗ C )( f ) (˜ ρ ( X )( m )) = ( M ⊗ C )( f )( m ⊗ m ) = M ( f ψ )( m ) ⊗ m ψ = ρ M ( Y ) ( M ( f )( m )) = ˜ ρ ( Y )( M ( f )( m )) (by (2.5))It may be verified easily that ˜ ρ ( X ) : M ( X ) −→ M ( X ) ⊗ C is right C -colinear. Thus, ˜ ρ : M −→ M ⊗ C is a morphism in M ( ψ ) C D . Therefore, we have the following commutative diagram M ( X ) ⊗ C υ ( M )( X ) −−−−−−→ M ( X ) ˜ ρ ( X ) ⊗ id C y y ˜ ρ ( X ) M ( X ) ⊗ C ⊗ C υ ( M⊗ C )( X ) −−−−−−−−→ M ( X ) ⊗ C Thus, we get υ ( M ⊗ C )( X ) ((˜ ρ ( X ) ⊗ id C )( m ⊗ c )) = υ ( M ⊗ C )( X )( m ⊗ m ⊗ c )= ˜ ρ ( X ) ( υ ( M )( X )( m ⊗ c )) = ρ M ( X ) ( υ ( M )( X )( m ⊗ c ))Now applying id M ( X ) ⊗ ε C on both sides, we obtain( βα )( υ )( M )( X )( m ⊗ c ) = ( id M ⊗ ε C )( X ) υ ( M ⊗ C )( X )( m ⊗ m ⊗ c ) = υ ( M )( X )( m ⊗ c ) Theorem 3.8.
Let F : M ( ψ ) C D −→ M od - D be the forgetful functor and G : M od - D −→ M ( ψ ) C D , N 7→ N ⊗ C be its right adjoint. Then, F is separable if and only if there exists θ ∈ V such that θ X ◦ ∆ C = ε C · id X ∀ X ∈ Ob ( D )13 roof. We first recall from (3.1) that the unit of the adjunction is given by µ ( M ) : M −→ M ⊗
C µ ( M )( X )( m ) = m ⊗ m for M ∈ Ob ( M ( ψ ) C D ) and m ∈ M ( X ). Suppose that F is separable. Then, by Theorem 3.2, thereexists υ ∈ V such that υ ◦ µ = 1 M ( ψ ) C D . Therefore, using Proposition 3.7, corresponding to υ ∈ V wecan obtain an element θ ∈ V given by θ X ( c ⊗ d ) = (cid:0) ( id h X ⊗ ε C ) υ ( h X ⊗ C ) (cid:1) ( X )( id X ⊗ c ⊗ d ) for each c, d ∈ C . Moreover, we have( θ X ◦ ∆ C )( c ) = (cid:0) ( id h X ⊗ ε C ) υ ( h X ⊗ C ) (cid:1) ( X )( id X ⊗ c ⊗ c )= (cid:0) ( id h X ⊗ ε C ) υ ( h X ⊗ C ) (cid:1) ( X ) (cid:0) µ ( h X ⊗ C )( X ) (cid:1) ( id X ⊗ c )= (cid:0) ( id h X ⊗ ε C )( id h X ⊗ C ) (cid:1) ( X )( id X ⊗ c ) = (cid:0) id h X ( X ) ⊗ ε C (cid:1) ( id X ⊗ c ) = ε C ( c ) id X for any c ∈ C . Conversely, suppose that θ ∈ V is such that θ X ◦ ∆ C = ε C · id X for every X ∈ Ob ( D ).Corresponding to θ ∈ V there exists υ ∈ V defined by υ ( M ) : M ⊗ C −→ M , m ⊗ c
7→ M (cid:0) θ X ( m ⊗ c ) (cid:1) ( m )for M ∈ Ob (cid:0) M ( ψ ) C D (cid:1) , X ∈ Ob ( D ) , m ∈ M ( X ) and c ∈ C . Further, we have( υ ◦ µ )( M )( X )( m ) = υ ( M )( X )( µ ( M )( X )( m )) = υ ( M )( X )( m ⊗ m )= M ( θ X (( m ) ⊗ m )) ( m ) = M ( θ X (( m ) ⊗ ( m ) )) ( m )= M (( θ X ◦ ∆ C )( m )) ( m )= M (( id X ) ε C ( m )) ( m ) = m This shows that υ ◦ µ = 1 M ( ψ ) C D . Hence, F is separable by Theorem 3.2.Next we investigate the separability of the functor G : M od - D −→ M ( ψ ) C D given by G ( N ) = N ⊗ C forany N ∈
M od - D . Since G is a right adjoint of F , it follows from Theorem 3.2 that the functor G isseparable if and only if there exists a natural transformation ω : 1 Mod - D −→ F G such that ν ◦ ω = 1 Mod - D ,where ν is the counit of the adjunction as explained in (3.2). We set W := N at (1 Mod - D , F G ) and proceedto give another interpretation of W .We define h : D op ⊗ D −→ V ect K as h ( X, Y ) : =
Hom D ( X, Y ) (cid:0) h ( φ ) (cid:1) ( f ) := φ ′′ f φ ′ (3.20)for any ( X, Y ) ∈ Ob ( D op ⊗ D ), φ := ( φ ′ , φ ′′ ) ∈ Hom D op ⊗D (cid:0) ( X, Y ) , ( X ′ , Y ′ ) (cid:1) and f ∈ Hom D ( X, Y ).Similarly, we define the functor h ⊗ C : D op ⊗ D −→ V ect K as( h ⊗ C )( X, Y ) : =
Hom D ( X, Y ) ⊗ C (cid:0) ( h ⊗ C )( φ ) (cid:1) ( f ⊗ c ) := φ ′′ f φ ′ ψ ⊗ c ψ (3.21)for any ( X, Y ) ∈ Ob ( D op ⊗ D ), φ := ( φ ′ , φ ′′ ) ∈ Hom D op ⊗D (cid:0) ( X, Y ) , ( X ′ , Y ′ ) (cid:1) , f ∈ Hom D ( X, Y ) and c ∈ C . By slight abuse of notation, we will make no distinction between functors D op ⊗ D −→ V ect K and functors D −→
M od - D . We observe that h ⊗ C : D op ⊗ D −→ V ect K corresponds to F ◦ ( h ⊗ C )when viewed as a functor from D −→
M od - D .Given a natural transformation η : h −→ h ⊗ C , it is easy to see that η ( − , Y ) : h Y = Hom D ( − , Y ) −→ h Y ⊗ C = Hom D ( − , Y ) ⊗ C is a morphism of right D -modules for each Y ∈ Ob ( D ). Similarly, for each X ∈ Ob ( D ), η ( X, − ) : X h = Hom D ( X, − ) −→ X h ⊗ C = Hom D ( X, − ) ⊗ C is a morphism of left D -modules.Throughout the rest of this section, we set W := N at ( h, h ⊗ C ), the K -space consisting of all naturaltransformations between the functors h and h ⊗ C .14 emma 3.9. Let η ∈ W . We set η ( X, X )( id X ) = P a X ⊗ c X for each X ∈ Ob ( D ) and η ( Y, Z )( g ) := P ˆ g ⊗ c g for any g ∈ Hom D ( Y, Z ) . Then, η ( Y, Z )( g ) = X ˆ g ⊗ c g = X a Z g ψ ⊗ c Zψ = X ga Y ⊗ c Y Proof.
Since η ( − , Z ) : h Z −→ h Z ⊗ C is a morphism of right D -modules for each Z ∈ Ob ( D ), we havethe following commutative diagram: h Z ( Z ) η ( Z,Z ) −−−−→ h Z ( Z ) ⊗ C h Z ( g ) y y ( h Z ⊗ C )( g ) h Z ( Y ) η ( Y,Z ) −−−−→ h Z ( Y ) ⊗ C This diagram alongwith Lemma 2.4 gives η ( Y, Z )( g ) = X (cid:0) ( h Z ⊗ C )( g ) (cid:1) ( a Z ⊗ c Z ) = X h Z ( g ψ )( a Z ) ⊗ c Zψ = X a Z g ψ ⊗ c Z ψ (3.22)Since η ( Y, − ) : Y h −→ Y h ⊗ C is a morphism of left D -modules, we also have the following commutativediagram: Y h ( Y ) η ( Y,Y ) −−−−→ Y h ( Y ) ⊗ C Y h ( g ) y y ( Y h ⊗ C )( g ) Y h ( Z ) η ( Y,Z ) −−−−→ Y h ( Z ) ⊗ C This gives η ( Y, Z )( g ) = ( Y h ⊗ C )( g ) (cid:16)X a Y ⊗ c Y (cid:17) = X ga Y ⊗ c Y (3.23)The result now follows from (3.22) and (3.23). Proposition 3.10.
The K -spaces W = N at (1 Mod - D , F G ) and W = N at ( h, h ⊗ C ) are isomorphic.Proof. We define a K -linear map γ : W −→ W by setting η = γ ( ω ) : h −→ h ⊗ C η ( X, Y ) := ω ( h Y )( X )for any ( X, Y ) ∈ Ob ( D op ⊗ D ). We now verify that the map is well-defined. Let φ := ( φ ′ , φ ′′ ) ∈ Hom D op ⊗D (cid:0) ( X, Y ) , ( X ′ , Y ′ ) (cid:1) . Since ω ( h Y ) : h Y −→ h Y ⊗ C is a morphism of right D -modules, we havethe following commutative diagram: h Y ( X ) ω ( h Y )( X ) −−−−−−→ h Y ( X ) ⊗ C h Y ( φ ′ ) y y ( h Y ⊗ C )( φ ′ ) h Y ( X ′ ) ω ( h Y )( X ′ ) −−−−−−−→ h Y ( X ′ ) ⊗ C (3.24)The morphism φ ′′ : Y −→ Y ′ in D induces a morphism φ ′′ : h Y −→ h Y ′ of right D -modules. Therefore,using the naturality of ω , we get the following commutative diagram: h Y ( X ′ ) ω ( h Y )( X ′ ) −−−−−−−→ h Y ( X ′ ) ⊗ C X ′ h ( φ ′′ )= h φ ′′ ( X ′ ) y y ( X ′ h ( φ ′′ ) ⊗ id C )=( h φ ′′ ( X ′ ) ⊗ id C ) h Y ′ ( X ′ ) ω ( h Y ′ )( X ′ ) −−−−−−−→ h Y ′ ( X ′ ) ⊗ C (3.25)We now observe that for f ∈ Hom D ( X, Y ), we have (cid:0) h ( φ ) (cid:1) ( f ) = φ ′′ f φ ′ = X ′ h ( φ ′′ ) (cid:0) h Y ( φ ′ )( f ) (cid:1)(cid:0) ( h ⊗ C )( φ ) (cid:1) ( g ⊗ c ) = φ ′′ gφ ′ ψ ⊗ c ψ = ( X ′ h ( φ ′′ ) ⊗ id C ) (cid:0) ( h Y ⊗ C )( φ ′ )( g ⊗ c ) (cid:1) h ( X, Y ) η ( X,Y ) −−−−−→ h ( X, Y ) ⊗ C h ( φ ) y y ( h ⊗ C )( φ ) h ( X ′ , Y ′ ) η ( X ′ ,Y ′ ) −−−−−−→ h ( X ′ , Y ′ ) ⊗ C This shows that η ∈ W .Conversely, let η ∈ W = N at ( h, h ⊗ C ). For any Y ∈ Ob ( D ), η ( − , Y ) : h Y −→ h Y ⊗ C (3.26)is a morphism of right D -modules. For any f ∈ Hom D ( X, Y ), the naturality of η gives us the followingcommutative diagram: h X = Hom D ( − , X ) η ( − ,X ) −−−−−→ Hom D ( − , X ) ⊗ C = h X ⊗ C h ( − ,f ) y y h ( − ,f ) ⊗ id C h Y = Hom D ( − , Y ) η ( − ,Y ) −−−−−→ Hom D ( − , Y ) ⊗ C = h Y ⊗ C (3.27)Now, for any M in M od - D , we know that M = colim y ∈ el ( M ) h | y | . Similarly, M ⊗ C = colim y ∈ el ( M ) ( h | y | ⊗ C )where the colimit is taken in M od - D . Thus, the morphisms as in (3.26) induce a morphism ω ( M ) : M −→ M ⊗ C of right D -modules. Moreover, for any morphism M ζ −→ N in M od - D , the commutativediagrams as in (3.27) induce the following equality:( ζ ⊗ id C ) ◦ ω ( M ) = ω ( N ) ◦ ζ Therefore, for η ∈ W we have obtained a natural transformation ω : 1 Mod - D −→ F G in W . We willdenote this K -linear map by δ : W −→ W , i.e., δ ( η ) = ω determined by ω ( h Y ) := η ( − , Y ) for each Y ∈ Ob ( D ). It may be easily verified that the morphisms γ and δ are inverses of each other. Theorem 3.11.
Let F : M ( ψ ) C D −→ M od - D be the forgetful functor and G : M od - D −→ M ( ψ ) C D , N 7→ N ⊗ C be its right adjoint. Then G is separable if and only if there exists η ∈ W = N at ( h, h ⊗ C ) such that ( id h ⊗ ε C ) η = id h (3.28) Proof.
Suppose that G is separable. Then, by Theorem 3.2, there exists ω ∈ W = N at (1 Mod - D , F G )such that ν ◦ ω = 1 Mod - D , where ν is the counit of the adjunction. Using Proposition 3.10, correspondingto ω ∈ W , there exists an element η ∈ W given by η ( X, Y ) = ω ( h Y )( X ) for every ( X, Y ) ∈ Ob ( D op ⊗D ).The condition (3.28) now follows from the definition of the counit in (3.2).Conversely, let η ∈ W be such that ( id h ⊗ ε C ) η = id h . We consider ω : 1 Mod - D −→ F G givenby ω ( h Y ) := η ( − , Y ) for each Y ∈ Ob ( D ). Then, ( id h Y ⊗ ε C ) ω ( h Y ) = ( id h Y ⊗ ε C ) η ( − , Y ) = id h Y .Since F is a left adjoint and it is clear from the definition that G preserves colimits, we obtain that( id N ⊗ ε C ) ω ( N ) = id N for any N ∈
M od - D , i.e, ( id ⊗ ε C ) ω = 1 Mod - D . Therefore, G is separable byTheorem 3.2. Let F : A −→ B be a functor which has a right adjoint G : B −→ A . Then, the pair (
F, G ) is called aFrobenius pair if G is both a right and a left adjoint of F . We recall the following characterization forFrobenius pairs (see [8, § Theorem 3.12.
Let F : A −→ B be a functor which has a right adjoint G . Then, ( F, G ) is a Frobeniuspair if and only if there exist υ ∈ N at ( GF, A ) and ω ∈ N at (1 B , F G ) such that F ( υ ( M )) ◦ ω ( F ( M )) = id F ( M ) (3.29) υ ( G ( N )) ◦ G ( ω ( N )) = id G ( N ) (3.30) for all M ∈ A and N ∈ B . emma 3.13. For any ω ∈ W = N at (1 Mod - D , F G ) , N ∈ Comod - C and Y ∈ Ob ( D ) , we have ω ( N ⊗ h Y ) = id N ⊗ ω ( h Y ) .Proof. For each n ∈ N , we define ζ n : h Y −→ N ⊗ h Y by ζ n ( X )( f ) := n ⊗ f for any X ∈ Ob ( D ) and f ∈ Hom D ( X, Y ). It may be easily verified that ζ n is a morphism of right D -modules. Therefore, using the naturality of ω , we have the following commutative diagram: h Y ( X ) ω ( h Y )( X ) −−−−−−→ h Y ( X ) ⊗ C ζ n ( X ) y y ζ n ( X ) ⊗ id C N ⊗ h Y ( X ) ω ( N ⊗ h Y )( X ) −−−−−−−−−→ N ⊗ h Y ( X ) ⊗ C Let f ∈ Hom D ( X, Y ). We set ω ( h Y )( X )( f ) = P ˆ f ⊗ c f . Then, we have ω ( N ⊗ h Y )( X )( n ⊗ f ) = ( ζ n ( X ) ⊗ id C ) ω ( h Y )( X )( f ) = P ( ζ n ( X ) ⊗ id C )( ˆ f ⊗ c f )= P n ⊗ ˆ f ⊗ c f = ( id N ⊗ ω ( h Y )) ( X )( n ⊗ f )The result follows. Theorem 3.14.
Let F : M ( ψ ) C D −→ M od - D be the forgetful functor and G : M od - D −→ M ( ψ ) C D , N 7→ N ⊗ C be its right adjoint. Then, ( F , G ) is a Frobenius pair if and only if there exist θ ∈ V and η ∈ W such that the following conditions hold: ε C ( d ) f = X ˆ f ◦ θ X ( c f ⊗ d ) (3.31) ε C ( d ) f = X ˆ f ψ ◦ θ X ( d ψ ⊗ c f ) (3.32) for any f ∈ Hom D ( X, Y ) , d ∈ C and η ( X, Y )( f ) = P ˆ f ⊗ c f .Proof. Suppose there exist θ ∈ V and η ∈ W such that (3.31) and (3.32) hold. Then, using theisomorphisms V ∼ = V and W ∼ = W as in Propositions 3.7 and 3.10, there exist υ ∈ V and ω ∈ W corresponding to θ ∈ V and η ∈ W respectively. We also know by Proposition 2.9 that the collection { N ⊗ h Y } , where N ranges over all (isomorphisms classes of) finite dimensional C -comodules and Y rangesover all objects in D , forms a generating set for M ( ψ ) C D . Therefore, we first verify the condition (3.29)for M = N ⊗ h Y ∈ M ( ψ ) C D , where N ∈ Comod - C and Y ∈ Ob ( D ). For any n ⊗ f ∈ N ⊗ Hom D ( X, Y ),we have (cid:0) F ( υ ( N ⊗ h Y )) ◦ ω ( F ( N ⊗ h Y )) (cid:1) ( X )( n ⊗ f )= υ ( N ⊗ h Y )( X ) (cid:0) id N ⊗ ω ( h Y ) (cid:1) ( X )( n ⊗ f ) (by Lemma 3 . υ ( N ⊗ h Y )( X ) ( id N ⊗ η ( X, Y )) ( n ⊗ f )= P υ ( N ⊗ h Y )( X )( n ⊗ ˆ f ⊗ c f )= P ( N ⊗ h Y ) (cid:16) θ X (( n ⊗ ˆ f ) ⊗ c f ) (cid:17) ( n ⊗ ˆ f ) (by (3.17))= P ( N ⊗ h Y )( θ X ( n ψ ⊗ c f ))( n ⊗ ˆ f ψ ) (by (2.9))= P n ⊗ ˆ f ψ ◦ θ X (cid:0) n ψ ⊗ c f (cid:1) (by (2.8))= n ⊗ ε C ( n ) f (by (3.32))= n ⊗ f (3.33)This proves (3.29) for the generators of M ( ψ ) C D . As explained in the proof of Proposition 2.9, for any M in M ( ψ ) C D , there is an epimorphism M m ∈ el ( M ) η m : M m ∈ el ( M ) V m ⊗ h | m | −→ M M ( ψ ) C D . The morphism η := L m ∈ el ( M ) η m induces the following commutative diagram: L F ( V m ⊗ h | m | ) L F ( υ ( V m ⊗ h | m | )) ω ( F ( V m ⊗ h | m | )) −−−−−−−−−−−−−−−−−−−−−−−−→ L F ( V m ⊗ h | m | ) F ( η ) y y F ( η ) F ( M ) F ( υ ( M )) ω ( F ( M )) −−−−−−−−−−−−→ F ( M ) (3.34)From (3.33), it follows that F (cid:0) υ (cid:0) V m ⊗ h | m | (cid:1)(cid:1) ω (cid:0) F (cid:0) V m ⊗ h | m | (cid:1)(cid:1) = id F ( V m ⊗ h | m | ) for each m ∈ el ( M ).Thus, by the commutative diagram (3.34), we have (cid:0) F ( υ ( M )) ◦ ω ( F ( M )) (cid:1) ◦ F ( η ) = F ( η ) (3.35)Since F is a left adjoint, it preserves epimorphisms. Since η is an epimorphism, so is F ( η ). Therefore,(3.35) implies that F ( υ ( M )) ◦ ω ( F ( M )) = id F ( M ) . This proves (3.29) for any
M ∈ Ob ( M ( ψ ) C D ).Next, we verify the condition (3.30). From the definition, it is clear that G preserves colimits. Since any D -module may be expressed as the colimit of representable functors, it is enough to verify the condition(3.30) for representable functors. For any f ⊗ d ∈ h Y ( X ) ⊗ C , we have (cid:0) υ ( G ( h Y )) ( X ) ◦ G ( ω ( h Y )) ( X ) (cid:1) ( f ⊗ d ) = (cid:0) υ ( G ( h Y )) ( X ) ◦ ( ω ( h Y ) ⊗ id C )( X ) (cid:1) ( f ⊗ d )= (cid:0) υ ( h Y ⊗ C ) ( X ) ◦ ( η ( X, Y ) ⊗ id C ) (cid:1) ( f ⊗ d )= P υ ( h Y ⊗ C ) ( X )( ˆ f ⊗ c f ⊗ d )= P ( h Y ⊗ C ) (cid:16) θ X (cid:16) ( ˆ f ⊗ c f ) ⊗ d (cid:17)(cid:17) ( ˆ f ⊗ c f ) (by (3.17))= P ( h Y ⊗ C ) (cid:0) θ X (cid:0) c f ⊗ d (cid:1)(cid:1) ( ˆ f ⊗ c f ) (by (2.6))= P h Y (cid:16)(cid:0) θ X ( c f ⊗ d ) (cid:1) ψ (cid:17) ( ˆ f ) ⊗ c f ψ (by (2.4))= P ˆ f ◦ (cid:0) θ X ( c f ⊗ d ) (cid:1) ψ ⊗ c f ψ = P ˆ f ◦ (cid:0) θ X ( c f ⊗ d ) (cid:1) ⊗ d (by (3.12))= ε C ( d ) f ⊗ d (by (3.31))= f ⊗ d This proves (3.30). Therefore, ( F , G ) is a Frobenius pair.Conversely, suppose ( F , G ) is a Frobenius pair. Then, there exist υ ∈ V and ω ∈ W satisfying (3.29)and (3.30). Then, using the isomorphisms V ∼ = V and W ∼ = W as in Propositions 3.7 and 3.10, thereexist θ ∈ V and η ∈ W corrresponding to υ ∈ V and ω ∈ W respectively. We will now verify theconditions (3.31) and (3.32). Taking M = C ⊗ h Y in (3.29), for any d ∈ C and f ∈ Hom D ( X, Y ) wehave d ⊗ f = (cid:0) υ ( C ⊗ h Y ) ( X ) ◦ ω ( C ⊗ h Y ) ( X ) (cid:1) ( d ⊗ f )= (cid:0) υ ( C ⊗ h Y ) ( X ) ◦ ( id C ⊗ ω h Y ( X )) (cid:1) ( d ⊗ f ) (by Lemma 3 . υ ( C ⊗ h Y ) ( X ) ( d ⊗ η ( X, Y )( f ))= P υ ( C ⊗ h Y ) ( X ) (cid:16) d ⊗ ˆ f ⊗ c f (cid:17) = P ( C ⊗ h Y ) (cid:16) θ X (cid:16) ( d ⊗ ˆ f ) ⊗ c f (cid:17)(cid:17) ( d ⊗ ˆ f ) (by (3.17))= P ( C ⊗ h Y ) (cid:16) θ X (cid:16) d ψ ⊗ c f (cid:17)(cid:17) ( d ⊗ ˆ f ψ ) (by (2.9))= P d ⊗ ˆ f ψ ◦ θ X ( d ψ ⊗ c f ) (by (2.8))Applying ε C ⊗ id h Y ( X ) on both sides, we get ε C ( d ) f = P ε C ( d ) ˆ f ψ ◦ θ X ( d ψ ⊗ c f )= P ε C ( d ψ ) ˆ f ψ ψ ◦ θ X ( d ψ ⊗ c f ) (by (2.2))= P ε C (( d ψ ) ) ˆ f ψ ◦ θ X (( d ψ ) ⊗ c f ) (by (2.3))= P ˆ f ψ ◦ θ X ( d ψ ⊗ c f ))18his proves (3.32). Now, taking N = h Y in (3.30), we have f ⊗ d = (cid:0) υ ( G ( h Y )) ( X ) ◦ G ( ω ( h Y )) ( X ) (cid:1) ( f ⊗ d )= ( υ ( h Y ⊗ C ) ( X ) ◦ ( ω ( h Y ) ⊗ id C ) ( X )) ( f ⊗ d )= ( υ ( h Y ⊗ C )( X ) ◦ ( η ( X, Y ) ⊗ id C )) ( f ⊗ d )= P υ ( h Y ⊗ C ) ( X )( ˆ f ⊗ c f ⊗ d )= P ( h Y ⊗ C ) (cid:16) θ X (cid:16) ( ˆ f ⊗ c f ) ⊗ d (cid:17)(cid:17) ( ˆ f ⊗ c f ) (by (3.17))= P ( h Y ⊗ C ) (cid:0) θ X (cid:0) c f ⊗ d (cid:1) (cid:1) ( ˆ f ⊗ c f ) (by (2.6))= P h Y (cid:0) (cid:0) θ X ( c f ⊗ d ) (cid:1) ψ (cid:1) ( ˆ f ) ⊗ c f ψ (by (2.4))= P ˆ f ◦ (cid:0) θ X ( c f ⊗ d ) (cid:1) ψ ⊗ c f ψ = P ˆ f ◦ θ X ( c f ⊗ d ) ⊗ d (by (3.12))Applying id h Y ( X ) ⊗ ε C on both sides, we get (3.31). This proves the result. We continue with ( D , C, ψ ) being an entwining structure. For each Y ∈ Ob ( D ), we obtain an object Hom ( C, h Y ) in M od - D by setting Hom ( C, h Y )( X ) := Hom K ( C, h Y ( X )) Hom ( C, h Y )( g ) : Hom K (cid:0) C, h Y ( X ) (cid:1) −→ Hom K (cid:0) C, h Y ( X ′ ) (cid:1) given by Hom ( C, h Y )( g )( φ )( x ) = ( φ · g )( x ) := φ ( x ) g (3.36)for any X ∈ Ob ( D ), g ∈ Hom D ( X ′ , X ), φ ∈ Hom K (cid:0) C, h Y ( X ) (cid:1) and x ∈ C . Using (3.36), we now definea functor Hom ( C, h ) :
D −→
M od - D as follows: Hom ( C, h )( Y ) := Hom ( C, h Y ) (cid:0) Hom ( C, h )( f ) (cid:1) ( Z ) : (cid:0) Hom ( C, h Y ) (cid:1) ( Z ) −→ (cid:0) Hom ( C, h X ) (cid:1) ( Z ) given by (cid:0) Hom ( C, h )( f ) (cid:1) ( Z )( φ )( x ) = ( f · φ )( x ) := f ψ ◦ φ ( x ψ ) (3.37)for any f ∈ Hom D ( Y, X ), φ ∈ Hom K (cid:0) C, h Y ( Z ) (cid:1) and x ∈ C .For the rest of this section, we assume that C is finite dimensional. Then, for each Z ∈ Ob ( D ), we havean isomorphism Hom K (cid:0) C, h Y ( Z ) (cid:1) ∼ = C ∗ ⊗ h Y ( Z ) (3.38)Let { d i } ≤ i ≤ k be a basis for C and { d ∗ i } ≤ i ≤ k be its dual basis. Lemma 3.15.
Let C be a finite dimensional coalgebra. Then, we have a functor C ∗ ⊗ h : D −→
M od - D Y C ∗ ⊗ h Y (3.39) Proof.
For each Y ∈ Ob ( D ), it is clear that C ∗ ⊗ h Y ∈ M od - D . We consider f ∈ Hom D ( Y, X ) andan element c ∗ ⊗ g ∈ C ∗ ⊗ h Y ( Z ). By the isomorphism in (3.38), c ∗ ⊗ g corresponds to the element φ c ∗ ⊗ g ∈ Hom K (cid:0) C, h Y ( Z ) (cid:1) given by φ c ∗ ⊗ g ( x ) = c ∗ ( x ) g for each x ∈ C . From the action in (3.37), theelement f · φ c ∗ ⊗ g ∈ (cid:0) Hom ( C, h X ) (cid:1) ( Z ) is given by( f · φ c ∗ ⊗ g )( x ) = f ψ ◦ φ c ∗ ⊗ g ( x ψ ) = c ∗ ( x ψ )( f ψ ◦ g )Again, using the isomorphism in (3.38), the element in C ∗ ⊗ h X ( Z ) corresponding to f · φ c ∗ ⊗ g is given by P ki =1 c ∗ ( d ψi ) d ∗ i ⊗ f ψ g . It may be easily verified that ( C ∗ ⊗ h )( f ) : C ∗ ⊗ h Y −→ C ∗ ⊗ h X is a morphismof right D -modules. The result now follows.Since C is a coalgebra, its vector space dual C ∗ is an algebra with the convolution product ( c ∗ • d ∗ )( x ) := P c ∗ ( x ) d ∗ ( x ) for c ∗ , d ∗ ∈ C ∗ and x ∈ C . Let N be any left C ∗ -module. Then, we have a K -linear map ρ : N −→ Hom ( C ∗ , N ) defined by ρ ( n )( c ∗ ) := c ∗ n for n ∈ N and c ∗ ∈ C ∗ .In general, there is an embedding N ⊗ C ֒ → Hom ( C ∗ , N ) given by ( n ⊗ x )( c ∗ ) := c ∗ ( x ) n for x ∈ C . Since C is finite dimensional, this embedding is also a surjection. This gives us a K -linear map ρ : N −→ N ⊗ C N a right C -comodule (see, for instance, [17, § ρ ( n ) = P ki =1 d ∗ i n ⊗ d i . Inparticular, C ∗ becomes a right C -comodule with ρ C ∗ ( c ∗ ) = k X i =1 d ∗ i • c ∗ ⊗ d i (3.40)Considering the element ε C ∈ C ∗ , the coassociativity of the coaction ρ C ∗ may be used to verify that k X j =1 k X i =1 ( d ∗ i • d ∗ j ) ⊗ d i ⊗ d j = k X j =1 d ∗ j ⊗ ∆( d j ) (3.41) Proposition 3.16.
Let C be a finite dimensional coalgebra. Then, we have a functor: C ∗ ⊗ h : D −→ M ( ψ ) C D Y C ∗ ⊗ h Y Proof.
From (3.40), we know that C ∗ is a right C -comodule. Applying Lemma 2.5, it follows that each C ∗ ⊗ h Y is an object in M ( ψ ) C D . Accordingly, the right C -comodule structure on C ∗ ⊗ h Y ( Z ) for any Z ∈ Ob ( D ) is given by the following composition: σ rC ∗ ⊗ h Y ( Z ) : C ∗ ⊗ h Y ( Z ) ρ C ∗ ⊗ id −−−−−→ C ∗ ⊗ C ⊗ h Y ( Z ) id ⊗ ψ ZY −−−−−→ C ∗ ⊗ h Y ( Z ) ⊗ C Explicitly, we have σ rC ∗ ⊗ h Y ( Z ) ( c ∗ ⊗ g ) = k P i =1 d ∗ i • c ∗ ⊗ g ψ ⊗ d ψi for each c ∗ ⊗ g ∈ C ∗ ⊗ h Y ( Z ). We consider f ∈ Hom D ( Y, X ). By Lemma 3.15, this induces a morphism C ∗ ⊗ h Y −→ C ∗ ⊗ h X in M od - D . In orderto show that C ∗ ⊗ h : D −→ M ( ψ ) C D is a functor, it therefore suffices to show that each morphism C ∗ ⊗ h Y ( Z ) −→ C ∗ ⊗ h X ( Z ) ( c ∗ ⊗ g ) k X j =1 c ∗ ( d ψj ) d ∗ j ⊗ f ψ g (3.42)is right C -colinear. For any c ∗ ⊗ g ∈ C ∗ ⊗ h Y ( Z ), we have σ rC ∗ ⊗ h X ( Z ) ( f · ( c ∗ ⊗ g )) = P kj =1 σ rC ∗ ⊗ h X ( Z ) (cid:16) c ∗ ( d ψj ) d ∗ j ⊗ f ψ g (cid:17) = P ki =1 P kj =1 c ∗ ( d ψj ) d ∗ i • d ∗ j ⊗ ( f ψ g ) ψ ⊗ d ψi = P ki =1 P kj =1 c ∗ ( d ψj ) d ∗ i • d ∗ j ⊗ f ψψ g ψ ⊗ d ψi ψ (by (2.1))= P kj =1 c ∗ ( d j ψ ) d ∗ j ⊗ f ψψ g ψ ⊗ d j ψψ (by (3.41))= P kj =1 c ∗ ( d j ψ ) d ∗ j ⊗ f ψψ g ψ ⊗ (cid:16)P ki =1 d ∗ i ( d j ψ ) d ψi (cid:17) = P kj =1 P ki =1 d ∗ i ( d j ψ ) c ∗ ( d j ψ ) d ∗ j ⊗ f ψψ g ψ ⊗ d ψi = P kj =1 P ki =1 d ∗ i (( d j ψ ) ) c ∗ (( d j ψ ) ) d ∗ j ⊗ f ψ g ψ ⊗ d ψi (by (2.3))= P kj =1 P ki =1 ( d ∗ i • c ∗ )( d ψj ) d ∗ j ⊗ f ψ g ψ ⊗ d ψi = ( f ⊗ id C ) · (cid:0) P ki =1 ( d ∗ i • c ∗ ) ⊗ g ψ ⊗ d ψi (cid:1) = ( f ⊗ id C ) · (cid:16) σ rC ∗ ⊗ h Y ( Z ) ( c ∗ ⊗ g ) (cid:17) Since C is finite dimensional, the right C -comodule structure on C ∗ ⊗ h Y ( X ) induces a right C -comodulestructure on Hom ( C, h Y ( X )) for each X, Y ∈ Ob ( D ) which we now explain. Let φ ∈ Hom ( C, h Y ( X )).Then, φ corresponds to the element P ≤ i ≤ k d ∗ i ⊗ φ ( d i ) ∈ C ∗ ⊗ h Y ( X ). We know by Proposition 3.16that σ rC ∗ ⊗ h Y ( X ) k X i =1 d ∗ i ⊗ φ ( d i ) ! = k X j =1 k X i =1 d ∗ j • d ∗ i ⊗ ( φ ( d i )) ψ ⊗ d ψj P ki =1 d ∗ j • d ∗ i ⊗ ( φ ( d i )) ψ ⊗ d ψj ∈ C ∗ ⊗ h Y ( X ) ⊗ C corresponds to the element φ ⊗ φ ∈ Hom ( C, h Y ( X )) ⊗ C given by φ ( x ) ⊗ φ = P kj =1 P ki =1 ( d ∗ j • d ∗ i )( x )( φ ( d i )) ψ ⊗ d ψj = P kj =1 P ki =1 d ∗ j ( x ) d ∗ i ( x )( φ ( d i )) ψ ⊗ d ψj = ψ ( x ⊗ φ ( x )) (3.43)for x ∈ C . It now follows from (3.36), (3.37), (3.43) and Proposition 3.16 that we have a functor Hom ( C, h ) :
D −→ M ( ψ ) C D Y Hom K ( C, h Y ) (3.44)We also recall from (3.3) and (3.4), the functor h ⊗ C : D −→ M ( ψ ) C D , defined as follows:( h ⊗ C )( Y ) := h Y ⊗ C ( h ⊗ C )( f )( Z )( g ⊗ c ) := f g ⊗ c for f ∈ Hom D ( Y, X ) and g ⊗ c ∈ h Y ( Z ) ⊗ C . We now set V := N at ( h ⊗ C, C ∗ ⊗ h ). Proposition 3.17.
Let C be a finite dimensional coalgebra. Then, V = N at ( G F , M ( ψ ) C D ) ∼ = V ∼ = V = N at ( h ⊗ C, C ∗ ⊗ h ) Proof.
Since C is finite dimensional, we know that C ∗ ⊗ h Y ( X ) ∼ = Hom K (cid:0) C, h Y ( X ) (cid:1) for each X, Y ∈ Ob ( D ). We first define a K -linear map Υ XY : h Y ( X ) ⊗ C −→ C ∗ ⊗ h Y ( X ) given by (cid:0) Υ XY ( f ⊗ c ) (cid:1) ( d ) := f ψ ◦ θ X ( d ψ ⊗ c ) (3.45)for any f ∈ Hom D ( X, Y ) and c, d ∈ C . In other words, we haveΥ XY ( f ⊗ c ) = k X i =1 d ∗ i ⊗ (cid:0) f ψ ◦ θ X ( d ψi ⊗ c ) (cid:1) (3.46)where { d i } ≤ i ≤ k is a basis for C and { d ∗ i } ≤ i ≤ k is its dual basis.We now define α ′ : V −→ V by setting α ′ ( θ ) = Υ with Υ : h ⊗ C −→ C ∗ ⊗ h defined as follows:Υ Y : h Y ⊗ C −→ C ∗ ⊗ h Y Υ Y ( X ) := Υ XY for any X, Y ∈ Ob ( D ). We now verify that α ′ is a well-defined map. For this, we first check thatΥ Y : h Y ⊗ C −→ C ∗ ⊗ h Y is a morphism in M ( ψ ) C D for every Y ∈ Ob ( D ). For any g ∈ Hom D ( X ′ , X ),we need to show that the following diagram commutes: h Y ( X ) ⊗ C Υ Y ( X ) −−−−−→ C ∗ ⊗ h Y ( X ) ( h Y ⊗ C )( g ) y y ( C ∗ ⊗ h Y )( g ) h Y ( X ′ ) ⊗ C Υ Y ( X ′ ) −−−−−→ C ∗ ⊗ h Y ( X ′ )For any f ⊗ c ∈ h Y ( X ) ⊗ C , we have( C ∗ ⊗ h Y )( g )Υ Y ( X )( f ⊗ c ) = k P i =1 ( C ∗ ⊗ h Y )( g ) (cid:0) d ∗ i ⊗ f ψ ◦ θ X ( d ψi ⊗ c ) (cid:1) = k P i =1 d ∗ i ⊗ (cid:0) f ψ ◦ θ X ( d ψi ⊗ c ) (cid:1) ◦ g (by (2.8))= k P i =1 d ∗ i ⊗ f ψ g ψψ ◦ θ X ′ ( d ψi ψ ⊗ c ψ ) (by (3.11))= k P i =1 d ∗ i ⊗ ( f g ψ ) ψ ◦ θ X ′ ( d ψi ⊗ c ψ ) (by (2.1))= Υ Y ( X ′ )( f g ψ ⊗ c ψ ) = Υ Y ( X ′ )( h Y ⊗ C )( g )( f ⊗ c )21his shows that Υ Y is a morphism of right D -modules for every Y ∈ Ob ( D ). Next we verify thatΥ Y ( X ) : h Y ( X ) ⊗ C −→ C ∗ ⊗ h Y ( X ) is right C -colinear for every X, Y ∈ Ob ( D ). We have σ rC ∗ ⊗ h Y ( X ) (cid:0) Υ Y ( X )( f ⊗ c ) (cid:1) = k P i =1 σ rC ∗ ⊗ h Y ( X ) (cid:0) d ∗ i ⊗ f ψ ◦ θ X ( d ψi ⊗ c ) (cid:1) = k P i,j =1 d ∗ j • d ∗ i ⊗ (cid:0) f ψ ◦ θ X ( d ψi ⊗ c ) (cid:1) ψ ⊗ d ψj (by (3.40))= k P i =1 d ∗ i ⊗ (cid:16) f ψ ◦ θ X (cid:16) d i ψ ⊗ c (cid:17)(cid:17) ψ ⊗ d i ψ (by (3.41))= k P i =1 d ∗ i ⊗ f ψψ ◦ (cid:16) θ X (cid:16) d i ψ ⊗ c (cid:17)(cid:17) ψ ⊗ d i ψψ (by (2.1))= k P i =1 d ∗ i ⊗ f ψ ◦ (cid:16) θ X (cid:16) ( d ψi ) ⊗ c (cid:17)(cid:17) ψ ⊗ ( d ψi ) ψ (by (2.3))= k P i =1 d ∗ i ⊗ f ψ ◦ θ X ( d ψi ⊗ c ) ⊗ c (by (3.12))= Υ Y ( X )( f ⊗ c ) ⊗ c = (Υ Y ( X ) ⊗ id C ) (cid:16) π r h Y ( X ) ⊗ C ( f ⊗ c ) (cid:17) Finally, we verify that Υ is a natural transformation from h ⊗ C to C ∗ ⊗ h , i.e., the following diagramcommutes for any g ∈ Hom D ( Y, Y ′ ): h Y ⊗ C Υ Y −−−−→ C ∗ ⊗ h Y ( h ⊗ C )( g ) y y ( C ∗ ⊗ h )( g ) h Y ′ ⊗ C Υ Y ′ −−−−→ C ∗ ⊗ h Y ′ For any f ⊗ c ∈ h Y ( X ) ⊗ C , we have( C ∗ ⊗ h )( g )( X )Υ Y ( X )( f ⊗ c ) = k P i =1 ( C ∗ ⊗ h )( g )( X ) (cid:0) d ∗ i ⊗ f ψ ◦ θ X ( d ψi ⊗ c ) (cid:1) (by (3.46))= k P i,j =1 d ∗ i ( d ψj ) d ∗ j ⊗ g ψ f ψ θ X ( d ψi ⊗ c ) (by Lemma 3 . k P j =1 d ∗ j ⊗ g ψ f ψ ◦ θ X (cid:18) n P i =1 d ∗ i ( d j ψ ) d iψ ⊗ c (cid:19) = k P j =1 d ∗ j ⊗ g ψ f ψ ◦ θ X ( d ψj ψ ⊗ c )= k P j =1 d ∗ j ⊗ ( gf ) ψ ◦ θ X ( d ψj ⊗ c ) (by (2.1))= Υ Y ′ ( X )( gf ⊗ c ) = Υ Y ′ ( X )( h ⊗ C )( g )( X )( f ⊗ c )This proves that Υ ∈ V .For the converse, we first observe that the functors C ∗ ⊗ h and Hom ( C, h ) are isomorphic which followsfrom (3.38). We define β ′ : V −→ V by setting β ′ (Υ) = θ with θ X : C ⊗ C −→ End D ( X ) defined asfollows: θ X ( c ⊗ d ) := (cid:0) Υ XX ( id X ⊗ d ) (cid:1) ( c )for any X ∈ Ob ( D ) and c, d ∈ C . We will now verify that θ satisfies (3.11) and (3.12). For each X ∈ Ob ( D ), we know that Υ X : h X ⊗ C −→ Hom ( C, h X ) is a morphism of right D -modules. Therefore,for any f ∈ Hom D ( Y, X ), we have the following commutative diagram: h X ( X ) ⊗ C Υ X ( X ) −−−−−→ Hom K ( C, h X ( X )) ( h X ⊗ C )( f ) y y Hom ( C, h X )( f ) h X ( Y ) ⊗ C Υ X ( Y ) −−−−→ Hom K ( C, h X ( Y )) (3.47)22ince Υ : h ⊗ C −→ Hom ( C, h ) is a natural transformation, the following diagram also commutes forany f ∈ Hom D ( Y, X ): h Y ⊗ C Υ Y −−−−→ Hom ( C, h Y ) ( h ⊗ C )( f ) y y Hom ( C,h )( f ) h X ⊗ C Υ X −−−−→ Hom ( C, h X ) (3.48)Therefore, we have θ X ( c ⊗ d ) ◦ f = (cid:0) (Υ XX ( id X ⊗ d )) ( c ) (cid:1) ◦ f = (cid:0) (Υ XX ( id X ⊗ d )) · f (cid:1) ( c ) (by (3.36))= (Υ X ( Y )( h X ⊗ C )( f )( id X ⊗ d )) ( c ) (by (3.47))= (cid:0) Υ X ( Y ) h X ( f ψ )( id X ) ⊗ d ψ (cid:1) ( c )= Υ Y X (cid:0) f ψ ⊗ d ψ (cid:1) ( c )= Υ Y X ◦ (cid:0) ( h ⊗ C )( f ψ )( Y )( id Y ⊗ d ψ ) (cid:1) ( c )= (cid:0) f ψ · Υ Y Y ( id Y ⊗ d ψ ) (cid:1) ( c ) (by (3.48))= f ψψ ◦ (cid:0) Υ Y Y ( id Y ⊗ d ψ ) (cid:1) ( c ψ ) (by (3.37))= f ψψ ◦ (cid:0) θ Y ( c ψ ⊗ d ψ ) (cid:1) This proves (3.11). Further, we have (cid:0) θ X ( c ⊗ d ) (cid:1) ψ ⊗ c ψ = ψ (cid:0) c ⊗ θ X ( c ⊗ d ) (cid:1) = ψ (cid:0) c ⊗ (Υ XX ( id X ⊗ d )) ( c ) (cid:1) = (cid:0) Υ XX ( id X ⊗ d ) (cid:1) ( c ) ⊗ (Υ XX ( id X ⊗ d ) (cid:1) (by (3.43))= (cid:0) Υ XX ( id X ⊗ d ) (cid:1) ( c ) ⊗ ( id X ⊗ d ) (Υ XX is C -colinear)= (cid:0) Υ XX ( id X ⊗ d ) (cid:1) ( c ) ⊗ d (by (2.6))= θ X ( c ⊗ d ) ⊗ d This proves (3.12). It remains to show that α ′ and β ′ are inverses of each other. For every θ ∈ V and c, d ∈ C , it follows from (3.45) that (cid:0) ( β ′ ◦ α ′ )( θ ) (cid:1) X ( c ⊗ d ) = ( α ′ ( θ )) XX ( id X ⊗ d )( c ) = θ X ( c ⊗ d )Finally, for any Υ ∈ V , f ∈ Hom D ( X, Y ) and c, d ∈ C , we have (cid:0) ( α ′ ◦ β ′ )(Υ) (cid:1) XY ( f ⊗ c )( d ) = k P i =1 d ∗ i ( d ) f ψ ◦ (cid:0) ( β ′ (Υ)) X ( d ψi ⊗ c ) (cid:1) = k P i =1 d ∗ i ( d ) f ψ ◦ (cid:0) Υ XX ( id X ⊗ c )( d ψi ) (cid:1) = f ψ ◦ (cid:0) Υ XX ( id X ⊗ c )( d ψ ) (cid:1) = (cid:0) f · (Υ XX ( id X ⊗ c )) (cid:1) ( d ) (by (3.37))= (Υ XY ( f ⊗ c )) ( d ) (by (3.48))This proves the result. Proposition 3.18.
Let C be a finite dimensional coalgebra. Then, we have isomorphisms W = N at (1 Mod - D , F G ) ∼ = W = N at ( h, h ⊗ C ) ∼ = W := N at ( C ∗ ⊗ h, h ⊗ C ) Proof.
Given an η : h −→ h ⊗ C , we want to define Φ : C ∗ ⊗ h −→ h ⊗ C . For each Y ∈ Ob ( D ), we firstdefine a K -linear map Φ Y Y : C ∗ ⊗ h Y ( Y ) −→ h Y ( Y ) ⊗ C by the following composition: C ∗ ⊗ h Y ( Y ) id C ∗ ⊗ η ( Y,Y ) −−−−−−−−−→ C ∗ ⊗ h Y ( Y ) ⊗ C id C ∗⊗ h Y ( Y ) ⊗ ∆ C −−−−−−−−−−−−→ C ∗ ⊗ h Y ( Y ) ⊗ C ⊗ C τ ⊗ id C y h Y ( Y ) ⊗ C ⊗ ( C ∗ ⊗ C ) ev −−−−−→ h Y ( Y ) ⊗ C Y Y ( c ∗ ⊗ id Y ) = P a Y ⊗ c ∗ ( c Y ) c Y , where P a Y ⊗ c Y = η ( Y, Y )( id Y ) as in the notation of Lemma3.9. We observe that an element c ∗ ⊗ f ∈ C ∗ ⊗ h Y ( X ) may be written as c ∗ ⊗ f = ( C ∗ ⊗ h Y )( f )( c ∗ ⊗ id Y ).For each X ∈ Ob ( D ), we now define Φ XY : C ∗ ⊗ h Y ( X ) −→ h Y ( X ) ⊗ C as follows:Φ XY ( c ∗ ⊗ f ) := ( h Y ⊗ C )( f ) (Φ Y Y ( c ∗ ⊗ id Y )) = X a Y f ψ ⊗ c ∗ ( c Y ) c Y ψ (3.49)for any c ∗ ⊗ f ∈ C ∗ ⊗ h Y ( X ).We define γ ′ : W −→ W by setting γ ′ ( η ) = Φ with Φ : C ∗ ⊗ h −→ h ⊗ C given byΦ Y : C ∗ ⊗ h Y −→ h Y ⊗ C Φ Y ( X ) := Φ XY for every X, Y ∈ Ob ( D ). We now verify that γ ′ is a well-defined map. For this, we first check thatΦ Y : C ∗ ⊗ h Y −→ h Y ⊗ C is a morphism of right D -modules for every Y ∈ Ob ( D ), i.e., the followingdiagram commutes for any g ∈ Hom D ( X ′ , X ): C ∗ ⊗ h Y ( X ) Φ Y ( X ) −−−−→ h Y ( X ) ⊗ C ( C ∗ ⊗ h Y )( g ) y y ( h Y ⊗ C )( g ) C ∗ ⊗ h Y ( X ′ ) Φ Y ( X ′ ) −−−−−→ h Y ( X ′ ) ⊗ C We haveΦ Y ( X ′ )( C ∗ ⊗ h Y )( g )( c ∗ ⊗ f ) = Φ Y ( X ′ )( c ∗ ⊗ f g ) = P a Y ( f g ) ψ ⊗ c ∗ ( c Y ) c Y ψ = P a Y f ψ g ψ ⊗ c ∗ ( c Y ) c Y ψψ = ( h Y ⊗ C )( g ) (Φ Y ( X )( c ∗ ⊗ f ))Next we verify that Φ Y ( X ) : C ∗ ⊗ h Y ( X ) −→ h Y ( X ) ⊗ C is right C -colinear for any X, Y ∈ Ob ( D ):(Φ Y ( X ) ⊗ id C ) (cid:16) σ rC ∗ ⊗ h Y ( X ) ( c ∗ ⊗ f ) (cid:17) = P ki =1 Φ Y ( X ) (cid:0) d ∗ i • c ∗ ⊗ f ψ (cid:1) ⊗ d ψi = P ki =1 P a Y f ψψ ⊗ ( d ∗ i • c ∗ )( c Y ) c Y ψ ⊗ d ψi = P ki =1 P a Y f ψψ ⊗ d ∗ i ( c Y ) c ∗ ( c Y ) c Y ψ ⊗ d ψi = P a Y f ψψ ⊗ c ∗ ( c Y ) c Y ψ ⊗ c Y ψ = P a Y f ψψ ⊗ c ∗ ( c Y )( c Y ) ψ ⊗ ( c Y ) ψ = P a Y f ψ ⊗ c ∗ ( c Y )( c Y ψ ) ⊗ ( c Y ψ ) = π r h Y ( X ) ⊗ C (Φ Y ( X )( c ∗ ⊗ f ))It follows that Φ Y : C ∗ ⊗ h Y −→ h Y ⊗ C is a morphism in M ( ψ ) C D . To show that Φ ∈ N at ( C ∗ ⊗ h, h ⊗ C ),it remains to verify that the following diagram commutes: C ∗ ⊗ h Y Φ Y −−−−→ h Y ⊗ C ( C ∗ ⊗ h )( g ) y y ( h ⊗ C )( g ) C ∗ ⊗ h Z Φ Z −−−−→ h Z ⊗ C for any g ∈ Hom D ( Y, Z ). For any X ∈ Ob ( D ) and c ∗ ⊗ f ∈ C ∗ ⊗ h Y ( X ), we haveΦ Z ( X )( C ∗ ⊗ h )( g )( X )( c ∗ ⊗ f ) = P ki =1 Φ Z ( X ) (cid:16) c ∗ ( d ψi ) d ∗ i ⊗ g ψ f (cid:17) = P ki =1 P c ∗ ( d ψi ) a Z ( g ψ f ) ψ ⊗ d ∗ i ( c Z ) c Z ψ = P ki =1 P c ∗ ( d ψi ) a Z g ψψ f ψ ⊗ d ∗ i ( c Z ) c Z ψ ψ = P c ∗ ( c Z ψ ) a Z g ψψ f ψ ⊗ c Z ψ ψ = P c ∗ (cid:0) ( c Z ψ ) (cid:1) a Z g ψ f ψ ⊗ ( c Z ψ ) ψ = P c ∗ (cid:0) c Y (cid:1) ga Y f ψ ⊗ ( c Y ) ψ (by Lemma 3 . h ⊗ C )( g )( X )Φ Y ( X )( c ∗ ⊗ f )Conversely, we define δ ′ : W −→ W by setting δ ′ (Φ) = η with η : h −→ h ⊗ C given by η ( X, Y )( f ) := Φ Y ( X ) (cid:0) ε C ⊗ f (cid:1) (3.50)24or any ( X, Y ) ∈ Ob ( D op ⊗ D ) and f ∈ Hom D ( X, Y ). We now verify that η ∈ W . Let φ : ( X, Y ) −→ ( X ′ , Y ′ ) be a morphism in D op ⊗ D given by φ ′ : X ′ −→ X and φ ′′ : Y −→ Y ′ in D . Then, using thefact that Φ Y : C ∗ ⊗ h Y −→ h Y ⊗ C is a morphism of right D -modules, we have( h Y ⊗ C )( φ ′ ) η ( X, Y )( f ) = ( h Y ⊗ C )( φ ′ )Φ Y ( X )( ε C ⊗ f ) = Φ Y ( X ′ )( C ∗ ⊗ h Y )( φ ′ )( ε C ⊗ f )= Φ Y ( X ′ )( ε C ⊗ f φ ′ ) = η ( X ′ , Y ) ( h Y ( φ ′ )( f ))for any f ∈ Hom D ( X, Y ). This shows that the following diagram commutes: h Y ( X ) η ( X,Y ) −−−−−→ h Y ( X ) ⊗ C h Y ( φ ′ ) y y ( h Y ⊗ C )( φ ′ ) h Y ( X ′ ) η ( X ′ ,Y ) −−−−−→ h Y ( X ′ ) ⊗ C (3.51)Now using the naturality of Φ : C ∗ ⊗ h −→ h ⊗ C , we also have( X ′ h ⊗ C )( φ ′′ ) η ( X ′ , Y )( g ) = ( h ⊗ C )( φ ′′ )Φ Y ( X ′ )( ε C ⊗ g )= Φ Y ′ ( X ′ )( C ∗ ⊗ h )( φ ′′ )( ε C ⊗ g )= Φ Y ′ ( X ′ ) (cid:0) P ki =1 ε C ( d ψi ) d ∗ i ⊗ φ ′′ ψ g (cid:1) = Φ Y ′ ( X ′ ) (cid:0) P ki =1 ε C ( d i ) d ∗ i ⊗ φ ′′ g (cid:1) = Φ Y ′ ( X ′ ) (cid:0) ε C ⊗ φ ′′ g (cid:1) = η ( X ′ , Y ′ )( φ ′′ g ) = η ( X ′ , Y ′ )( X ′ h ( φ ′′ )( g ))for any g ∈ h Y ( X ′ ). Thus, we get the following commutative diagram: h Y ( X ′ ) η ( X ′ ,Y ) −−−−−→ h Y ( X ′ ) ⊗ C X ′ h ( φ ′′ ) y y ( X ′ h ⊗ C )( φ ′′ ) h Y ′ ( X ′ ) η ( X ′ ,Y ′ ) −−−−−−→ h Y ′ ( X ′ ) ⊗ C (3.52)It now follows from (3.51) and (3.52) that the following diagram commutes: h ( X, Y ) η ( X,Y ) −−−−−→ h ( X, Y ) ⊗ C h ( φ ) y y ( h ⊗ C )( φ ) h ( X ′ , Y ′ ) η ( X ′ ,Y ′ ) −−−−−−→ h ( X ′ , Y ′ ) ⊗ C This shows that η ∈ W . It remains to check that γ ′ and δ ′ are inverses of each other. First we verifythat (cid:0) ( δ ′ ◦ γ ′ )( η ) (cid:1) ( X, Y ) = η ( X, Y ) for all
X, Y ∈ Ob ( D ). For this, we set Φ = γ ′ ( η ). Then, for any f ∈ Hom D ( X, Y ), we have(( δ ′ ◦ γ ′ )( η )) ( X, Y )( f ) = Φ Y ( X )( ε C ⊗ f )= ( h Y ⊗ C )( f )Φ Y ( Y )( ε C ⊗ id Y ) (by (3.49))= P ( h Y ⊗ C )( f ) (cid:0) a Y ⊗ ε ( c Y ) c Y (cid:1) = P ( h Y ⊗ C )( f )( a Y ⊗ c Y )= P a Y f ψ ⊗ c Y ψ = η ( X, Y )( f ) (by Lemma 3 . (cid:0) ( γ ′ ◦ δ ′ )(Φ) (cid:1) Y ( X ) = Φ Y ( X ) for any X, Y ∈ Ob ( D ). Since C ∗ ⊗ h Y ( X ) and h Y ( X ) ⊗ C are right C -comodules for any X, Y ∈ Ob ( D ), they are also left C ∗ -modules. The left actionsare respectively given by d ∗ ( c ∗ ⊗ f ) := k X i =1 d ∗ ( d ψi )( d ∗ i • c ∗ ) ⊗ f ψ (3.53) d ∗ ( f ⊗ x ) := d ∗ ( x )( f ⊗ x ) (3.54)25or any d ∗ , c ∗ ∈ C ∗ , f ∈ h Y ( X ) and x ∈ C . Moreover, since Φ Y ( X ) : C ∗ ⊗ h Y ( X ) −→ h Y ( X ) ⊗ C isright C -colinear, it is also left C ∗ -linear. We now set η = δ ′ (Φ). Then, for any c ∗ ⊗ f ∈ C ∗ ⊗ h Y ( X ), wehave (( γ ′ ◦ δ ′ )(Φ)) Y ( X )( c ∗ ⊗ f )= ( γ ′ ( η )) Y ( X )( c ∗ ⊗ f )= P a Y f ψ ⊗ c ∗ ( c Y ) c Y ψ = P ( h Y ⊗ C )( f ) ( c ∗ ( c Y )( a Y ⊗ c Y ))= ( h Y ⊗ C )( f ) ( c ∗ ( P a Y ⊗ c Y )) (by (3.54))= ( h Y ⊗ C )( f ) (cid:0) c ∗ (Φ Y ( Y )( ε C ⊗ id Y )) (cid:1) = ( h Y ⊗ C )( f ) (Φ Y ( Y )( c ∗ ( ε C ⊗ id Y ))) (since Φ Y ( X ) is C ∗ -linear)= ( h Y ⊗ C )( f )Φ Y ( Y )( c ∗ ⊗ id Y ) (by (3.53))= Φ Y ( X )( C ∗ ⊗ h Y )( f )( c ∗ ⊗ id Y ) (Φ Y is a morphism of right D -modules)= Φ Y ( X )( c ∗ ⊗ f )This proves the result. Theorem 3.19.
Let ( D , C, ψ ) be an entwining structure and assume that C is a finite dimensionalcoalgebra. Let F : M ( ψ ) C D −→ M od - D be the functor forgetting the C -coaction and G : M od - D −→ M ( ψ ) C D given by N 7→ N ⊗ C be its right adjoint. Then, the following statements are equivalent:(i) ( F , G ) is a Frobenius pair.(ii) There exist η ∈ W and θ ∈ V such that the corresponding morphisms γ ′ ( η ) = Φ : C ∗ ⊗ h −→ h ⊗ C and α ′ ( θ ) = Υ : h ⊗ C −→ C ∗ ⊗ h given by Φ XY ( c ∗ ⊗ f ) = P a Y f ψ ⊗ c ∗ ( c Y ) c Y ψ Υ XY ( f ⊗ d ) = P ki =1 d ∗ i ⊗ f ψ ◦ θ X ( d ψi ⊗ d ) where f ∈ h Y ( X ) , c ∗ ∈ C ∗ and d ∈ C , are inverses of each other.(iii) C ∗ ⊗ h and h ⊗ C are isomorphic as objects of the category D M ( ψ ) C D of functors from D to M ( ψ ) C D .Proof. (i) ⇒ (ii) By assumption, there exist η ∈ W and θ ∈ V satisfying (3.31) and (3.32). Then, α ′ ( θ ) = Υ and γ ′ ( η ) = Φ are morphisms in D M ( ψ ) C D in the notation of Proposition 3.17 and Proposition3.18. Since Υ XY : h Y ( X ) ⊗ C −→ C ∗ ⊗ h Y ( X ) and Φ XY : C ∗ ⊗ h Y ( X ) −→ h Y ( X ) ⊗ C are right C -colinear, they are also left C ∗ -linear. Using this fact and (3.53), we haveΥ XY (Φ XY ( c ∗ ⊗ f )) = Υ XY (Φ XY (( C ∗ ⊗ h Y )( f )( c ∗ ⊗ id Y )))= Υ XY (( h Y ⊗ C )( f ) (Φ Y Y ( c ∗ ⊗ id Y )))= ( C ∗ ⊗ h Y )( f ) (Υ Y Y (Φ Y Y ( c ∗ ⊗ id Y )))= ( C ∗ ⊗ h Y )( f ) (Υ Y Y (Φ Y Y ( c ∗ • ε C ⊗ id Y )))= ( C ∗ ⊗ h Y )( f ) ( c ∗ · (Υ Y Y (Φ Y Y ( ε C ⊗ id Y ))))= ( C ∗ ⊗ h Y )( f ) ( c ∗ · (Υ Y Y ( η ( Y, Y )( id Y ))))= ( C ∗ ⊗ h Y )( f ) (cid:16) c ∗ · (cid:16)P ki =1 P d ∗ i ⊗ ( a Y ) ψ ◦ θ X ( d ψi ⊗ c Y ) (cid:17)(cid:17) (by (3.46))= ( C ∗ ⊗ h Y )( f ) (cid:16) c ∗ · (cid:16)P ki =1 ε C ( d i ) d ∗ i ⊗ id Y (cid:17)(cid:17) (by (3.32))= ( C ∗ ⊗ h Y )( f ) ( c ∗ • ε C ⊗ id Y ) = c ∗ ⊗ f for any c ∗ ⊗ f ∈ C ∗ ⊗ h Y ( X ). Thus, Υ ◦ Φ = id C ∗ ⊗ h .Using the naturality of Υ and Φ, we haveΦ XY (Υ XY ( f ⊗ c )) = Φ XY (Υ XY (( h ⊗ C )( f )( X )( id X ⊗ c )))= ( h ⊗ C )( f )( X ) (Φ XX (Υ XX ( id X ⊗ c )))= ( h ⊗ C )( f )( X ) (cid:16) Φ XX (cid:16)P ki =1 d ∗ i ⊗ θ X ( d i ⊗ c ) (cid:17)(cid:17) = ( h ⊗ C )( f )( X ) (cid:16)P ki =1 P a X ( θ X ( d i ⊗ c )) ψ ⊗ d ∗ i ( c X ) c X ψ (cid:17) = ( h ⊗ C )( f )( X ) P a X ( θ X ( c X ⊗ c )) ψ ⊗ c X ψ = ( h ⊗ C )( f )( X ) P a X ◦ θ X ( c X ⊗ c ) ⊗ c (by (3.12))= ( h ⊗ C )( f )( X ) ( ε C ( c ) id X ⊗ c ) (by (3.31))= f ⊗ c f ⊗ c ∈ h Y ( X ) ⊗ C . Thus, Φ ◦ Υ = id h ⊗ C . This proves ( ii ).(ii) ⇒ (iii) is obvious since both Φ and Υ are morphisms in D M ( ψ ) C D .(iii) ⇒ (i) Let Φ : C ∗ ⊗ h −→ h ⊗ C denote the isomorphism in D M ( ψ ) C D . We consider the followingmorphism of ( D op ⊗ D )-modulesΛ : h −→ C ∗ ⊗ h Λ Y ( X )( f ) := ε C ⊗ f for any f ∈ Hom D ( X, Y ). We now set η = Φ ◦ Λ ∈ W and θ = β ′ (Φ − ) ∈ V where β ′ is as inProposition 3.17. If η ( X, Y )( f ) = P ˆ f ⊗ c f , then ε C ⊗ f = Φ − XY (Φ XY ( ε C ⊗ f )) = Φ − XY ( η ( X, Y )( f ))= P Φ − XY ( ˆ f ⊗ c f )= P ( α ′ ( θ )) XY ( ˆ f ⊗ c f )= P P ki =1 d ∗ i ⊗ ˆ f ψ ◦ θ X ( d ψi ⊗ c f ) (by (3.46)) (3.55)Using the isomorphism as in (3.38) and evaluating the equality in (3.55) at d ∈ C , we get (3.32). Wealso have id X ⊗ d = Φ XX (Φ − XX ( id X ⊗ d )) = Φ XX (cid:0) ( α ′ ( θ )) XX ( id X ⊗ d ) (cid:1) = P ki =1 Φ XX ( d ∗ i ⊗ θ X ( d i ⊗ d )) (by (3.46))= P ki =1 Φ XX (( C ∗ ⊗ h X )( θ X ( d i ⊗ d ))( d ∗ i ⊗ id X ))= P ki =1 ( h X ⊗ C )( θ X ( d i ⊗ d )) (Φ XX ( d ∗ i ⊗ id X ))= P ki =1 ( h X ⊗ C )( θ X ( d i ⊗ d )) (Φ XX ( d ∗ i · ( ε C ⊗ id X ))) (by (3.53))= P ki =1 ( h X ⊗ C )( θ X ( d i ⊗ d )) ( d ∗ i · Φ XX ( ε C ⊗ id X )) (since Φ XX is C ∗ -linear)= P ki =1 ( h X ⊗ C )( θ X ( d i ⊗ d )) ( d ∗ i · ( η ( X, X )( id X )))= P ki =1 P ( h X ⊗ C )( θ X ( d i ⊗ d )) ( d ∗ i · ( a X ⊗ c X ))= P ki =1 P ( h X ⊗ C )( θ X ( d i ⊗ d )) ( d ∗ i ( c X )( a X ⊗ c X ))= P ( h X ⊗ C )( θ X ( c X ⊗ d ))( a X ⊗ c X )= P a X ◦ ( θ X ( c X ⊗ d )) ψ ⊗ c X ψ = P a X ◦ θ X ( c X ⊗ d ) ⊗ d (by (3.12))By applying the map id h X ( X ) ⊗ ε C , we obtain ε C ( d ) · id X = X a X (cid:0) θ X ( c X ⊗ d ) (cid:1) (3.56)Now using Lemma 3.9 and (3.56), we obtain P ˆ f ⊗ ( θ X ( c f ⊗ d )) = P ( id h Y ( X ) ⊗ θ X )( ˆ f ⊗ c f ⊗ d ) = ( id h Y ( X ) ⊗ θ X ) ( η ( X, Y )( f ) ⊗ d )= P ( id h Y ( X ) ⊗ θ X ) ( f a X ⊗ c X ⊗ d )for any f ∈ Hom D ( X, Y ) and d ∈ C . Applying to both sides the composition Hom D ( X, Y ) ⊗ Hom D ( X, X ) −→ Hom D ( X, Y ), we obtain P ˆ f ◦ ( θ X ( c f ⊗ d )) = P f a X ◦ θ X ( c X ⊗ d ) = ε C ( d ) f . Thisproves (3.31). Therefore, ( F , G ) is a Frobenius pair by Theorem 3.14. This completes the proof. Let D be a small K -linear category. Let ( D , C, ψ ) be a right-right entwining structure. We denote by D M D the category of D - D bimodules, i.e., the category whose objects are functors from D op ⊗ D to V ect K and whose morphisms are natural transformations between these functors. We recall the functors h and h ⊗ C in D M D from (3.20) and (3.21) respectively: h ( X, Y ) =
Hom D ( X, Y ) (cid:0) h ( φ ) (cid:1) ( f ) = φ ′′ f φ ′ (4.1)( h ⊗ C )( X, Y ) =
Hom D ( X, Y ) ⊗ C (cid:0) ( h ⊗ C )( φ ) (cid:1) ( f ⊗ c ) = φ ′′ f φ ′ ψ ⊗ c ψ (4.2)for any ( X, Y ) ∈ Ob ( D op ⊗ D ), φ := ( φ ′ , φ ′′ ) ∈ Hom D op ⊗D (cid:0) ( X, Y ) , ( X ′ , Y ′ ) (cid:1) and f ∈ Hom D ( X, Y ), c ∈ C . We refer, for instance, to [19, § D M D a monoidalcategory with h ∈ D M D as the unit object. 27 efinition 4.1. A D -coring C is a coalgebra object in the monoidal category D M D . Explicitly, a D -coring is a functor C : D op ⊗ D −→ V ect K with two morphisms ∆ C : C −→ C ⊗ D C , ε C : C −→ h satisfying the coassociativity and counit axioms in D M D . A right C -comodule consists of a right D -module M equipped with a morphism ρ M : M −→ M ⊗ D C of right D -modules satisfying ( id M ⊗ D ∆ C ) ◦ ρ M = ( ρ M ⊗ D id C ) ◦ ρ M ( id M ⊗ D ε C ) ◦ ρ M = id M (4.3) A morphism η : ( M , ρ M ) −→ ( N , ρ N ) of right C -comodules is a morphism η : M −→ N of right D -modules satisfying ρ N ◦ η = ( η ⊗ D id C ) ◦ ρ M The category of right C -comodules will be denoted by Comod - C . Lemma 4.2.
Let ( D , C, ψ ) be a right-right entwining structure. Then, the functor h ⊗ C is a D -coring.Proof. It may be verified that ( h ⊗ C ) ⊗ D ( h ⊗ C ) ∼ = h ⊗ C ⊗ C . This gives us morphisms id h ⊗ ∆ C : h ⊗ C −→ h ⊗ C ⊗ C ∼ = ( h ⊗ C ) ⊗ D ( h ⊗ C ) id h ⊗ ε C : h ⊗ C −→ h (4.4)in D M D . Using the coassociativity and counitality of the K -coalgebra C , it may be verified that id h ⊗ ∆ C and id h ⊗ ε C satisfy the coassociativity and counit axioms in the category D M D . Thus, h ⊗ C is a coalgebraobject in D M D . Proposition 4.3.
Let ( D , C, ψ ) be a right-right entwining structure. Then, the category M ( ψ ) C D ofentwined modules is identical to the category Comod - ( h ⊗ C ) .Proof. Let
M ∈ M ( ψ ) C D . It may be verified that M ⊗ C ∼ = M ⊗ D ( h ⊗ C ) as right D -modules. Then,by Lemma 2.4, M ⊗ C ∈ M ( ψ ) C D and we have ρ M ( X ) ( M ( f )( m )) = M ( f ψ )( m ) ⊗ m ψ = ( M ⊗ C )( f )( m ⊗ m )for any f ∈ Hom D ( X, Y ) and m ∈ M ( Y ). We thus obtain a morphism ρ M : M −→ M ⊗ C ∼ = M ⊗ D ( h ⊗ C ) of right D -modules given by ρ M ( X ) := ρ M ( X ) for each X ∈ Ob ( D ).Applying (4.4), we have M ⊗ D ( h ⊗ C ) id M ⊗ D ∆ ( h ⊗ C ) = id M ⊗ id h ⊗ ∆ C −−−−−−−−−−−−−−−−−−−−→ M ⊗ D ( h ⊗ C ) ⊗ D ( h ⊗ C ) ∼ = y ∼ = y M ⊗ C id M ⊗ ∆ C −−−−−−→ M ⊗ C ⊗ C (4.5)and M ⊗ D ( h ⊗ C ) id M ⊗ ε ( h ⊗ C ) = id M ⊗ id h ⊗ ε C −−−−−−−−−−−−−−−−−−→ M ⊗ D h ∼ = y ∼ = y M ⊗ C id M ⊗ ε C −−−−−−→ M (4.6)The conditions in (4.3) now follow from the fact that ρ M ( X ) is a C -coaction for each X ∈ Ob ( D ).Therefore, M is a right ( h ⊗ C )-comodule.Conversely, let N ∈
Comod -( h ⊗ C ). Then, N is a right D -module with a given morphism ρ N : N −→N ⊗ D ( h ⊗ C ) ∼ = N ⊗ C of right D -modules satisfying the conditions in (4.3). Thus, for each Y ∈ Ob ( D ),we have a morphism ρ N ( Y ) : N ( Y ) −→ N ( Y ) ⊗ C which satisfies( id N ( Y ) ⊗ ∆ C ) ◦ ρ N ( Y ) = ( ρ N ( Y ) ⊗ id C ) ◦ ρ N ( Y ) ( id N ( Y ) ⊗ ε C ) ◦ ρ N ( Y ) = id N ( Y ) (4.7)In (4.7), we have identified id N ⊗ ∆ C = id N ⊗ ∆ h ⊗ C and id N ⊗ ε C = id N ⊗ ε ( h ⊗ C ) as in (4.5) and (4.6)respectively. Therefore, ρ N ( Y ) defines a right C -comodule structure on N ( Y ) for every Y ∈ Ob ( D ).Since ρ N is a morphism of right D -modules, we also have ρ N ( X )( N ( f )( n )) = ( N ⊗ C )( f )( n ⊗ n ) = N ( f ψ )( n ) ⊗ n ψ (4.8)for any f ∈ Hom D ( X, Y ) and n ∈ N ( Y ). Therefore, N ∈ M ( ψ ) C D .28 emma 4.4. Let i : E −→ D be an inclusion of small K -linear categories. Then, the functor h ⊗ E h : E op ⊗ E −→ V ect K is a D -coring, where h is the D - D -bimodule as in (4.1) .Proof. It is immediate that the functor h ⊗ E h is a D - D -bimodule. We need to show that h ⊗ E h is acoalgebra object in D M D . We now define ∆ : h ⊗ E h −→ ( h ⊗ E h ) ⊗ D ( h ⊗ E h ) ∼ = ( h ⊗ E h ) ⊗ E h as follows:for ( X, Y ) ∈ Ob ( D op ⊗ D ), we set∆( X, Y ) : h Y ⊗ E X h −→ ( h ⊗ E h )( − , Y ) ⊗ E h ( X, − ) ∼ = h Y ⊗ E h ⊗ E X h f ⊗ f ′ f ⊗ id Z ⊗ f ′ (4.9)for any f ⊗ f ′ ∈ h Y ( Z ) ⊗ X h ( Z ) and Z ∈ Ob ( E ). It is easy to check that ∆( X, Y ) is well-defined. Also,it can be verified that for any morphism ( φ ′ , φ ′′ ) : ( X, Y ) −→ ( X ′ , Y ′ ) in D op ⊗ D , the following diagramcommutes: h Y ⊗ E X h ∆( X,Y ) −−−−−→ h Y ⊗ E h ⊗ E X h h φ ′′ ⊗ E φ ′ h y y h φ ′′ ⊗ E id h ⊗ E φ ′ h h Y ′ ⊗ E X ′ h ∆( X ′ ,Y ′ ) −−−−−−→ h Y ′ ⊗ E h ⊗ E X ′ h Thus, ∆ is a morphism of D - D -bimodules. The map ε : h ⊗ E h −→ h is defined by composition. It maybe verified that ∆ and ε satisfy the coassociativity and counit axioms respectively.Let D be a small K -linear category and let C be a K -coalgebra. We consider the category D M C ofleft-right Doi-Hopf modules (compare Example 2.3). Explicitly, an object in D M C consists of a left D -module M with a given right C -comodule structure on M ( X ) for each X ∈ Ob ( D ) such that thefollowing compatibility condition holds: (cid:0) M ( f )( m ) (cid:1) ⊗ (cid:0) M ( f )( m ) (cid:1) = M ( f )( m ) ⊗ m for each f ∈ Hom D ( X, Y ) and m ∈ M ( X ). A morphism η : M −→ N in D M C is a left D -modulemorphism such that each η ( X ) : M ( X ) −→ N ( X ) is right C -colinear. By definition, ( h ⊗ C )( X, − ) = X h ⊗ C is a left D -module for each X ∈ Ob ( D ). The map id ⊗ ∆ C : Hom D ( X, Y ) ⊗ C −→ Hom D ( X, Y ) ⊗ C ⊗ C gives a right C -comodule structure on ( X h ⊗ C )( Y ) for each Y ∈ Ob ( D ). Clearly, X h ⊗ C ∈ D M C .From this point onwards, we suppose additionally that each Hom D ( X, Y ) has a given right C -comodulestructure denoted by ρ XY : Hom D ( X, Y ) −→ Hom D ( X, Y ) ⊗ C Definition 4.5.
Let
E ⊆ D be the subcategory with Ob ( E ) = Ob ( D ) and Hom E ( X, Y ) =
Hom
CMod - D ( h X , h Y ) = { η ∈ Hom
Mod - D ( h X , h Y ) | η is objectwise C -colinear } = { g ∈ Hom D ( X, Y ) | ρ ZY ( gf ) = ( Z h ⊗ C )( g )( ρ ZX ( f )) ∀ f ∈ Hom D ( Z, X ) } We will say that E is the subcategory of C -coinvariants of D . Example 4.6.
Let H be a Hopf algebra over K and let D be a right co- H -category. In this case,the subcategory E of H -coinvariants of D is given by setting Ob ( E ) = Ob ( D ) and Hom E ( X, Y ) =
Hom D ( X, Y ) coH . It follows that the right C -comodule structures ρ XY : Hom D ( X, Y ) −→ Hom D ( X, Y ) ⊗ C induce amorphism X h −→ X h ⊗ C of left E -modules for each X ∈ Ob ( D ). Further, for every Y ∈ Ob ( D ), thisinduces a morphism( h ⊗ E X h )( Y ) = h Y ⊗ E X h −→ h Y ⊗ E X h ⊗ C = ( h ⊗ E X h )( Y ) ⊗ Cf ⊗ f ′ f ⊗ ρ XZ ( f ′ ) (4.10)where f ∈ Hom D ( Z, Y ) , f ′ ∈ Hom D ( X, Z ) and Z ∈ Ob ( E ). It may be easily verified that the coactionin (4.10) makes h ⊗ E X h an object of D M C .We obtain therefore canonical morphisms of K -vector spaces given by the following composition { can XY : h Y ⊗ E X h −→ h Y ⊗ E ( X h ⊗ C ) −→ h Y ⊗ D ( X h ⊗ C ) ∼ = Hom D ( X, Y ) ⊗ C } ( X,Y ) ∈ Ob ( D ) For each X ∈ Ob ( D ), this induces a morphism in D M C as follows can X : h ⊗ E X h −→ X h ⊗ C can X ( Y ) := can XY efinition 4.7. Let C be a K -coalgebra and D be a small K -linear category such that Hom D ( X, Y ) hasa right C -comodule structure for every X, Y ∈ Ob ( D ) . Let E be a K -linear subcategory of D . Then, D is called a C -Galois extension of E if(i) Ob ( E ) = Ob ( D ) and Hom E ( X, Y ) =
Hom
CMod - D ( h X , h Y ) .(ii) The induced canonical morphism can X : h ⊗ E X h −→ X h ⊗ C is an isomorphism in D M C for each X ∈ Ob ( D ) . Let D be a C -Galois extension of E . For each X ∈ Ob ( D ), we define τ X : C −→ h X ⊗ E X h τ X ( c ) := can − XX ( id X ⊗ c ) (4.11)We refer to these as the translation maps of the Galois extension. Lemma 4.8.
Let D be a C -Galois extension of E . Let { τ X : C −→ h X ⊗ E X h } X ∈ Ob ( D ) be the associatedtranslation maps. We use the notation τ X ( c ) = c (1) ⊗ c (2) (summation omitted). Then,(i) τ X is right C -colinear i.e., c (1) ⊗ c (2)0 ⊗ c (2)1 = ( c ) (1) ⊗ ( c ) (2) ⊗ c .(ii) For any f ∈ Hom D ( X, Y ) , we have f ( f ) (1) ⊗ ( f ) (2) = id Y ⊗ f ∈ h Y ⊗ E X h .(iii) c (1) c (2) = ε C ( c ) · id X .Proof. The C -colinearity of τ X follows from the C -colinearity of can − XX . Explicitly, for any c ∈ C , wehave c (1) ⊗ c (2)0 ⊗ c (2)1 = ( id ⊗ ρ ) τ X ( c ) = ( id ⊗ ρ ) can − XX ( id X ⊗ c )= ( can − XX ⊗ id C )( id ⊗ ∆ C )( id X ⊗ c )= ( can − XX ⊗ id C )( id X ⊗ c ⊗ c )= can − XX ( id X ⊗ c ) ⊗ c = τ X ( c ) ⊗ c = ( c ) (1) ⊗ ( c ) (2) ⊗ c This proves (i). Since can − X : X h ⊗ C −→ h ⊗ E X h is a morphism of left D -modules for each X ∈ Ob ( D ),we also have f ( f ) (1) ⊗ ( f ) (2) = ( h ⊗ E X h )( f ) ( τ X ( f ))= ( h ⊗ E X h )( f ) (cid:0) can − XX ( id X ⊗ f ) (cid:1) = can − XY (( X h ⊗ C )( f )( id X ⊗ f ))= can − XY ( f ⊗ f ) = id Y ⊗ f This proves (ii). Again using the definition of can XX and τ X , we have ( can XX ◦ τ X )( c ) = id X ⊗ c . Thus, can XX ( c (1) ⊗ c (2) ) = c (1) c (2)0 ⊗ c (2)1 = id X ⊗ c Now, by applying the map id ⊗ ε C to both sides, we get (iii). Theorem 4.9.
Let D be a C -Galois extension of E . We denote by ρ XY : Hom D ( X, Y ) −→ Hom D ( X, Y ) ⊗ C the right C -comodule structure maps. Then, there exists a unique right-right entwining structure ( D , C, ψ ) which makes h Y an object in M ( ψ ) C D for every Y ∈ Ob ( D ) with its canonical D -module struc-ture and right C -coactions { ρ XY } X ∈ Ob ( D ) .This entwining structure ( D , C, ψ ) is given by ψ XY : C ⊗ Hom D ( X, Y ) τ Y ⊗ id −−−−→ h Y ⊗ E Y h ⊗ Hom D ( X, Y ) −→ h Y ⊗ E X h can XY −−−−→ Hom D ( X, Y ) ⊗ C Proof.
Using Lemma 4.8, the proof will follow essentially in the same way as that of [6, Theorem 2.7].
Lemma 4.10.
Let D be a C -Galois extension of E . Then, h ⊗ E h ∼ = h ⊗ C as D -corings. roof. We define can : h ⊗ E h −→ h ⊗ C by setting can ( X, Y ) := can XY for each ( X, Y ) ∈ Ob ( D op ⊗ D ).We first verify that can is a morphism of D - D -bimodules. Clearly, can ( X, − ) = can X which, by definition,is a morphism of left D -modules. Therefore, it suffices to show that can ( − , Y ) is a morphism of right D -modules, i.e., the following diagram commutes for any g ∈ Hom D ( Z, Z ′ ):( h Y ⊗ E h )( Z ′ ) can ( Z ′ ,Y ) −−−−−−−→ ( h Y ⊗ C )( Z ′ ) ( h Y ⊗ E h )( g ) y y ( h Y ⊗ C )( g ) ( h Y ⊗ E h )( Z ) can ( Z,Y ) −−−−−−→ ( h Y ⊗ C )( Z )By Theorem 4.9, we know that h W is an object in M ( ψ ) C D for each W ∈ Ob ( D ). Thus, for any f ∈ h W ( Z ′ ), we have( f g ) ⊗ ( f g ) = ρ ZW ( f g ) = ρ ZW ( h W ( g )( f )) = ( h W ⊗ C )( g )( f ⊗ f ) = f g ψ ⊗ f ψ Therefore, for any f ′ ⊗ f ∈ h Y ( W ) ⊗ E Z ′ h ( W ), we obtain can ( Z, Y ) (( h Y ⊗ E h )( g )( f ′ ⊗ f )) = can ( Z, Y )( f ′ ⊗ f g ) = f ′ ◦ ( f g ) ⊗ ( f g ) = f ′ f g ψ ⊗ f ψ = ( h Y ⊗ C )( g ) ( can ( Z ′ , Y )( f ′ ⊗ f ))It remains to verify that can is also a coalgebra morphism. First, we show that the following diagramcommutes: h ⊗ E h can −−−−→ h ⊗ C ∆ h ⊗E h y y ∆ h ⊗ C ( h ⊗ E h ) ⊗ D ( h ⊗ E h ) can ⊗ D can −−−−−−−→ ( h ⊗ C ) ⊗ D ( h ⊗ C )For any ( X, Y ) ∈ Ob ( D op ⊗ D ) and w ⊗ w ′ ∈ h Y ( W ) ⊗ X h ( W ), we have∆ h ⊗ C ( X, Y ) ( can XY ( w ⊗ w ′ )) = ∆ h ⊗ C ( X, Y )( ww ′ ⊗ w ′ ) = ( ww ′ ⊗ w ′ ) ⊗ D ( id X ⊗ w ′ )= ( ww ′ ⊗ w ′ ) ⊗ D ( id X ⊗ w ′ )= ( X h ⊗ C )( w )( ρ XW ( w ′ )) ⊗ D ( id X ⊗ w ′ )= ( X h ⊗ C )( w ) ( h W ⊗ C )( w ′ )( ρ W W ( id W ))) ⊗ D ( id X ⊗ w ′ )= ( h Y ⊗ C )( w ′ ) (( W h ⊗ C )( w )( ρ W W ( id W ))) ⊗ D ( id X ⊗ w ′ )= (( W h ⊗ C )( w )( ρ W W ( id W ))) · w ′ ⊗ D ( id X ⊗ w ′ )= (( W h ⊗ C )( w )( ρ W W ( id W ))) ⊗ D ( w ′ ⊗ w ′ )= ( w ◦ id W ⊗ id W ) ⊗ D ( w ′ ⊗ w ′ )= can W Y ( w ⊗ id W ) ⊗ D can XW ( id W ⊗ w ′ )It may be verified easily that can is compatible with counits. Since can is a morphism in the categoryof D - D -bimodules and can ( X, Y ) = can XY is an isomorphism for each ( X, Y ) ∈ Ob ( D op ⊗ D ), it followsthat can is an isomorphism with inverse given by can − ( X, Y ) := can − XY . This proves the result. Definition 4.11.
Let D be a small K -linear category such that Hom D ( X, Y ) is a right C -comodule forevery X, Y ∈ Ob ( D ) . Let Φ XY : C −→ Hom D ( X, Y ) and Φ Y Z : C −→ Hom D ( Y, Z ) be two C -comodulemaps. Then, their convolution product is given by Φ Y Z ∗ Φ XY : C −→ Hom D ( X, Z ) , c Φ Y Z ( c ) ◦ Φ XY ( c ) A collection of right C -comodule maps Φ = { Φ XY : C −→ Hom D ( X, Y ) } X,Y ∈ Ob ( D ) is said to be convo-lution invertible if there exists a collection Φ ′ = { Φ ′ XY : C −→ Hom D ( X, Y ) } X,Y ∈ Ob ( D ) of C -comodulemaps such that (Φ XY ∗ Φ ′ Y X )( c ) = ε C ( c ) · id Y = (Φ ′ XY ∗ Φ Y X )( c ) for every c ∈ C . Theorem 4.12.
Let C be a K -coalgebra and D be a small K -linear category such that Hom D ( X, Y ) hasa right C -comodule structure ρ XY for every X, Y ∈ Ob ( D ) . Let E be the subcategory of C -coinvariants f D . If there exists a convolution invertible collection Φ = { Φ XY : C −→ Hom D ( X, Y ) } X,Y ∈ Ob ( D ) ofright C -comodule maps, then the following are equivalent:(i) D is a C -Galois extension of E .(ii) There exists a right-right entwining structure ( D , C, ψ ) such that h Y is an object in M ( ψ ) C D for every Y ∈ Ob ( D ) with its canonical D -module structure and right C -coactions { ρ XY } X ∈ Ob ( D ) .(iii) For any f ∈ Hom D ( X, Y ) , the morphism f ◦ Φ ′ ZX ( f ) ∈ Hom E ( Z, Y ) for every Z ∈ Ob ( D ) , where Φ ′ is the convolution inverse of Φ .Proof. By Theorem 4.9, we have ( i ) ⇒ ( ii ). To prove ( ii ) ⇒ ( iii ), we will use the equality( X h ⊗ C ) (Φ ′ XY ( c )) ( ρ XX ( id X )) = ψ XY ( c ⊗ Φ ′ XY ( c )) (4.12)for any c ∈ C . We first give a proof of this. Since h Y ∈ M ( ψ ) C D , we have ρ XY ( f ) = ρ XY ( h Y ( f )( id Y )) = h Y ( f ψ )( id Y ) ⊗ id Y ψ = id Y f ψ ⊗ id Y ψ = ( X h ⊗ C )( id Y )( ψ XY ( id Y ⊗ f )) (4.13)for any f ∈ Hom D ( X, Y ). Also, for any c ∈ C , we have( X h ⊗ C )( id X )( ψ XX ( id X ⊗ Φ Y X ( c )Φ ′ XY ( c ))) = ( X h ⊗ C )( id X )( ψ XX ( id X ⊗ ε C ( c ) id X ))= ε C ( c ) id X ⊗ id X (4.14)Now, using (4.14), we have( X h ⊗ C ) (Φ ′ XY ( c )) ( ρ XX ( id X ))= ( X h ⊗ C ) (Φ ′ XY ( c )) ( ε C ( c ) id X ⊗ id X )= ( X h ⊗ C ) (Φ ′ XY ( c )) (( X h ⊗ C )( id X )( ψ XX ( id X ⊗ Φ Y X ( c )Φ ′ XY ( c )))) (using (4.14))= ( X h ⊗ C ) (Φ ′ XY ( c )) (cid:16) ( X h ⊗ C )( id X ) (cid:16) (Φ Y X ( c )) ψ (Φ ′ XY ( c )) ψ ⊗ id X ψψ (cid:17)(cid:17) (using (2.1))= ( X h ⊗ C ) (Φ ′ XY ( c )) (cid:16) id X ◦ (Φ Y X ( c )) ψ (Φ ′ XY ( c )) ψ ⊗ id X ψψ (cid:17) = ( X h ⊗ C ) (Φ ′ XY ( c )) (cid:16) ( h X ⊗ C )(Φ ′ XY ( c ))( id X ◦ (Φ Y X ( c )) ψ ⊗ id X ψ ) (cid:17) = ( X h ⊗ C ) (Φ ′ XY ( c )) (cid:16) ( h X ⊗ C )(Φ ′ XY ( c )) (cid:16) ( Y h ⊗ C )( id X ) ( ψ Y X ( id X ⊗ Φ Y X ( c ))) (cid:17)(cid:17) = ( X h ⊗ C ) (Φ ′ XY ( c )) (( h X ⊗ C )(Φ ′ XY ( c )) ( ρ Y X (Φ Y X ( c )))) (using (4.13))= ( X h ⊗ C ) (Φ ′ XY ( c )) (( h X ⊗ C )(Φ ′ XY ( c ))(Φ Y X ( c ) ⊗ c )) (since Φ Y X is C -colinear)= Φ ′ XY ( c )Φ Y X ( c )(Φ ′ XY ( c )) ψ ⊗ c ψ = ε C ( c ) id Y (Φ ′ XY ( c )) ψ ⊗ c ψ = ( h Y ⊗ C )(Φ ′ XY ( c ))( ε C ( c ) id Y ⊗ c )= ( h Y ⊗ C )(Φ ′ XY ( c ))( id Y ⊗ c )= (Φ ′ XY ( c )) ψ ⊗ c ψ = ψ XY ( c ⊗ Φ ′ XY ( c ))This proves the equality (4.12).For any f ∈ Hom D ( X, Y ), consider the morphism f ◦ Φ ′ ZX ( f ) : Z −→ Y in D . Then f ◦ Φ ′ ZX ( f )induces a morphism of right D -modules h Z −→ h Y which we denote by ˜ f . We now verify that the map˜ f ( X ′ ) : h Z ( X ′ ) −→ h Y ( X ′ ) is right C -colinear for each X ′ ∈ Ob ( D ). Since h Y is an object in M ( ψ ) C D for every Y ∈ Ob ( D ), the following diagram commutes for any g ∈ Hom D ( X ′ , Z ): h Y ( Z ) ρ XY −−−−→ h Y ( Z ) ⊗ C h Y ( g ) y y ( h Y ⊗ C )( g ) h Y ( X ′ ) ρ X ′ Y −−−−→ h Y ( X ′ ) ⊗ C (4.15)32hus, we have ρ X ′ Y ( ˜ f ( X ′ )( g )) = ρ X ′ Y ( f ◦ Φ ′ ZX ( f ) ◦ g ) = ρ X ′ Y ( h Y (Φ ′ ZX ( f ) ◦ g )( f ))= ( h Y ⊗ C )(Φ ′ ZX ( f ) ◦ g )( ρ XY ( f )) (using (4.15))= ( h Y ⊗ C )( g )(( h Y ⊗ C )(Φ ′ ZX ( f ))( f ⊗ f ))= ( h Y ⊗ C )( g ) (cid:16) f (Φ ′ ZX ( f )) ψ ⊗ f ψ (cid:17) = ( h Y ⊗ C )( g ) (cid:0) ( Z h ⊗ C )( f ) ( ψ ZX ( f ⊗ Φ ′ ZX ( f ))) (cid:1) = ( h Y ⊗ C )( g ) (cid:0) ( Z h ⊗ C )( f )( Z h ⊗ C ) ((Φ ′ ZX )( f )) ( ρ ZZ ( id Z )) (cid:1) (using (4.12))= ( h Y ⊗ C )( g ) (( Z h ⊗ C )( f ◦ Φ ′ ZX ( f ))( ρ ZZ ( id Z )))= ( h ⊗ C ) ( g, f ◦ Φ ′ ZX ( f )) ( ρ ZZ ( id Z ))= ( X ′ h ⊗ C ) ( f ◦ Φ ′ ZX ( f )) (( h Z ⊗ C )( g )( ρ ZZ ( id Z )))= ( ˜ f ( X ′ ) ⊗ id C ) (( h Z ⊗ C )( g )( ρ ZZ ( id Z )))= ( ˜ f ( X ′ ) ⊗ id C )( ρ X ′ Z ( g ))Therefore, ˜ f ∈ Hom
CMod - D ( h Z , h Y ) = Hom E ( Z, Y ).For ( iii ) ⇒ ( i ), we start by showing that can XY : h Y ⊗ E X h −→ Hom D ( X, Y ) ⊗ C is an isomorphismfor each X, Y ∈ Ob ( D ). We define can − XY : Hom D ( X, Y ) ⊗ C −→ h Y ⊗ E X h by can − XY ( f ⊗ c ) := f ◦ Φ ′ Y X ( c ) ⊗ E Φ XY ( c ) ∈ h Y ⊗ E X h (4.16)for any f ∈ Hom D ( X, Y ) and c ∈ C . Then, using the C -colinearity of Φ XY , we have( can XY ◦ can − XY )( f ⊗ c ) = f ◦ Φ ′ Y X ( c ) ◦ (Φ XY ( c )) ⊗ (Φ XY ( c )) = f ◦ Φ ′ Y X ( c ) ◦ Φ XY ( c ) ⊗ c = f ⊗ c On the other hand, by assumption, we obtain( can − XY ◦ can XY )( g ⊗ E g ′ ) = gg ′ Φ ′ Y X ( g ′ ) ⊗ E Φ XY ( g ′ ) = g ⊗ E g ′ Φ ′ Y X ( g ′ )Φ XY ( g ′ ) = g ⊗ E g ′ for any g ⊗ E g ′ ∈ h Y ⊗ E X h . From the definition in (4.16), it is clear that setting can − X ( Y ) := can − XY for each Y ∈ Ob ( D ) determines a morphism in D M C which is inverse to can X . This completes theproof. Example 4.13.
Let H be a Hopf algebra over K . If C is a left H -module category, then the smashproduct category C H (see [16] ) is a right co- H -category with the right H -coaction determined by f h f h ⊗ h on each Hom C H ( X, Y ) =
Hom C ( X, Y ) ⊗ H . By definition, we know that Ob ( C ) = Ob ( C H ) .It is easy to see that Hom C ( X, Y ) =
Hom C H ( X, Y ) coH .We claim that C H is an H -Galois extension of C . We first observe that for any f h ∈ Hom C H ( Z, Y ) and f ′ h ′ ∈ Hom C H ( X, Z ) , we have ( f h ) ⊗ C ( f ′ h ′ ) = ( f h )( f ′ H ) ⊗ C ( id X h ′ ) Thus, can XY : h Y ⊗ C X h −→ Hom C H ( X, Y ) ⊗ H has the following form can XY (( f h ) ⊗ C ( f ′ h ′ )) = ( f h )( f ′ H )( id X h ′ ) ⊗ h ′ for each X, Y ∈ Ob ( C H ) , Then, it may be verified that for each X, Y ∈ Ob ( C H ) , can XY is anisomorphism with inverse can − XY : Hom C H ( X, Y ) ⊗ H −→ h Y ⊗ C X h determined by can − XY (( g k ) ⊗ k ′ ) := ( g k )( id X S ( k ′ )) ⊗ C ( id X k ′ ) Proposition 4.14.
Let D be a C -Galois extension of E . If there exists a convolution invertible collection Φ = { Φ XY : C −→ Hom D ( X, Y ) } X,Y ∈ Ob ( D ) of right C -comodule maps, then Hom D ( X, − ) ∼ = Hom E ( X, − ) ⊗ C ∈ E M C for each X ∈ Ob ( E ) = Ob ( D ) . roof. Let Φ ′ be the convolution inverse of Φ. Given f ∈ Hom D ( X, Y ), it follows from Theorem 4.12that f ◦ Φ ′ ZX ( f ) ∈ Hom E ( Z, Y ) for every Z ∈ Ob ( D ). We define η : Hom D ( X, − ) −→ Hom E ( X, − ) ⊗ C η ( Y )( f ) := f ◦ Φ ′ XX ( f ) ⊗ f Using Definition 4.5, we see that ρ XY ′ ( gf ) = gf ⊗ f for any g ∈ Hom E ( Y, Y ′ ). Hence, we have( gf ) ⊗ ( gf ) ⊗ ( gf ) = ( id ⊗ ∆ C )(( gf ) ⊗ ( gf ) ) = ( ρ XY ′ ⊗ id C )( gf ⊗ f ) = gf ⊗ f ⊗ f (4.17)Using (4.17), it may be easily seen that η is a morphism of left E -modules. Using the coassociativityof the C -coactions { ρ XY } X,Y ∈ Ob ( D ) , it is also clear that η is objectwise C -colinear. Therefore, η is amorphism in E M C .Conversely, we define ζ : Hom E ( X, − ) ⊗ C −→ Hom D ( X, − ) given by ζ ( Y )( f ′ ⊗ c ) := f ′ ◦ Φ XX ( c ) for Y ∈ Ob ( E ). It is immediate that ζ is a morphism of left E -modules. Moreover, ρ XY ( f ′ ◦ Φ XX ( c )) = ( f ′ ◦ Φ XX ( c )) ⊗ ( f ′ ◦ Φ XX ( c )) = f ′ ◦ (Φ XX ( c )) ⊗ (Φ XX ( c )) = f ′ ◦ Φ XX ( c ) ⊗ c where the last equality follows from the fact that Φ XX is C -colinear. It follows that ζ ( Y ) is C -colinearfor each Y ∈ Ob ( E ) and hence ζ is a morphism in E M C . It may be verified that ζ is the inverse of η . Definition 4.15.
Let D be a small K -linear category and E be a K -subcategory. Let ( C , ∆ C , ε C ) be a D -coring. Then, a collection G ( C , E ) = { s X ∈ C ( X, X ) } X ∈ Ob ( E ) is said to be group-like for C with respect to E if(i) ∆ C ( X, X )( s X ) = s X ⊗ s X and ε C ( s X ) = id X for any X ∈ Ob ( E ) ,(ii) For any f ∈ Hom E ( X, Y ) , we have f · s X = C ( − , f )( X )( s X ) = C ( f, − )( Y )( s Y ) = s Y · f (4.18) Example 4.16. (i) If E is a subcategory of D , then the collection { id X ⊗ id X ∈ h X ⊗ E X h } X ∈ Ob ( E ) isgroup-like for h ⊗ E h with respect to E .(ii) Let D be a C -Galois extension of E . Then h ⊗ C is a D -coring (by Theorem 4.9 and Lemma 4.2)and the collection { id X ⊗ id X ∈ Hom D ( X, X ) ⊗ C } X ∈ Ob ( E ) is group-like for h ⊗ C with respect to E .Since h Y ∈ M ( ψ ) C D for each Y ∈ Ob ( D ) , we have ρ XY ( f ) = ρ XY ( h Y ( f )( id Y )) = h Y ( f ψ )( id Y ) ⊗ id Y ψ = id Y f ψ ⊗ id Y ψ = ( id Y ⊗ id Y ) · f for any f ∈ Hom D ( X, Y ) . But, if f ∈ Hom E ( X, Y ) , then we also have ρ XY ( f ) = ρ XY ( f ◦ id X ) = f · ρ XX ( id X ) = f ◦ id X ⊗ id X = f · ( id X ⊗ id X ) Proposition 4.17.
Let
E ⊆ D be a subcategory and C be a D -coring. Let { s X } X ∈ Ob ( E ) be a group-likecollection for C with respect to E . For a right C -comodule ( N , ρ N ) , the E -submodule N co C : E op −→ V ect K of coinvariants of N is given by: N co C ( X ) := { n ∈ N ( X ) | ρ N ( X )( n ) = n ⊗ s X }N co C ( f )( n ′ ) := N ( f )( n ′ ) for any X ∈ Ob ( E ) , f ∈ Hom E ( X, Y ) and n ′ ∈ N co C ( Y ) .Proof. We will show that for any f ∈ Hom E ( X, Y ), the morphism N co C ( f ) : N co C ( Y ) −→ N co C ( X )is well-defined. Since ρ N : N −→ N ⊗ D C is a morphism of right D -modules, we have the followingcommutative diagram: N ( Y ) ρ N ( Y ) −−−−→ N ⊗ D C ( Y, − ) N ( f ) y y ( N ⊗ D C )( f )= id N ⊗ C ( f, − ) N ( X ) ρ N ( X ) −−−−→ N ⊗ D C ( X, − )34et n ′ ∈ N co C ( Y ) so that ρ N ( Y )( n ′ ) = n ′ ⊗ s Y . Since f ∈ Hom E ( X, Y ), using (4.18) we have ρ N ( X ) ( N ( f )( n ′ )) = ( id N ⊗ C ( f, − )) ( n ′ ⊗ D s Y ) = n ′ ⊗ D s Y · f = n ′ ⊗ D f · s X = N ( f )( n ′ ) ⊗ D s X . This shows that N ( f )( n ′ ) = N co C ( f )( n ′ ) ∈ N co C ( X ). The result follows.The next result shows that in the case of a C -Galois extension E ⊆ D , we recover the notion of coinvariantsas in Definition (4.5).
Lemma 4.18.
Let D be a C -Galois extension of E . Consider the collection { id X ⊗ id X ∈ Hom D ( X, X ) ⊗ C } X ∈ Ob ( D ) which is group-like for h ⊗ C with respect to E . Then, ( Hom D ( − , Y )) co ( h ⊗ C ) ( X ) = Hom E ( X, Y ) for any X , Y ∈ Ob ( D ) = Ob ( E ) .Proof. Since D is a C -Galois extension of E , we know that there is a canonical entwining ( D , C, ψ ) suchthat h Y ∈ M ( ψ ) C D . Using Proposition 4.3, h Y may be treated as an object of Comod -( h ⊗ C ). Let g ∈ ( Hom D ( − , Y )) co ( h ⊗ C ) ( X ). Then, ρ XY ( g ) = g ◦ id X ⊗ id X . Using the fact that h Y ∈ M ( ψ ) C D wehave ρ ZY ( gf ) = ( h Y ⊗ C )( f )( ρ XY ( g )) = ( h Y ⊗ C )( f ) ( g ◦ id X ⊗ id X )= g ◦ id X ◦ f ψ ⊗ id X ψ = ( Z h ⊗ C )( g )( id X ◦ f ψ ⊗ id X ψ ) = ( Z h ⊗ C )( g ) ρ ZX ( f )for any f ∈ Hom D ( Z, X ). Therefore, g ∈ Hom E ( X, Y ). The converse follows directly using the Definition(4.5).
Lemma 4.19.
Let D be a C -Galois extension of E and let ( D , C, ψ ) be the canonical entwining structureassociated to it. We denote by ρ XY : Hom D ( X, Y ) −→ Hom D ( X, Y ) ⊗ C the right C -comodule structuremaps. Then, for any M ∈
M od - E , we may obtain an object M ⊗ E h ∈ M ( ψ ) C D by setting ( M ⊗ E h )( Y ) := M ⊗ E Y h ( M ⊗ E h )( f )( m ⊗ g ) := m ⊗ gf for f ∈ Hom D ( X, Y ) and m ⊗ g ∈ M ( Z ) ⊗ Y h ( Z ) . In fact, this determines a functor from M od - E to M ( ψ ) C D .Proof. Clearly,
M ⊗ E h ∈ M od - D . For each Y ∈ Ob ( D ), it may be verified that M ⊗ E Y h has a right C -comodule structure given by M ⊗ E Y h id ⊗ ρ −−−→ M ⊗ E Y h ⊗ C m ⊗ g m ⊗ ρ Y Z ( g )for any g ∈ Hom D ( Y, Z ) and m ∈ M ( Z ). By Theorem 4.9, h Z is an object in M ( ψ ) C D for every Z ∈ Ob ( D ) with its canonical D -module structure and right C -coactions { ρ XZ } X ∈ Ob ( D ) . Therefore, wehave ρ XZ ( h Z ( f )( g ))) = ( gf ) ⊗ ( gf ) = g f ψ ⊗ g ψ for any f ∈ Hom D ( X, Y ). Consequently, we have( id ⊗ ρ XZ ) (( M ⊗ E h )( f )( m ⊗ g )) = m ⊗ ( gf ) ⊗ ( gf ) = m ⊗ g f ψ ⊗ g ψ (4.19)This shows that M ⊗ E h ∈ M ( ψ ) C D . Lemma 4.20.
Let D be a C -Galois extension of E . If there exists a convolution invertible collection Φ = { Φ XY : C −→ Hom D ( X, Y ) } X,Y ∈ Ob ( D ) of right C -comodule maps, then(i) Hom D ( X, − ) is flat as a left E -module.(ii) L X ∈ Ob ( D ) Hom D ( X, − ) is faithfully flat as a left E -module.(iii) For any M ∈
M od - E , there is a monomorphism M ֒ → M ⊗ E h in M od - E given by m m ⊗ id X for any m ∈ M ( X ) .Proof. (i) Let i : M ֒ → M be a monomorphism of right E -modules. By Proposition 4.14, it followsthat the induced map M ⊗ E Hom D ( X, − ) −→ M ⊗ E Hom D ( X, − ) coincides with the map M ( X ) ⊗ C i ( X ) ⊗ id C −−−−−−→ M ( X ) ⊗ C for each X ∈ Ob ( E ) = Ob ( D ). Since i ( X ) ⊗ id C is clearly a monomorphism, itfollows that Hom D ( X, − ) is flat as a left E -module.35ii) This is clear from the fact that M ( X ) ⊗ C = M ⊗ E Hom D ( X, − ) = 0 ⇒ M ( X ) = 0.(iii) Since L Y ∈ Ob ( D ) Hom D ( Y, − ) is faithfully flat as a left E -module, it is enough to prove that for each Y ∈ Ob ( D ), we have a monomorphism M ⊗ E Hom D ( Y, − ) −→ M ⊗ E h ⊗ E Hom D ( Y, − ) M ( X ) ⊗ Y h ( X ) ∋ m ⊗ f m ⊗ id X ⊗ f (4.20)This is true because the morphism in (4.20) has a section M ⊗ E h ⊗ E Hom D ( Y, − ) −→ M ⊗ E Hom D ( Y, − ) m ′ ⊗ g ′ ⊗ f ′ m ⊗ g ′ f ′ (4.21)for any m ′ ∈ M ( Z ) and g ′ ⊗ f ′ ∈ X h ( Z ) ⊗ Y h ( X ). Theorem 4.21.
Let D be a C -Galois extension of E and let ( D , C, ψ ) be the canonical entwiningstructure associated to it. Suppose there exists a convolution invertible collection Φ = { Φ XY : C −→ Hom D ( X, Y ) } X,Y ∈ Ob ( D ) of right C -comodule maps. Then, the categories M ( ψ ) C D and Mod- E are equiv-alent.Proof. We consider the collection { id X ⊗ id X ∈ Hom D ( X, X ) ⊗ C } X ∈ Ob ( E ) which is group-like for thecoring h ⊗ C with respect to E . We define F : M od - E −→ M ( ψ ) C D M 7→ M ⊗ E h G : M ( ψ ) C D −→ M od - E N 7→ N co ( h ⊗ C ) Using Lemma 4.19 and Proposition 4.17, we see that the functors F and G are well-defined. We nowverify that G ◦ F ∼ = id Mod - E i.e., ( M ⊗ E h ) co ( h ⊗ C ) ∼ = M for any M ∈
M od - E .From Lemma 4.10, we know that h ⊗ C ∼ = h ⊗ E h as D -corings. Under this isomorphism, the collection { id X ⊗ id X ∈ Hom D ( X, X ) ⊗ C } X ∈ Ob ( E ) maps to the collection { id X ⊗ id X ∈ h X ⊗ X h } X ∈ Ob ( E ) whichis group-like for h ⊗ E h with respect to E . Therefore, it suffices to show that M ∼ = (
M ⊗ E h ) co ( h ⊗ E h ) .By Lemma (4.20)(iii), we have an inclusion i : M −→ M ⊗ E h of right E -modules. It is clear that i ( M ) ⊆ ( M ⊗ E h ) co ( h ⊗ E h ) . By definition, ˜ ρ = ρ M⊗ E h : M ⊗ E h −→ ( M ⊗ E h ) ⊗ D ( h ⊗ E h ) is determinedby ˜ ρ ( X )( m ⊗ f ) = m ⊗ E id Y ⊗ E f ∀ m ⊗ f ∈ M ( Y ) ⊗ X h ( Y )for each X ∈ Ob ( D ). The coinvariants ( M ⊗ E h ) co ( h ⊗ E h ) : E op −→ V ect K are given by( M ⊗ E h ) co ( h ⊗ E h ) ( X ) = { P Y ∈ Ob ( E ) m Y ⊗ f Y ∈ M ⊗ X h | ˜ ρ ( X )( P m Y ⊗ f Y ) = P m Y ⊗ E f Y ⊗ E id X } For P m Y ⊗ f Y ∈ ( M ⊗ E h ) co ( h ⊗ E h ) ( X ), we now have˜ ρ ( X )( X m Y ⊗ f Y ) = X m Y ⊗ E f Y ⊗ E id X = X m Y ⊗ E id Y ⊗ E f Y ∈ ( M ⊗ E h ) ⊗ E X h (4.22)We set P := ( M ⊗ E h ) / M ∈
M od - E and consider the following short exact sequence:0 −→ M i −→ M ⊗ E h η −→ P −→ η induces the morphism η ⊗ id h : ( M ⊗ E h ) ⊗ E h −→ P ⊗ E h of right E -modules which for each X ∈ Ob ( D ) is given by( η ⊗ id h )( X ) : ( M ⊗ E h ) ⊗ E X h −→ P ⊗ E X h m ′ ⊗ f ′ ⊗ g ′ η ( Y )( m ′ ⊗ f ′ ) ⊗ g ′ where m ′ ∈ M ( Z ), f ′ ∈ Hom D ( Y, Z ), g ′ ∈ Hom D ( X, Y ) and
Y, Z ∈ Ob ( E ). Applying ( η ⊗ id h )( X ) to(4.22), we obtain X η ( X )( m Y ⊗ E f Y ) ⊗ E id X = X η ( Y )( m Y ⊗ E id Y ) ⊗ E f Y = X η ( Y )( i ( Y )( m Y )) ⊗ E f Y = 0 (4.23)Applying Lemma (4.20)(iii) to the inclusion P ֒ → P ⊗ E h , it follows from (4.23) that P η ( X )( m Y ⊗ E f Y ) =0 for every X ∈ Ob ( E ). Therefore, P m Y ⊗ f Y ∈ i ( M )( X ). This proves that M ∼ = (
M ⊗ E h ) co ( h ⊗ E h ) .36t remains to show that F ◦ G ∼ = id M ( ψ ) C D . Let N ∈ M ( ψ ) C D ∼ = Comod -( h ⊗ C ). Then, N is a right D -module with a given morphism ρ N : N −→ N ⊗ D ( h ⊗ C ) ∼ = N ⊗ D ( h ⊗ E h ) ∼ = N ⊗ E h in M ( ψ ) C D . By definition, N co ( h ⊗ C ) is the equalizer of the following morphisms0 −→ N co ( h ⊗ C ) −→ N j / / ρ N / / N ⊗ E h (4.24)where j is given by j ( X ) : N ( X ) −→ N ⊗ E X h n n ⊗ id X for every X ∈ Ob ( D ). By Lemma 4.20(i), it follows that N co ( h ⊗ C ) ⊗ E X h is the equalizer of the followingmorphisms 0 −→ N co ( h ⊗ C ) ⊗ E X h −→ N ⊗ E X h j ⊗ id / / ρ N ⊗ id / / N ⊗ E h ⊗ E X h (4.25)Comparing with (4.9), we observe that j ⊗ id = id N ⊗ E ∆ h ⊗ E h ( X, − ). Using the coassociativity of ρ N : N −→ N ⊗ E h , it follows from (4.25) that ρ N ( X ) factorises through N co ( h ⊗ C ) ⊗ E X h , which isdenoted by ρ ′N ( X ) : N ( X ) −→ N co ( h ⊗ C ) ⊗ E X h ⊆ N ⊗ E X h .We claim that ρ ′N : N −→ N co ( h ⊗ C ) ⊗ E h is an isomorphism in M ( ψ ) C D . From the counit property, weknow that ( id N ⊗ D ε h ⊗ E h ) ◦ ρ N = id N . Hence, ρ N is a monomorphism and so is ρ ′N . It remains to showthat ρ ′N ( X ) is an epimorphism for each X ∈ Ob ( D ). For each X ∈ Ob ( D ), we define ζ ( X ) : N co ( h ⊗ C ) ⊗ E X h −→ N ( X ) X Y ∈ Ob ( D ) n Y ⊗ f Y X Y ∈ Ob ( D ) N ( f Y )( n Y )Since ρ ′N is a morphism of right D -modules, we now have ρ ′N ( X ) ( ζ ( X ) ( n Y ⊗ f Y )) = ρ ′N ( X )( N ( f Y )( n Y )) = ( N co ( h ⊗ C ) ⊗ E h )( f Y )( ρ ′N ( Y )( n Y ))= ( N co ( h ⊗ C ) ⊗ E h )( f Y )( n Y ⊗ id Y ) = n Y ⊗ f Y This shows that F ◦ G ∼ = id M ( ψ ) C D . References [1] J. Y. Abuhlail,
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