aa r X i v : . [ m a t h . C T ] A ug Entwined modules over representations of categories
Abhishek Banerjee ∗† Abstract
We introduce a theory of modules over a representation of a small category taking values in entwining structuresover a semiperfect coalgebra. This takes forward the aim of developing categories of entwined modules to thesame extent as that of module categories as well as the philosophy of Mitchell of working with rings with severalobjects. The representations are motivated by work of Estrada and Virili, who developed a theory of modulesover a representation taking values in small preadditive categories, which were then studied in the same spiritas sheaves of modules over a scheme. We also describe, by means of Frobenius and separable functors, how ourtheory relates to that of modules over the underlying representation taking values in small K -linear categories. MSC(2020) Subject Classification:
Keywords:
Rings with several objects, entwined modules, separable functors, Frobenius pairs
The purpose of this paper is to study modules over representations of a small category taking values in spaces thatbehave like quotients of categorified fiber bundles. Let H be a Hopf algebra having a coaction ρ ∶ A Ð→ A ⊗ H on analgebra A such that A becomes an H -comodule algebra. Let B denote the algebra of coinvariants of this coaction.Suppose that the inclusion B ↪ A is faithfully flat and the canonical morphism can ∶ A ⊗ B A Ð→ A ⊗ H x ⊗ y ↦ x ⋅ ρ ( y ) is an isomorphism. This datum is the algebraic counterpart of a principal fiber bundle given by the quotient of anaffine algebraic group scheme acting freely on an affine scheme over a field K (see, for instance, [25], [29]). If H hasbijective antipode, then modules over the algebra B of coinvariants may be recovered as “ ( A, H ) -Hopf modules” (seeSchneider [29]).These ( A, H ) -Hopf modules may be rolled into the more general concept of modules over an ‘entwining structure’consisting of an algebra R , a coalgebra C and a morphism ψ ∶ C ⊗ R Ð→ R ⊗ C satisfying certain conditions.Entwining structures were introduced by Brzezi´nski and Majid [9]. It was soon realized (see Brzezi´nski [5]) thatentwining structures provide a single formalism that unifies relative Hopf modules, Doi-Hopf modules, Yetter-Drinfeldmodules and several other concepts such as coalgebra Galois extensions. As pointed out in Brzezi´nski [7], an entwiningstructure ( R, C, ψ ) behaves like a single bialgebra, or more generally a comodule algebra over a bialgebra. Accordingly,the investigation of entwining structures as well as the modules over them has emerged as an object of study in itsown right (see, for instance, [1], [3], [4], [5], [8], [12], [13], [14], [21], [22], [28]). ∗ Department of Mathematics, Indian Institute of Science, Bangalore, India. Email: [email protected] † partially supported by SERB Matrics fellowship MTR/2017/000112
1e consider an entwining structure consisting of a small K -linear category R , a coalgebra C and a family ofmorphisms ψ = { ψ rs ∶ C ⊗ R( r, s ) Ð→ R( r, s ) ⊗ C } r,s ∈R satisfying certain conditions (see Definition 2.1). This is in keeping with the general philosophy of Mitchell [24],where a small K -linear category is viewed as a K -algebra with several objects. In fact, we consider the category E nt of such entwining structures. When the coalgebra C is fixed, we have the subcategory E nt C . Given an entwiningstructure (R , C, ψ ) , we have a category M C R ( ψ ) of modules over it (see our earlier work in [4]). These entwinedmodules over (R , C, ψ ) may be seen as modules over a certain categorical quotient space of R , which need not existin an explicit sense, but is studied only through its category of modules.We work with representations R ∶ X Ð→ E nt C of a small category X taking values in E nt C , where C is a fixedcoalgebra. This is motivated by the work of Estrada and Virili [19], who introduced a theory of modules overa representation A ∶ X Ð→ Add , where
Add is the category of small preadditive categories. The modules over A ∶ X Ð→ Add were studied in the spirit of sheaves of modules over a scheme, or more generally, a ringed space. Byconsidering small preaditive categories, the authors in [19] also intended to take Mitchell’s idea one step forward: fromreplacing rings with small preadditive categories to replacing ring representations by representations taking valuesin small preadditive categories. In this paper, we develop a theory of modules over a representation R ∶ X Ð→ E nt C taking values in entwining structures. We also describe, by means of Frobenius and separable functors, how thistheory relates to that of modules over the underlying representation taking values in small K -linear categories.This paper has two parts. In the first part, we introduce and develop the properties of the category M od C − R ofmodules over R ∶ X Ð→ E nt C . For this, we have to combine techniques on comodules along with adapting themethods of Estrada and Virili [19]. When R ∶ X Ð→ E nt C is a flat representation (see Section 6), we also considerthe subcategory Cart − R of cartesian entwined modules over R . In the analogy with sheaves of modules over ascheme, the cartesian objects may be seen as similar to quasi-coherent sheaves.Let L in be the category of small K -linear categories. In the second part, we consider the underlying representation R ∶ X Ð→ E nt C Ð→ L in , which we continue to denote by R . Accordingly, we have a category M od − R of modulesover R ∶ X Ð→ E nt C Ð→ L in in the sense of Estrada and Virili [19]. We study the relation between M od C − R and M od − R by describing Frobenius and separability conditions for a pair of adjoint functors between them (seeSection 7) F ∶ M od C − R Ð→ M od − R G ∶ M od − R Ð→ M od C − R Here, the left adjoint F may be thought of as an ‘extension of scalars’ and the right adjoint G as a ‘restriction ofscalars.’The idea is as follows: as mentioned before, modules over an entwining structure (R , C, ψ ) may be seen as modulesover a certain categorical quotient space of R , which behaves like a subcategory of R . Again, this “subcategory”of R need not exist in an explicit sense, but is studied only through the category of modules M C R ( ψ ) . Accordingly,a representation R ∶ X Ð→ E nt C taking values in E nt C may be thought of as a subfunctor of the underlyingrepresentation R ∶ X Ð→ E nt C Ð→ L in . We want to understand the properties of the inclusion of this “subfunctor”:in particular, whether it behaves like a separable, split or Frobenius extension of rings. We recall here (see [10,Theorem 1.2]) that if R Ð→ S is an extension of rings, these properties may be expressed in terms of the functors F ∶ M od − R Ð→ M od − S (extension of scalars) and G ∶ M od − S Ð→ M od − R (restriction of scalars) as follows R Ð→ S split extension ⇔ F ∶ M od − R Ð→ M od − S separable R Ð→ S separable extension ⇔ G ∶ M od − S Ð→ M od − R separable R Ð→ S Frobenius extension ⇔ ( F, G ) Frobenius pair of functorsWe now describe the paper in more detail. Throughout,we let K be a field. We begin in Section 2 by describingthe categories of entwining structures and entwined modules. For a morphism ( α, γ ) ∶ (R , C, ψ ) Ð→ (S , D, ψ ′ ) of2ntwining structures, we describe ‘extension of scalars’ and ‘restriction of scalars’ on categories of entwined modules.Our first result is as follows. Theorem 1. (see 2.3, 2.4 and 2.5) Let ( α, γ ) ∶ (R , C, ψ ) Ð→ (S , D, ψ ′ ) be a morphism of entwining structures.(1) There is a functor ( α, γ ) ∗ ∶ M C R ( ψ ) Ð→ M D S ( ψ ′ ) of extension of scalars.(2) Suppose that the coalgebra map γ ∶ C Ð→ D is also a monomorphism of vector spaces. Then, there is a functor ( α, γ ) ∗ ∶ M D S ( ψ ′ ) Ð→ M C R ( ψ ) of restriction of scalars. Further, there is an adjunction of functors which is given bynatural isomorphisms M D S ( ψ ′ )(( α, γ ) ∗ M , N ) = M C R ( ψ )(M , ( α, γ ) ∗ N ) for any M ∈ M C R ( ψ ) and N ∈ M D S ( ψ ′ ) . In Section 3, we give conditions for the category M C R ( ψ ) of modules over an entwining structure (R , C, ψ ) to haveprojective generators. We recall that a K -coalgebra C is said to be right semiperfect if the category of right C -comodules has enough projectives. Theorem 2. (see 3.5) Let (R , C, ψ ) be an entwining structure and let C be a right semiperfect K -coalgebra. Then,the category M C R ( ψ ) of entwined modules is a Grothendieck category with a set of projective generators. In Section 4, we fix a coalgebra C . We introduce the category M od C − R of modules over a representation R ∶ X Ð→ E nt C , which is our main object of study. Our first purpose is to show that M od C − R is a Grothendieck category. Theorem 3. (see 4.9) Let C be a right semiperfect coalgebra over a field K . Let R ∶ X Ð→ E nt C be an entwined C -representation of a small category X . Then, the category M od C − R of entwined modules over R is a Grothendieckcategory. Given R ∶ X Ð→ E nt C , we have an entwining structure ( R x , C, ψ x ) for each x ∈ X . Our next aim is to giveconditions for M od C − R to have projective generators. For this, we will construct an extension functor ex Cx and anevaluation functor ev Cx relating the categories M od C − R and M C R x ( ψ x ) at each x ∈ X . Theorem 4. (see 5.3 and 5.5) Let C be a right semiperfect coalgebra over a field K . Let X be a poset and let R ∶ X Ð→ E nt C be an entwined C -representation of X .(1) For each x ∈ X , there is an extension functor ex Cx ∶ M C R x Ð→ M od C − R which is left adjoint to an evaluationfunctor ev Cx ∶ M od C − R Ð→ M C R x ( ψ x ) .(2) The family { ex Cx ( V ⊗ H r ) ∣ x ∈ X , r ∈ R x , V ∈ P roj f ( C )} is a set of projective generators for M od C − R , where P roj f ( C ) is the set of isomorphism classes of finite dimensional projective C -comodules. We introduce the category of cartesian entwined modules in Section 6. Here, we will assume that X is a poset and R ∶ X Ð→ E nt C is a flat representation, i.e., for any morphism α ∶ x Ð→ y in X , the functor α ∗ ∶ = R ∗ α ∶ M C R x ( ψ x ) Ð→ M C R y ( ψ y ) is exact. We then apply induction on N × M or ( X ) to show that any cartesian entwined module may beexpressed as a sum of submodules whose cardinality is ≤ κ ∶ = sup {∣ N ∣ , ∣ C ∣ , ∣ K ∣ , ∣ M or ( X )∣ , ∣ M or ( R x )∣ , x ∈ X } . Theorem 5. (see 6.10) Let C be a right semiperfect coalgebra over a field K . Let X be a poset and let R ∶ X Ð→ E nt C be an entwined C -representation of X . Suppose that R is flat. Then, Cart C − R is a Grothendieck category. In the next three sections, we study separability and Frobenius conditions for functors relating
M od C − R to thecategory M od − R of modules over the underlying representation R ∶ X Ð→ E nt C Ð→ L in . For this, we have toadapt the techniques from [10] as well as our earlier work in [4]. For more on Frobenius and separability conditionsfor Doi-Hopf modules and modules over entwining structures of algebras, we refer the reader to [6], [15], [16], [17].At each x ∈ X , we have functors F x ∶ M C R x ( ψ x ) Ð→ M R x and G x ∶ M R x Ð→ M C R x ( ψ x ) which combine to givefunctors F ∶ M od C − R Ð→ M od − R and G ∶ M od − R Ð→ M od C − R respectively. We will also need to consider aspace V of elements θ = { θ x ( r ) ∶ C ⊗ C Ð→ R x ( r, r )} x ∈ X ,r ∈ R x and a space W of elements η = { η x ( s, r ) ∶ R x ( s, r ) Ð→ R x ( s, r ) ⊗ C } x ∈ X ,r,s ∈ R x satisfying certain conditions (see Sections 7 and 8).3 heorem 6. (see 7.2, 7.3 and 7.7) Let X be a poset, C be a right semiperfect K -coalgebra and R ∶ X Ð→ E nt C bean entwined C -representation.(1) The forgetful functor F ∶ M od C − R Ð→ M od − R has a right adjoint G ∶ M od − R Ð→ M od C − R .(2) A natural transformation υ ∈ N at ( G F , Mod C − R ) corresponds to a collection of natural transformations { υ x ∈ N at ( G x F x , M C R x ( ψ x ) )} x ∈ X such that for any α ∶ x Ð→ y in X and object M ∈ M od C − R , we have M α ○ υ x ( M x ) = α ∗ υ y ( M y ) ○ G x F x ( M α ) .(3) The space N at ( G F , Mod C − R ) is isomorphic to V . The main results in Sections 7 and 8 give necessary and sufficient conditions for the forgetful functor F ∶ M od C − R Ð→ M od − R and its right adjoint G ∶ M od − R Ð→ M od C − R to be separable. In Section 9, we give necessary andsufficient conditions for ( F , G ) to be a Frobenius pair, i.e., G is both a left and a right adjoint of F . Theorem 7. (see 7.8, 7.9 and 7.10) Let X be a partially ordered set. Let C be a right semiperfect K -coalgebra andlet R ∶ X Ð→ E nt C be an entwined C -representation.(1) The functor F ∶ M od C − R Ð→ M od − R is separable if and only if there exists θ ∈ V such that θ x ( r )( c ⊗ c ) = ε C ( c ) ⋅ id r for every x ∈ X , r ∈ R x and c ∈ C .(2) Suppose additionally that the representation R ∶ X Ð→ E nt C is flat. Then, we have(a) The functor F ∶ M od C − R Ð→ M od − R restricts to a functor F c ∶ Cart C − R Ð→ Cart − R . Moreover, F c has a right adjoint G c ∶ Cart − R Ð→ Cart C − R .(b) Suppose there exists θ ∈ V such that θ x ( r )( c ⊗ c ) = ε C ( c ) ⋅ id r for every x ∈ X , r ∈ R x and c ∈ C . Then, F c ∶ Cart C − R Ð→ Cart − R is separable. Theorem 8. (see 8.2 and 8.3) Let X be a partially ordered set, C be a right semiperfect K -coalgebra and let R ∶ X Ð→ E nt C be an entwined C -representation.(1) The spaces N at ( Mod − R , F G ) and W are isomorphic.(2) The functor G ∶ M od − R Ð→ M od C − R is separable if and only if there exists η ∈ W such that id = ( id ⊗ ε C ) ○ η x ( s, r ) for each x ∈ X and s , r ∈ R x . Theorem 9. (see 9.1, 9.4, 9.5) Let X be a partially ordered set, C be a right semiperfect K -coalgebra and let R ∶ X Ð→ E nt C be an entwined C -representation.(1) ( F , G ) is a Frobenius pair if and only if there exist θ ∈ V and η ∈ W such that ε C ( d ) f = ∑ ̂ f ○ θ x ( r )( c f ⊗ d ) and ε C ( d ) f = ∑ ̂ f ψ x ○ θ x ( r )( d ψ x ⊗ c f ) for every x ∈ X , r ∈ R x , f ∈ R x ( r, s ) and d ∈ C , where η x ( r, s )( f ) = ̂ f ⊗ c f .(2) Suppose additionally that the representation R ∶ X Ð→ E nt C is flat. Then, G ∶ M od − R Ð→ M od C − R restrictsto a functor G c ∶ Cart − R Ð→ Cart C − R . Further, ( F c , G c ) is a Frobenius pair of adjoint functors between Cart C − R and Cart − R . We conclude in Section 10 by giving examples of how to construct entwined representations and describe modulesover them. In particular, we show how to construct entwined representations using B -comodule categories, where B is a bialgebra. Let K be a field and let V ect K be the category of vector spaces over K . Let R be a small K -linear category. Thecategory of right R -modules will be denoted by M R . For any object r ∈ R , we denote by H r ∶ R op Ð→ V ect K theright R -module represented by r and by r H ∶ R Ð→ V ect K the left R -module represented by r . Given a K -coalgebra C , the category of right C -comodules will be denoted by Comod − C .4 efinition 2.1. (see [4, § ) Let R be a small K -linear category and C be a K -coalgebra. An entwining structure ( R , C, ψ ) over K is a collection of K -linear morphisms ψ = { ψ rs ∶ C ⊗ R ( r, s ) Ð→ R ( r, s ) ⊗ C } r,s ∈R satisfying the following conditions ( gf ) ψ ⊗ c ψ = g ψ f ψ ⊗ c ψψ ε C ( c ψ )( f ψ ) = ε C ( c ) ff ψ ⊗ ∆ C ( c ψ ) = f ψψ ⊗ c ψ ⊗ c ψ ψ ( c ⊗ id r ) = id r ⊗ c (2.1) for each f ∈ R ( r, s ) , g ∈ R ( s, t ) and c ∈ C . Here, we have suppressed the summation and written ψ ( c ⊗ f ) simply as f ψ ⊗ c ψ .A morphism ( α, γ ) ∶ ( R , C, ψ ) Ð→ ( S , D, ψ ′ ) of entwining structures consists of a functor α ∶ R Ð→ S and a counitalcoalgebra map γ ∶ C Ð→ D such that α ( f ψ ) ⊗ γ ( c ψ ) = α ( f ) ψ ′ ⊗ γ ( c ) ψ ′ for any c ⊗ f ∈ C ⊗ R ( r, s ) , where r, s ∈ R .We will denote by E nt the category of entwining structures over K . If M is a right R -module, m ∈ M ( r ) and f ∈ R ( s, r ) , the element M ( f )( r ) ∈ M ( s ) will often be denoted by mf .If α ∶ R Ð→ S is a functor of small K -linear categories, there is an obvious functor α ∗ ∶ M S Ð→ M R of restriction ofscalars. For the sake of convenience, we briefly recall here the well known extension of scalars α ∗ ∶ M R Ð→ M S . For M ∈ M R , the module α ∗ ( M ) ∈ M S is determined by setting α ∗ ( M )( s ) ∶ = ( ⊕ r ∈R M ( r ) ⊗ S ( s, α ( r ))) / V (2.2)for s ∈ S , where V is the subspace generated by elements of the form ( m ′ ⊗ α ( g ) f ) − ( m ′ g ⊗ f ) (2.3)for m ′ ∈ M ( r ′ ) , g ∈ R ( r, r ′ ) , f ∈ S ( s, α ( r )) and r , r ′ ∈ R .On the other hand, if γ ∶ C Ð→ D is a morphism of coalgebras and N is a right C -comodule, there is an obviouscorestriction of scalars γ ∗ ∶ Comod − C Ð→ Comod − D . The functor γ ∗ has a well known right adjoint γ ∗ ∶ Comod − D Ð→ Comod − C , known as the coinduction functor, given by the cotensor product N ↦ N ◻ D C (see, for instance,[11, § N ◻ D N ′ of a right D -comodule ( N, ρ ∶ N Ð→ N ⊗ D ) with a left D -comodule ( N ′ , ρ ′ ∶ N ′ Ð→ D ⊗ N ′ ) is given by the equalizer N ◻ D N ′ ∶ = Eq ⎛⎝ N ⊗ N ′ ρ ⊗ id −−−−−− → −−−−−− → id ⊗ ρ ′ N ⊗ D ⊗ N ′ ⎞⎠ (2.4)In other words, an element ∑ n i ⊗ n ′ i ∈ N ⊗ N ′ lies in N ◻ D N ′ if and only if ∑ n i ⊗ n i ⊗ n ′ i = ∑ n i ⊗ n ′ i ⊗ n ′ i .However, we will continue to suppress the summation and write an element of N ◻ D N ′ simply as n ⊗ n ′ . We willnow consider modules over an entwining structure ( R , C, ψ ) . Definition 2.2. (see [4, Definition 2.2] ) Let M be a right R -module with a given right C -comodule structure ρ M( s ) ∶M ( s ) Ð→ M ( s ) ⊗ C on M ( s ) for each s ∈ R . Then, M is said to be an entwined module over ( R , C, ψ ) if thefollowing compatibility condition holds: ρ M( s ) ( mf ) = ( mf ) ⊗ ( mf ) = m f ψ ⊗ m ψ (2.5) for every f ∈ R ( s, r ) and m ∈ M ( r ) . A morphism η ∶ M Ð→ N of entwined modules is a morphism η ∶ M Ð→ N in M R such that η ( r ) ∶ M ( r ) Ð→ N ( r ) is C -colinear for each r ∈ R . The category of entwined modules over ( R , C, ψ ) will be denoted by M C R ( ψ ) . roposition 2.3. Let ( α, γ ) ∶ ( R , C, ψ ) Ð→ ( S , D, ψ ′ ) be a morphism of entwining structures. Then, there is afunctor ( α, γ ) ∗ ∶ M C R ( ψ ) Ð→ M D S ( ψ ′ ) .Proof. We take M ∈ M C R ( ψ ) . Then, M ∈ M R and we consider N ∶ = α ∗ ( M ) ∈ M S . For s ∈ S , we consider an element m ⊗ f ∈ N ( s ) , where m ∈ M ( r ) and f ∈ S ( s, α ( r )) for some r ∈ R . We claim that the morphism ρ N ( s ) ∶ N ( s ) Ð→ N ( s ) ⊗ D ( m ⊗ f ) ↦ ( m ⊗ f ) ⊗ ( m ⊗ f ) ∶ = ( m ⊗ f ψ ′ ) ⊗ γ ( m ) ψ ′ (2.6)makes N ( s ) a right D -comodule. Here, the association m ↦ m ⊗ m comes from the C -comodule structure ρ M( r ) ∶M ( r ) Ð→ M ( r ) ⊗ C of M ( r ) .First, we show that ρ N ( s ) is well defined. For this, we consider m ′ ∈ M ( r ′ ) , g ∈ R ( r, r ′ ) and f ∈ S ( s, α ( r )) . We have ( m ′ g ⊗ f ) ⊗ ( m ′ g ⊗ f ) = (( m ′ g ) ⊗ f ψ ′ ) ⊗ γ (( m ′ g ) ) ψ ′ = ( m ′ g ψ ⊗ f ψ ′ ) ⊗ γ ( m ′ ψ ) ψ ′ = ( m ′ ⊗ α ( g ψ ) f ψ ′ ) ⊗ γ ( m ′ ψ ) ψ ′ = ( m ′ ⊗ α ( g ) ψ ′ f ψ ′ ) ⊗ γ ( m ′ ) ψ ′ ψ ′ = ( m ′ ⊗ ( α ( g ) f ) ψ ′ ) ⊗ γ ( m ′ ) ψ ′ (2.7)From the properties of entwining structures, it may be easily verified that the structure maps in (2.6) are coassociativeand counital, giving a right D -comodule structure on N ( s ) . We now consider f ′ ∈ S ( s ′ , s ) . Then, we have ( m ⊗ f f ′ ) ⊗ ( m ⊗ f f ′ ) = ( m ⊗ ( f f ′ ) ψ ′ ) ⊗ γ ( m ) ψ ′ = ( m ⊗ f ψ ′ ) f ′ ψ ′ ⊗ γ ( m ) ψ ′ ψ ′ = ( m ⊗ f ) f ′ ψ ′ ⊗ ( m ⊗ f ) ψ ′ (2.8)This shows that N ∈ M D S ( ψ ′ ) . Proposition 2.4.
Let ( α, γ ) ∶ ( R , C, ψ ) Ð→ ( S , D, ψ ′ ) be a morphism of entwining structures. Suppose additionallythat γ ∶ C Ð→ D is a monomorphism of vector spaces. Then, there is a functor ( α, γ ) ∗ ∶ M D S ( ψ ′ ) Ð→ M C R ( ψ ) .Proof. We take N ∈ M D S ( ψ ′ ) and set M ( r ) ∶ = N ( α ( r )) ◻ D C for each r ∈ R . For f ∈ R ( r ′ , r ) , we define M ( f ) ∶ M ( r ) Ð→ M ( r ′ ) n ⊗ c ↦ ( n ⊗ c ) ⋅ f ∶ = nα ( f ψ ) ⊗ c ψ (2.9)To show that this morphism is well defined, we need to check that M ( f )( n ⊗ c ) ∈ M ( r ′ ) = N ( α ( r ′ )) ◻ D C . Since n ⊗ c ∈ N ( α ( r )) ◻ D C , we know that n ⊗ n ⊗ c = n ⊗ γ ( c ) ⊗ c (2.10)6n particular, it follows that n ⊗ c ⊗ f ∈ Eq ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ N ( α ( r )) ⊗ C ⊗ R ( r ′ , r ) N ( α ( r )) ⊗ D ⊗ C ⊗ R ( r ′ , r ) id ⊗ id ⊗ ψ ×××Ö N ( α ( r )) ⊗ D ⊗ R ( r ′ , r ) ⊗ C id ⊗ id ⊗ α ⊗ id ×××Ö N ( α ( r )) ⊗ D ⊗ S ( α ( r ′ ) , α ( r )) ⊗ C id ⊗ ψ ′ ⊗ id ×××Ö N ( α ( r )) ⊗ S ( α ( r ′ ) , α ( r )) ⊗ D ⊗ C ×××Ö N ( α ( r ′ )) ⊗ D ⊗ C ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (2.11)From (2.11), it follows that n α ( f ψ ) ψ ′ ⊗ n ψ ′ ⊗ c ψ = nα ( f ψ ) ψ ′ ⊗ γ ( c ) ψ ′ ⊗ c ψ (2.12)Applying (2.10) and (2.12), we now see that ( nα ( f ψ )) ⊗ ( nα ( f ψ )) ⊗ c ψ = n α ( f ψ ) ψ ′ ⊗ n ψ ′ ⊗ c ψ = nα ( f ψ ) ψ ′ ⊗ γ ( c ) ψ ′ ⊗ c ψ = nα ( f ψψ ) ⊗ γ ( c ψ ) ⊗ c ψ = nα ( f ψ ) ⊗ γ ( c ψ ) ⊗ c ψ (2.13)From the definition, we may easily verify that the structure maps in (2.9) make M into a right R -module. To showthat M is entwined, it remains to check that nα ( f ψ ) ⊗ c ψ ⊗ c ψ = (( n ⊗ c ) ⋅ f ) ⊗ (( n ⊗ c ) ⋅ f ) = ( n ⊗ c ) ⋅ f ψ ⊗ ( n ⊗ c ) ψ = nα ( f ψψ ) ⊗ c ψ ⊗ c ψ (2.14)in N ( α ( r ′ )) ⊗ C ⊗ C . Since γ ∶ C Ð→ D is a monomorphism and all tensor products are taken over the field K , itsuffices to show that nα ( f ψ ) ⊗ γ ( c ψ ) ⊗ c ψ = nα ( f ψψ ) ⊗ γ ( c ψ ) ⊗ c ψ ∈ N ( α ( r ′ )) ⊗ D ⊗ C (2.15)Using (2.13) and the fact that ( α, γ ) is a morphism of entwining structures, the right hand side of (2.15) becomes nα ( f ψψ ) ⊗ γ ( c ψ ) ⊗ c ψ = nα ( f ψ ) ψ ′ ⊗ γ ( c ) ψ ′ ⊗ c ψ = n α ( f ψ ) ψ ′ ⊗ n ψ ′ ⊗ c ψ (2.16)From (2.13), we already know that nα ( f ψ ) ⊗ c ψ ∈ N ( α ( r ′ )) ◻ D C . As such, we have nα ( f ψ ) ⊗ γ ( c ψ ) ⊗ c ψ = ( nα ( f ψ )) ⊗ ( nα ( f ψ )) ⊗ c ψ = n α ( f ψ ) ψ ′ ⊗ n ψ ′ ⊗ c ψ (2.17)where the second equality follows from (2.13). From (2.16) and (2.17), the result of (2.15) is now clear.7 heorem 2.5. Let ( α, γ ) ∶ ( R , C, ψ ) Ð→ ( S , D, ψ ′ ) be a morphism of entwining structures such that γ ∶ C Ð→ D isa monomorphism of vector spaces. Then, there is an adjuction of functors M D S ( ψ ′ )(( α, γ ) ∗ M , N ) = M C R ( ψ )( M , ( α, γ ) ∗ N ) (2.18) for M ∈ M C R ( ψ ) and N ∈ M D S ( ψ ′ ) .Proof. We consider a morphism η ∶ ( α, γ ) ∗ M Ð→ N in M D S ( ψ ′ ) . Then, η corresponds to a morphism η ∶ α ∗ M Ð→ N in M S such that η ( s ) ∶ α ∗ M ( s ) Ð→ N ( s ) is D -colinear for each s ∈ S . Accordingly, we have η ′ ∶ M Ð→ N in M R such that η ′ ( r ) ∶ M ( r ) Ð→ N ( α ( r )) is D -colinear for each r ∈ R . Here, M ( r ) is treated as a D -comodule viacorestriction of scalars. Therefore, we have morphisms η ′′ ( r ) ∶ M ( r ) Ð→ N ( α ( r )) ◻ D C of C -comodules for each r ∈ R . Together, these determine a morphism M Ð→ ( α, γ ) ∗ N in M R . These arguments can be easily reversed andhence the result. Let ( R , C, ψ ) be an entwining structure. In [4, Proposition 2.9], it was shown that the category M C R ( ψ ) of entwinedmodules is a Grothendieck category. In this section, we will refine this result to give conditions for M C R ( ψ ) to havea collection of projective generators. Lemma 3.1.
Let G be a Grothendieck category. Fix a set of generators { G k } k ∈ K for G . Let Z ∈ G be an object. Let i X ∶ X ↪ Z , i Y ∶ Y ↪ Z be two subobjects of Z such that for any k ∈ K and any morphism f k ∶ G k Ð→ X , there exists g k ∶ G k Ð→ Y such that i Y ○ g k = i X ○ f k . Then, i X ∶ X ↪ Z factors through i Y ∶ Y ↪ Z , i.e., X is a subobject of Y .Proof. Since { G k } k ∈ K is a set of generators for G , we can choose (see [20, Proposition 1.9.1]) an epimorphism f ∶ ⊕ j ∈ J G j Ð→ X , corresponding to a collection of maps f j ∶ G j Ð→ X , with each G j a generator from the collection { G k } k ∈ K . Accordingly, we can choose morphisms g j ∶ G j Ð→ Y such that i Y ○ g j = i X ○ f j for each j ∈ J . Together,these { g j } j ∈ J determine a morphism g ∶ ⊕ j ∈ J G j Ð→ Y satisfying i Y ○ g = i X ○ f . Since i X , i Y are monomorphisms and f is an epimorphism, we have X = Im ( i X ) = Im ( i X ○ f ) = Im ( i Y ○ g ) = Im ( i Y ∣ Im ( g )) ⊆ Im ( i Y ) = Y (3.1) Lemma 3.2.
Let G be a Grothendieck category having a set of projective generators { G k } k ∈ K . Let f ∶ X Ð→ Y be amorphism in G . Let i ∶ X ′ ↪ X and j ∶ Y ′ ↪ Y be monomorphisms. Suppose that for any k ∈ K and any morphism f k ∶ G k Ð→ X ′ , there exists a morphism g k ∶ G k Ð→ Y ′ such that f ○ i ○ f k = j ○ g k ∶ G k Ð→ Y . Then, there exists f ′ ∶ X ′ Ð→ Y ′ such that j ○ f ′ = f ○ i .Proof. It suffices to show that Im ( f ○ i ) ⊆ Y ′ . We choose any k ∈ K and a morphism h k ∶ G k Ð→ Im ( f ○ i ) ↪ Y .Since G k is projective, we can choose f k ∶ G k Ð→ X ′ such that f ○ i ○ f k = h k . By assumption, we can now find g k ∶ G k Ð→ Y ′ such that f ○ i ○ f k = j ○ g k ∶ G k Ð→ Y . In particular, j ○ g k = h k . Applying Lemma 3.1, we obtain Im ( f ○ i ) ⊆ Y ′ . Lemma 3.3.
Let ( R , C, ψ ) be an entwining structure. Let V be a right C -comodule. Then, for any r ∈ R , the module V ⊗ H r given by ( V ⊗ H r )( r ′ ) = V ⊗ R ( r ′ , r )( V ⊗ H r )( f ) ∶ ( V ⊗ H r )( r ′ ) Ð→ ( V ⊗ H r )( r ′′ ) v ⊗ g ↦ v ⊗ gf (3.2) for r ′ ∈ R , f ∈ R ( r ′′ , r ′ ) is an entwined module in M C R ( ψ ) . Here, the right C -comodule structure on ( V ⊗ H r )( r ′ ) isgiven by taking v ⊗ g to v ⊗ g ψ ⊗ v ψ . roof. See [4, Lemma 2.5].For the rest of this section, we will assume that the coalgebra C is such that the category Comod − C of right C -comodules has enough projective objects. In other words, the coalgebra C is right semiperfect (see [18, Definition3.2.4]). Proposition 3.4.
Let ( R , C, ψ ) be an entwining structure with C a right semiperfect coalgebra. Let V be a projectiveright C -comodule. Then, for any r ∈ R , the module V ⊗ H r is a projective object of M C R ( ψ ) .Proof. We begin with a morphism ζ ∶ V ⊗ H r Ð→ M and an epimorphism η ∶ N Ð→ M in M C R ( ψ ) . In particular, weconsider the composition V Ð→ V ⊗ H r ( r ) Ð→ M ( r ) v ↦ v ⊗ id r ↦ ζ ( r )( v ⊗ id r ) (3.3)which is a morphism in Comod − C . Since V is projective, we can lift the map in (3.3) to a map T ∶ V Ð→ N ( r ) in Comod − C such that ( η ( r )( T ( v )) = ζ ( r )( v ⊗ id r ) for each v ∈ V .We now define ξ ∶ V ⊗ H r Ð→ N by setting for each s ∈ R ξ ( s ) ∶ V ⊗ H r ( s ) Ð→ N ( s ) v ⊗ g ↦ N ( g )( T ( v )) (3.4)We first check that ξ ∶ V ⊗ H r Ð→ N is a morphism in M R . Given g ′ ∈ R ( s ′ , s ) , we have N ( g ′ )( ξ ( s )( v ⊗ g )) = N ( gg ′ )( T ( v )) = ξ ( s ′ )( v ⊗ gg ′ ) = ξ ( s ′ )(( V ⊗ H r )( g ′ )( v ⊗ g )) (3.5)We also have, for v ⊗ g ∈ V ⊗ H r ( s ) , ξ ( s )( v ⊗ g ) ⊗ ξ ( s )( v ⊗ g ) = N ( g )( T ( v )) ⊗ N ( g )( T ( v )) = T ( v ) g ψ ⊗ T ( v ) ψ = T ( v ) g ψ ⊗ v ψ = N ( g ψ )( T ( v )) ⊗ v ψ = ( ξ ( s ) ⊗ id C )( v ⊗ g ψ ⊗ v ψ ) (3.6)This shows that ξ ( s ) ∶ V ⊗ H r ( s ) Ð→ N ( s ) is a morphism in Comod − C . Together with (3.5), it follows that ξ ∶ V ⊗ H r Ð→ N is a morphism in M C R ( ψ ) . Finally, we see that for v ⊗ g ∈ V ⊗ H r ( s ) , we have ( η ( s ) ○ ξ ( s ))( v ⊗ g ) = η ( s )( N ( g )( T ( v )))= M ( g )( η ( r )( T ( v )))= M ( g )( ζ ( r )( v ⊗ id r ))= ζ ( s )(( V ⊗ H r )( g )( v ⊗ id r ))= ζ ( s )( v ⊗ g ) (3.7)This gives us η ○ ξ = ζ ∶ V ⊗ H r Ð→ M . Hence the result. Theorem 3.5.
Let ( R , C, ψ ) be an entwining structure and let C be a right semiperfect K -coalgebra. Then, thecategory M C R ( ψ ) of entwined modules is a Grothendieck category with a set of projective generators.Proof. From [4, Proposition 2.9], we know that M C R ( ψ ) is a Grothendieck category. Let M be an object of M C R ( ψ ) .From the proof of [4, Proposition 2.9], we know that there exists an epimorphism η ′ ∶ ⊕ i ∈ I V ′ i ⊗ H r i Ð→ M (3.8)9here each r i ∈ R and each V ′ i is a finite dimensional C -comodule. Since Comod − C has enough projectives, itfollows from [18, Corollary 2.4.21] that we can choose for each V ′ i an epimorphism V i Ð→ V ′ i in Comod − C such that V i is a finite dimensional projective in Comod − C . This induces an epimorphism η ∶ ⊕ i ∈ I V i ⊗ H r i Ð→ M (3.9)The collection { V ⊗ H r } now gives a set of projective generators for M C R ( ψ ) , where r ∈ R and V ranges over(isomorphism classes of) finite dimensional projective C -comodules. We fix a K -coalgebra C which is right semiperfect. We consider the category E nt C whose objects are entwiningstructures ( R , C, ψ ) . A morphism in E nt C is a map ( α, id ) ∶ ( R , C, ψ ) Ð→ ( R ′ , C, ψ ′ ) of entwining structures, whichwe will denote simply by α . From Section 2, it follows that we have adjoint functors α ∗ = ( α, id C ) ∗ ∶ M C R ( ψ ) Ð→ M C R ′ ( ψ ′ ) α ∗ = ( α, id C ) ∗ ∶ M C R ′ ( ψ ′ ) Ð→ M C R ( ψ ) (4.1)We note in particular that the functors α ∗ = ( α, id C ) ∗ are exact. In fact, the functors α ∗ preserve both limits andcolimits. Definition 4.1.
Let X be a small category. Let C be a right semiperfect coalgebra over the field K . By an entwined C -representation of a small category, we will mean a functor R ∶ X Ð→ E nt C .In particular, for each object x ∈ X , we have an entwining structure ( R x , C, ψ x ) . Given a morphism α ∶ x Ð→ y in X , we have a morphism R α = ( R α , id C ) ∶ ( R x , C, ψ x ) Ð→ ( R y , C, ψ y ) of entwining structures. By abuse of notation, if R ∶ X Ð→ E nt C is an entwined C -representation, we will write α ∗ = R ∗ α ∶ M C R x ( ψ x ) Ð→ M C R y ( ψ y ) α ∗ = R α ∗ ∶ M C R y ( ψ y ) Ð→ M C R x ( ψ x ) (4.2)for any morphism α ∶ x Ð→ y in X . Also by abuse of notation, if f ∶ r ′ Ð→ r is a morphism in R x , we will oftendenote R α ( f ) ∶ R α ( r ′ ) Ð→ R α ( r ) in R y simply as α ( f ) ∶ α ( r ′ ) Ð→ α ( r ) . We will now consider modules over anentwined C -representation. Definition 4.2.
Let R ∶ X Ð→ E nt C be an entwined C -representation of a small category X . An entwined module M over R will consist of the following data(1) For each object x ∈ X , an entwined module M x ∈ M C R x ( ψ x ) .(2) For each morphism α ∶ x Ð→ y in X , a morphism M α ∶ M x Ð→ α ∗ M y in M C R x ( ψ x ) (equivalently, a morphism M α ∶ α ∗ M x Ð→ M y in M C R y ( ψ y ) ).Further, we suppose that M id x = id M x for each x ∈ X and that for any composable morphisms x α Ð→ y β Ð→ z , wehave α ∗ ( M β ) ○ M α = M βα ∶ M x Ð→ α ∗ M y Ð→ α ∗ β ∗ M z = ( βα ) ∗ M z . The latter condition may be expressed in anyof two equivalent ways M βα = α ∗ ( M β ) ○ M α ⇔ M βα = M β ○ β ∗ ( M α ) (4.3) A morphism η ∶ M Ð→ N of entwined modules over R consists of morphisms η x ∶ M x Ð→ N x in each M C R x ( ψ x ) such that the following diagram commutes M x η x ÐÐÐÐ→ N x M α ×××Ö ×××Ö N α α ∗ M y α ∗ η y ÐÐÐÐ→ α ∗ N y (4.4) for each α ∶ x Ð→ y in R . The category of entwined modules over R will be denoted by M od C − R . roposition 4.3. Let R ∶ X Ð→ E nt C be an entwined C -representation of a small category X . Then, M od C − R is an abelian category.Proof. Let η ∶ M Ð→ N be a morphism M od C − R . We define the kernel and cokernel of η by setting Ker ( η ) x ∶ = Ker ( η x ∶ M x Ð→ N x ) Cok ( η ) x ∶ = Cok ( η x ∶ M x Ð→ N x ) (4.5)for each x ∈ X . For α ∶ x Ð→ y in X , the morphisms Ker ( η ) α and Cok ( η ) α are induced in the obvious manner,using the fact that α ∗ ∶ M C R y ( ψ y ) Ð→ M C R x ( ψ x ) is exact. From this, it is also clear that Cok ( Ker ( η ) ↪ M ) = Ker ( N ↠ Cok ( η )) .We now let R ∶ X Ð→ E nt C be an entwined C -representation of a small category X and let M be an entwinedmodule over R . We consider some x ∈ X and a morphism η ∶ V ⊗ H r Ð→ M x (4.6)in M C R x ( ψ x ) , where V is a finite dimensional projective in Comod − C and r ∈ R x . For each y ∈ X , we now set N y ⊆ M y to be the image of the family of maps N y = Im ( ⊕ β ∈ X ( x,y ) β ∗ ( V ⊗ H r ) ⊕ β ∗ η ÐÐÐÐ→ β ∗ M x M β ÐÐÐÐ→ M y )= ∑ β ∈ X ( x,y ) Im ( β ∗ ( V ⊗ H r ) β ∗ η ÐÐÐÐ→ β ∗ M x M β ÐÐÐÐ→ M y ) (4.7)We denote by ι y the inclusion ι y ∶ N y ↪ M y . For each β ∈ X ( x, y ) , we denote by η ′ β ∶ β ∗ ( V ⊗ H r ) Ð→ N y thecanonical morphism induced from (4.7). Lemma 4.4.
For any α ∈ X ( y, z ) , β ∈ X ( x, y ) , the following composition β ∗ ( V ⊗ H r ) η ′ β ÐÐÐÐ→ N y ι y ÐÐÐÐ→ M y M α ÐÐÐÐ→ α ∗ M z (4.8) factors through α ∗ ( ι z ) ∶ α ∗ N z Ð→ α ∗ M z .Proof. Since ( α ∗ , α ∗ ) is an adjoint pair, it suffices to show that the composition α ∗ β ∗ ( V ⊗ H r ) α ∗ ( η ′ β ) ÐÐÐÐ→ α ∗ N y α ∗ ( ι y ) ÐÐÐÐ→ α ∗ M y M α ÐÐÐÐ→ M z (4.9)factors through ι z ∶ N z Ð→ M z . By definition, we know that the composition β ∗ ( V ⊗ H r ) η ′ β Ð→ N y ι y Ð→ M y factorsthrough β ∗ M x , i.e., we have ι y ○ η ′ β = M β ○ β ∗ η (4.10)Applying α ∗ , composing with M α and using (4.3), we get M α ○ α ∗ ( ι y ) ○ α ∗ ( η ′ β ) = M α ○ α ∗ ( M β ) ○ α ∗ ( β ∗ η ) = M αβ ○ α ∗ β ∗ η (4.11)From the definition in (4.7), it is now clear that the composition M α ○ α ∗ ( ι y ) ○ α ∗ ( η ′ β ) = M αβ ○ α ∗ β ∗ η factors through ι z ∶ N z Ð→ M z as M α ○ α ∗ ( ι y ) ○ α ∗ ( η ′ β ) = ι z ○ η ′ αβ . Proposition 4.5.
For any α ∈ X ( y, z ) , the morphism M α ∶ M y Ð→ α ∗ M z restricts to a morphism N α ∶ N y Ð→ α ∗ N z , giving us a commutative diagram M y M α ÐÐÐÐ→ α ∗ M zι y Õ××× Õ××× α ∗ ( ι z ) N y N α ÐÐÐÐ→ α ∗ N z (4.12)11 roof. We already know that ι z ∶ N z Ð→ M z is a monomorphism. Since α ∗ is a right adjoint, it follows that α ∗ ( ι z ) is also a monomorphism. Since C is right semiperfect, we know from Theorem 3.5 that M C R y ( ψ y ) is a Grothendieckcategory with projective generators { G k } k ∈ K .Using Lemma 3.2, it suffices to show that for any k ∈ K and anymorphism ξ k ∶ G k Ð→ N y , there exists ξ ′ k ∶ G k Ð→ α ∗ N z such that α ∗ ( ι z ) ○ ξ ′ k = M α ○ ι y ○ ξ k .From (4.7), we have an epimorphism ⊕ β ∈ X ( x,y ) η ′ β ∶ ⊕ β ∈ X ( x,y ) β ∗ ( V ⊗ H r ) Ð→ N y (4.13)Since G k is projective, we can lift ξ k ∶ G k Ð→ N y to a morphism ξ ′′ k ∶ G k Ð→ ⊕ β ∈ X ( x,y ) β ∗ ( V ⊗ H r ) such that ξ k = ⎛⎝ ⊕ β ∈ X ( x,y ) η ′ β ⎞⎠ ○ ξ ′′ k (4.14)From Lemma 4.4, we know that M α ○ ι y ○ η ′ β factors through α ∗ ( ι z ) ∶ α ∗ N z Ð→ α ∗ M z for each β ∈ X ( x, y ) . Theresult is now clear.Using the adjointness of ( α ∗ , α ∗ ) , we can also obtain a morphism N α ∶ α ∗ N y Ð→ N z for each α ∈ X ( y, z ) ,corresponding to the morphism N α ∶ N y Ð→ α ∗ N z in (4.12). The objects { N y ∈ M C R y ( ψ y )} y ∈ X , together with themorphisms { N α } α ∈ Mor ( X ) determine an object of M od C − R that we denote by N . Additionally, Proposition 4.5shows that we have an inclusion ι ∶ N ↪ M in M od C − R . Before we proceed further, we will describe the object N in a few more ways. Lemma 4.6.
Let η ′ ∶ V ⊗ H r Ð→ N x be the canonical morphism corresponding to the identity map in X ( x, x ) .Then, for any y ∈ X , we have N y = Im ⎛⎝ ⊕ β ∈ X ( x,y ) β ∗ ( V ⊗ H r ) ⊕ β ∗ η ′ ÐÐÐÐ→ β ∗ N x N β ÐÐÐÐ→ N y ⎞⎠ (4.15) Proof.
For any β ∈ X ( x, y ) , we consider the commutative diagram β ∗ ( V ⊗ H r ) β ∗ η ′ ÐÐÐÐ→ β ∗ N x N β ÐÐÐÐ→ N yβ ∗ ( ι x ) ×××Ö ×××Ö ι y β ∗ M x M β ÐÐÐÐ→ M y (4.16)By definition, we know that ι x ○ η ′ = η , which gives β ∗ ( ι x ) ○ β ∗ ( η ′ ) = β ∗ ( η ) . Composing with M β , we get Im ( M β ○ β ∗ ( η )) = Im ( M β ○ β ∗ ( ι x ) ○ β ∗ ( η ′ )) = Im ( ι y ○ N β ○ β ∗ η ′ ) ≅ Im ( N β ○ β ∗ η ′ ) (4.17)where the last isomorphism follows from the fact that ι y is monic. The result is now clear from the definition in(4.7). Lemma 4.7.
For any y ∈ X , we have N y = ∑ β ∈ X ( x,y ) Im ( β ∗ N x β ∗ ( ι x ) ÐÐÐÐ→ β ∗ M x M β ÐÐÐÐ→ M y ) (4.18)12 roof. For the sake of convenience, we set N ′ y ∶ = ∑ β ∈ X ( x,y ) Im ( β ∗ N x β ∗ ( ι x ) ÐÐÐÐ→ β ∗ M x M β ÐÐÐÐ→ M y ) From the commutative diagram in (4.16), we see that each of the morphisms β ∗ N x β ∗ ( ι x ) ÐÐÐÐ→ β ∗ M x M β ÐÐÐÐ→ M y factors through the subobject N y ⊆ M y . Hence, N ′ y ⊆ N y . On the other hand, it is clear that Im ( β ∗ ( V ⊗ H r ) β ∗ η ′ ÐÐÐÐ→ β ∗ N x β ∗ ( ι x ) ÐÐÐÐ→ β ∗ M x M β ÐÐÐÐ→ M y ) ⊆ Im ( β ∗ N x β ∗ ( ι x ) ÐÐÐÐ→ β ∗ M x M β ÐÐÐÐ→ M y ) Applying Lemma 4.6, it is now clear that N y ⊆ N ′ y . This proves the result.We now make a few conventions : if M is a module over a small K -linear category R , we denote by el ( M ) the union ⋃ r ∈R M ( r ) . The cardinality of el ( M ) will be denoted by ∣ M ∣ . If M is a module over an entwined C -representation R ∶ X Ð→ E nt C , we denote by el X ( M ) the union ⋃ x ∈ X el ( M x ) . The cardinality of el X ( M ) will be denoted by ∣ M ∣ . It is evident that if M ∈ M od C − R and N is either a quotient or a subobject of M , then ∣ N ∣ ≤ ∣ M ∣ .We now define the following cardinality κ = sup {∣ N ∣ , ∣ C ∣ , ∣ K ∣ , ∣ M or ( X )∣ , ∣ M or ( R x )∣ , x ∈ X } (4.19)We observe that ∣ β ∗ ( V ⊗ H r )∣ ≤ κ , where V is any finite dimensional C -comodule and β ∈ X ( x, y ) . Lemma 4.8.
We have ∣ N ∣ ≤ κ .Proof. We choose y ∈ X . From Lemma 4.6, we have N y = Im ⎛⎝ ⊕ β ∈ X ( x,y ) β ∗ ( V ⊗ H r ) ⊕ β ∗ η ′ ÐÐÐÐ→ β ∗ N x N β ÐÐÐÐ→ N y ⎞⎠ (4.20)Since N y is an epimorphic image of ⊕ β ∈ X ( x,y ) β ∗ ( V ⊗ H r ) , we have ∣ N y ∣ ≤ ∣ ⊕ β ∈ X ( x,y ) β ∗ ( V ⊗ H r )∣ ≤ κ (4.21)It follows that ∣ N ∣ = ∑ y ∈ X ∣ N y ∣ ≤ κ . Theorem 4.9.
Let C be a right semiperfect coalgebra over a field K . Let R ∶ X Ð→ E nt C be an entwined C -representation of a small category X . Then, the category M od C − R of entwined modules over R is a Grothendieckcategory.Proof. Since filtered colimits and finite limits in
M od C − R are computed pointwise, it is clear that they commutewith each other.We now consider an object M in M od C − R and an element m ∈ el X ( M ) . Then, m ∈ M x ( r ) for some x ∈ X and r ∈ R x . By [4, Lemma 2.8], we can find a finite dimensionsal C -subcomodule V ′ ⊆ M x ( r ) containing m and amorphism η ′ ∶ V ′ ⊗ H r Ð→ M x in M C R x ( ψ x ) such that η ′ ( r )( m ⊗ id r ) = m . Since C is semiperfect, we can choosea finite dimensional projective V in Comod − C along with an epimorphism V Ð→ V ′ . This induces a morphism η ∶ V ⊗ H r Ð→ M x in M C R x ( ψ x ) . Corresponding to η , we now define the subobject N ⊆ M as in (4.7). It is clearthat m ∈ el X ( N ) . By Lemma 4.8, we know that ∣ N ∣ ≤ κ .We now consider the set of isomorphism classes of objects in M od C − R having cardinality ≤ κ . From the above, itis clear that any object in M od C − R may be expressed as a sum of such objects. By choosing one object from eachsuch isomorphism class, we obtain a set of generators for M od C − R .13 Entwined representations of a poset and projective generators
In this section, the small category X will always be a partially ordered set. If x ≤ y in X , we will say that thereis a single morphism x Ð→ y in X . We continue with C being a right semiperfect coalgebra over the field K and R ∶ X Ð→ E nt C being an entwined C -representation of X . From Theorem 4.9, we know that M od C − R is aGrothendieck category.In this section, we will show that M od C − R has projective generators. For this, we will construct a pair of adjointfunctors ex Cx ∶ M C R x ( ψ x ) Ð→ M od C − R ev Cx ∶ M od C − R Ð→ M C R x ( ψ x ) (5.1)for each x ∈ X . Lemma 5.1.
Let X be a poset. Fix x ∈ X . Then, there is a functor ex Cx ∶ M C R x ( ψ x ) Ð→ M od C − R defined bysetting ex Cx ( M ) y ∶ = { α ∗ M if α ∈ X ( x, y ) if X ( x, y ) = φ (5.2) for each y ∈ X .Proof. It is immediate that each ex Cx ( M ) y ∈ M C R y ( ψ y ) . We consider β ∶ y Ð→ y ′ in X . If x /≤ y , we have0 = ex Cx ( M ) β ∶ = β ∗ ex Cx ( M ) y Ð→ ex Cx ( M ) y ′ in M C R y ′ ( ψ y ′ ) . Otherwise, we consider α ∶ x Ð→ y and α ′ ∶ x Ð→ y ′ .Then, we have id = ex Cx ( M ) β ∶ β ∗ ex Cx ( M ) y = β ∗ α ∗ M Ð→ α ′∗ M = ex Cx ( M ) y ′ which follows from the fact that β ○ α = α ′ . Given composable morphisms β , γ in X , it is now clear from thedefinitions that ex Cx ( M ) γβ = ex Cx ( M ) γ ○ γ ∗ ( ex Cx ( M ) β ) . Lemma 5.2.
Let X be a poset. Fix x ∈ X . Then, there is a functor ev Cx ∶ M od C − R Ð→ M C R x ( ψ x ) M ↦ M x (5.3) Additionally, ev Cx is exact.Proof. It is immediate that ev Cx is a functor. Since finite limits and finite colimits in M od C − R are computedpointwise, it follows that ev Cx is exact. Proposition 5.3.
Let X be a poset. Fix x ∈ X . Then, ( ex Cx , ev Cx ) is a pair of adjoint functors.Proof. For any M ∈ M C R x ( ψ x ) and N ∈ M od C − R , we will show that M od C − R ( ex Cx ( M ) , N ) ≅ M C R x ( ψ x )( M , ev Cx ( N )) (5.4)We begin with a morphism f ∶ M Ð→ N x in M C R x ( ψ x ) . Corresponding to f , we define η f ∶ ex Cx ( M ) Ð→ N in M od C − R by setting η fy ∶ ex Cx ( M ) y = α ∗ M α ∗ f ÐÐ→ α ∗ N x N α ÐÐ→ N y (5.5)whenever x ≤ y and α ∈ X ( x, y ) . Otherwise, we set 0 = η fy ∶ = ex Cx ( M ) y Ð→ N y . For β ∶ y Ð→ y ′ in X , we have toshow that the following diagram is commutative. β ∗ ex Cx ( M ) y β ∗ η fy ÐÐÐÐ→ β ∗ N yex Cx (M) β ×××Ö ×××Ö N β ex Cx ( M ) y ′ η fy ′ ÐÐÐÐ→ N y ′ (5.6)14f x /≤ y , then ex Cx ( M ) y = α ∶ x Ð→ y and α ′ = β ○ α ∶ x Ð→ y ′ .Then, (5.6) reduces to the commutative diagram β ∗ α ∗ M β ∗ ( N α ○ α ∗ f ) ÐÐÐÐÐÐÐ→ β ∗ N yid ×××Ö ×××Ö N β β ∗ α ∗ M = α ′∗ M N α ′ ○ α ′∗ ( f )= N β ○ β ∗ ( N α )○ β ∗ α ∗ f ÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐ→ N y ′ (5.7)Conversely, we take η ∶ ex Cx ( M ) Ð→ N in M od C − R . In particular, this determines f η = η x ∶ M Ð→ N x in M C R x ( ψ x ) .It may be easily verified that these two associations are inverse to each other. This proves the result. Corollary 5.4.
The functor ex Cx ∶ M C R x ( ψ x ) Ð→ M od C − R preserves projectives.Proof. From Proposition 5.3, we know that ( ex Cx , ev Cx ) is a pair of adjoint functors. From Lemma 5.2, we know thatthe right adjoint ev Cx is exact. It follows therefore that its left adjoint ex Cx preserves projective objects. Theorem 5.5.
Let C be a right semiperfect coalgebra over a field K . Let X be a poset and let R ∶ X Ð→ E nt C bean entwined C -representation of X . Then, M od C − R has projective generators.Proof. We denote by
P roj f ( C ) the set of isomorphism classes of finite dimensional projective C -comodules. We willshow that the family G = { ex Cx ( V ⊗ H r ) ∣ x ∈ X , r ∈ R x , V ∈ P roj f ( C )} (5.8)is a set of projective generators for M od C − R . From Proposition 3.4, we know that V ⊗ H r is projective in M C R x ( ψ x ) ,where r ∈ R x and V ∈ P roj f ( C ) . It now follows from Corollary 5.4 that each ex Cx ( V ⊗ H r ) is projective in M od C − R .It remains to show that G is a set of generators for M od C − R . For this, we consider a monomorphism ι ∶ N ↪ M in M od C − R such that N ⊊ M . Since kernels and cokernels in M od C − R are taken pointwise, it follows that thereis some x ∈ X such that ι x ∶ N x ↪ M x is a monomorphism with N x ⊊ M x .From the proof of Theorem 3.5, we know that { V ⊗ H r } r ∈ R x ,V ∈ P roj f ( C ) is a set of generators for M C R x ( ψ x ) . Accord-ingly, we can choose a morphism f ∶ V ⊗ H r Ð→ M x with r ∈ R x and V ∈ P roj f ( C ) such that f does not factor through ev Cx ( ι ) = ι x ∶ N x ↪ M x . Applying the adjunction ( ex Cx , ev Cx ) , we now obtain a morphism η ∶ ex Cx ( V ⊗ H r ) Ð→ M corresponding to f , which does not factor through ι ∶ N Ð→ M . It now follows (see, for instance, [20, § G is a set of generators for M od C − R . We continue with X being a poset, C being a right semiperfect K -coalgebra and R ∶ X Ð→ E nt C being an entwined C -representation of X . In this section, we will introduce the category of cartesian modules over R .Given a morphism α ∶ ( R , C, ψ ) Ð→ ( S , C, ψ ′ ) in E nt C , we already know that the left adjoint α ∗ is right exact. Wewill say that α ∶ ( R , C, ψ ) Ð→ ( S , C, ψ ′ ) is flat if α ∗ ∶ M C R ( ψ ) Ð→ M C S ( ψ ′ ) is exact. Accordingly, we will say that anentwined C -representation R ∶ X Ð→ E nt C is flat if α ∗ = R ∗ α ∶ M C R x ( ψ x ) Ð→ M C R y ( ψ y ) is exact for each α ∶ x Ð→ y in X . Definition 6.1.
Let R ∶ X Ð→ E nt C be an entwined C -representation of X . Suppose that R is flat. Let M be an entwined module over R . We will say that M is cartesian if for each α ∶ x Ð→ y in X , the morphism M α ∶ α ∗ M x Ð→ M y in M C R y ( ψ y ) is an isomorphism.We will denote by Cart C − R the full subcategory of M od C − R consisting of cartesian modules.
15t is clear that
Cart C − R is an abelian category, with filtered colimits and finite limits coming from M od C − R .We will now give conditions so that Cart − R is a Grothendieck category. For this, we will need some intermediateresults. First, we recall (see, for instance, [2]) that an object M in a Grothendieck category A is said to be finitelygenerated if the functor A ( M, ) satisfies lim Ð→ i ∈ I A ( M, M i ) = A ( M, lim Ð→ i ∈ I M i ) (6.1)where { M i } i ∈ I is any filtered system of objects in A connected by monomorphisms. Proposition 6.2.
Let ( R , C, ψ ) be an entwining structure with C a right semiperfect coalgebra. Let V be a finitedimensional projective right C -comodule. Then, for any r ∈ R , the module V ⊗ H r is a finitely generated projectiveobject in M C R ( ψ ) .Proof. From Proposition 3.4, we already know that V ⊗ H r is a projective object in M C R ( ψ ) . To show that it isfinitely generated, we consider a filtered system { M i } i ∈ I of objects in M C R ( ψ ) connected by monomorphisms and set M ∶ = lim Ð→ i ∈ I M i . Since M C R ( ψ ) is a Grothendieck category, we note that we have an inclusion η i ∶ M i ↪ M for each i ∈ I .We now take a morphism ζ ∶ V ⊗ H r Ð→ M in M C R ( ψ ) . We choose a basis { v , ..., v n } for V . For each 1 ≤ k ≤ n , wenow have a morphism in M R given by ζ k ∶ H r Ð→ V ⊗ H r H r ( s ) = R ( s, r ) ∋ f ↦ v k ⊗ f ∈ ( V ⊗ H r )( s ) (6.2)Then, each composition ζ ○ ζ k ∶ H r Ð→ M is a morphism in M R . Since H r is a finitely generated object in M R , wecan now choose j ∈ I such that every ζ ○ ζ k factors through η j ∶ M j ↪ M . We now construct the following pullbackdiagram in M C R ( ψ ) N ÐÐÐÐ→ M jι ×××Ö ×××Ö η j V ⊗ H r ζ ÐÐÐÐ→ M (6.3)Then, ι ∶ N Ð→ M is a monomorphism in M C R ( ψ ) . From the construction of finite limits in M C R ( ψ ) , it follows thatfor each s ∈ R , we have a pullback diagram in V ect K N ( s ) ÐÐÐÐ→ M j ( s ) ι ( s ) ×××Ö ×××Ö η j ( s ) ( V ⊗ H r )( s ) ζ ( s ) ÐÐÐÐ→ M ( s ) (6.4)By assumption, we know that ζ ( s )( v k ⊗ f ) ∈ Im ( η j ( s )) for any basis element v k and any f ∈ H r ( s ) . It follows that Im ( ζ ( s )) ⊆ Im ( η j ( s )) and hence the pullback N ( s ) = ( V ⊗ H r )( s ) . In other words, N = V ⊗ H r . The result is nowclear. Lemma 6.3.
Let α ∶ ( R , C, ψ ) Ð→ ( S , C, ψ ′ ) be a flat morphism in E nt C . Let M ∈ M C R ( ψ ) .(a) There exists a family { r i } i ∈ I of objects of R and a family { V i } i ∈ I of finite dimensional projective C -comodulessuch that there is an epimorphism in M C S ( ψ ′ ) η ∶ ⊕ i ∈ I ( V i ⊗ H α ( r i ) ) Ð→ α ∗ M (6.5)16 b) Let s ∈ S and let W be a finite dimensional projective in Comod − C . Let ζ ∶ W ⊗ H s Ð→ α ∗ M be a morphism in M C S ( ψ ′ ) . Then, there exists a finite set { r , ..., r n } of objects of R , a finite family { V , ..., V n } of finite dimensionalprojective C -comodules and a morphism η ′′ ∶ n ⊕ k = V k ⊗ H r k Ð→ M in M C R ( ψ ) such that ζ factors through α ∗ η ′′ .Proof. (a) From the proof of Theorem 3.5, we know that there exists an epimorphism in M C R ( ψ ) η ′ ∶ ⊕ i ∈ I V i ⊗ H r i Ð→ M (6.6)where each r i ∈ R and each V i is a finite dimensional projective C -comodule. Since α ∗ ∶ M C R ( ψ ) Ð→ M C S ( ψ ′ ) isa left adjoint, it induces an epimorphism α ∗ ( η ′ ) in M C S ( ψ ′ ) . From the definition in (2.2) and the construction inProposition 2.3, it is clear that α ∗ ( V i ⊗ H r i ) = V i ⊗ α ∗ H r i = V i ⊗ H α ( r i ) . This proves (a).(b) We consider the epimorphism α ∗ η ′ = η ∶ ⊕ i ∈ I ( V i ⊗ H α ( r i ) ) Ð→ α ∗ M constructed in (a). From Proposition 6.2, weknow that W ⊗ H s is a finitely generated projective object in M C S ( ψ ′ ) . As such ζ ∶ W ⊗ H s Ð→ α ∗ M can be liftedto a morphism ζ ′ ∶ W ⊗ H s Ð→ ⊕ i ∈ I ( V i ⊗ H α ( r i ) ) and ζ ′ factors through a finite direct sum of objects from the family { V i ⊗ H α ( r i ) } i ∈ I . The result is now clear. Lemma 6.4.
Let α ∶ ( R , C, ψ ) Ð→ ( S , C, ψ ′ ) be a flat morphism in E nt C . Let κ be any cardinal such that κ ≥ max { N , ∣ M or ( R )∣ , ∣ C ∣ , ∣ K ∣} (6.7) Let M ∈ M C R ( ψ ) and let A ⊆ el ( α ∗ M ) be a set of elements such that ∣ A ∣ ≤ κ . Then, there is a submodule N ↪ M in M C R ( ψ ) with ∣ N ∣ ≤ κ such that A ⊆ el ( α ∗ N ) .Proof. We consider some element a ∈ A ⊆ el ( α ∗ M ) . Then, we can choose a morphism ζ a ∶ W a ⊗ H s a Ð→ α ∗ M in M C S ( ψ ′ ) such that a ∈ el ( Im ( ζ a )) , where s a ∈ S and W a is a finite dimensional projective in Comod − C . UsingLemma 6.3(b), we can now choose a finite set { r a , ..., r an a } of objects of R , a finite family { V a , ..., V an a } of finitedimensional projective C -comodules and a morphism η a ′′ ∶ n a ⊕ k = V ak ⊗ H r ak Ð→ M in M C R ( ψ ) such that ζ a factorsthrough α ∗ η a ′′ . We now set N ∶ = Im ( η ′′ ∶ = ⊕ a ∈ A η a ′′ ∶ ⊕ a ∈ A n a ⊕ k = V ak ⊗ H r ak Ð→ M ) (6.8)Since α is flat and α ∗ is a left adjoint, we obtain α ∗ N = Im ( α ∗ η ′′ = ⊕ a ∈ A α ∗ η a ′′ ∶ ⊕ a ∈ A n a ⊕ k = V ak ⊗ H α ( r ak ) Ð→ α ∗ M ) (6.9)Since each a ∈ el ( Im ( ζ a )) and ζ a factors through α ∗ η a ′′ , we get A ⊆ el ( α ∗ N ) .It remains to show that ∣ N ∣ ≤ κ . Since N is a quotient of ⊕ a ∈ A n a ⊕ k = V ak ⊗ H r ak and ∣ A ∣ ≤ κ , it suffices to show that each ∣ V ak ⊗ H r ak ∣ ≤ κ . This is clear from the definition of κ , using the fact that each V ak is finite dimensional. Remark 6.5.
By considering α = id in Lemma 6.4, we obtain the following simple consequence: if A ⊆ el ( M ) is anysubset with ∣ A ∣ ≤ κ , there is a submodule N ↪ M in M C R ( ψ ) with ∣ N ∣ ≤ κ such that A ⊆ el ( N ) .17 emma 6.6. Let α ∶ ( R , C, ψ ) Ð→ ( S , C, ψ ′ ) be a flat morphism in E nt C and let M ∈ M C R ( ψ ) . Let κ be anycardinal such that κ ≥ max { N , ∣ M or ( R )∣ , ∣ M or ( S )∣ , ∣ C ∣ , ∣ K ∣} and let A ⊆ el ( M ) and B ⊆ el ( α ∗ M ) be subsets with ∣ A ∣ , ∣ B ∣ ≤ κ . Then, there exists a submodule N ⊆ M in M C R ( ψ ) such that(1) ∣ N ∣ ≤ κ , ∣ α ∗ N ∣ ≤ κ (2) A ⊆ el ( N ) and B ⊆ el ( α ∗ N ) .Proof. Applying Lemma 6.4 (and Remark 6.5), we obtain submodules N , N ⊆ M such that(1) ∣ N ∣ , ∣ N ∣ ≤ κ (2) A ⊆ el ( N ) , B ⊆ el ( α ∗ N ) .We set N ∶ = ( N + N ) ⊆ M . Then, ( N + N ) is a quotient of N ⊕ N and hence ∣ N ∣ ≤ κ . Also, it is clear that A ⊆ el ( N ) ⊆ el ( N ) . Since α is flat, we get B ⊆ el ( α ∗ N ) ⊆ el ( α ∗ N ) .It remains to show that ∣ α ∗ N ∣ ≤ κ . By the definition in (2.2), we know that α ∗ ( N )( s ) is a quotient of ( ⊕ r ∈R N ( r ) ⊗ S ( s, α ( r ))) (6.10)for each s ∈ S . Since κ ≥ ∣ M or ( R )∣ , ∣ M or ( S )∣ , it follows from (6.10) that ∣ α ∗ ( N )( s )∣ ≤ κ . Again since κ ≥ ∣ M or ( S )∣ ,we get ∣ α ∗ N ∣ ≤ κ .We will now show that Cart C − R has a generator when R ∶ X Ð→ E nt C is a flat representation of the poset X .This will be done using induction on N × M or ( X ) in a manner similar to the proof of [19, Proposition 3.25]. As inSection 4, we set κ = sup {∣ N ∣ , ∣ C ∣ , ∣ K ∣ , ∣ M or ( X )∣ , ∣ M or ( R x )∣ , x ∈ X } (6.11)Let M be a cartesian module over R ∶ X Ð→ E nt C . We now consider an element m ∈ el X ( M ) . Suppose that m ∈ M x ( r ) for some x ∈ X and r ∈ R x . As in the proof of Theorem 4.9, we fix a finite dimensional projective C -comodule V and a morphism η ∶ V ⊗ H r Ð→ M x in M C R x ( ψ x ) such that m is an element of the image of η .Corresponding to η , we define N ⊆ M as in (4.7). It is clear that m ∈ el X ( N ) . By Lemma 4.8, we know that ∣ N ∣ ≤ κ .Next, we choose a well ordering of the set M or ( X ) and consider the induced lexicographic ordering of N × M or ( X ) .Corresponding to each pair ( n, α ∶ y Ð→ z ) ∈ N × M or ( X ) , we will now define a subobject P ( n, α ) ↪ M in M od C − R satisfying the following conditions.(1) m ∈ el X ( P ( , α )) , where α is the least element of M or ( X ) .(2) P ( n, α ) ⊆ P ( m, β ) , whenever ( n, α ) ≤ ( m, β ) in N × M or ( X ) (3) For each ( n, α ∶ y Ð→ z ) ∈ N × M or ( X ) , the morphism P ( n, α ) α ∶ α ∗ P ( n, α ) y Ð→ P ( n, α ) z is an isomorphismin M C R z ( ψ z ) .(4) ∣ P ( n, α )∣ ≤ κ .For ( n, α ∶ y Ð→ z ) ∈ N × M or ( X ) , we start the process of constructing the module P ( n, α ) as follows: we set A ( w ) ∶ = ⎧⎪⎪⎨⎪⎪⎩ N w if n = α = α ⋃ ( m,β )<( n,α ) P ( m, β ) w otherwise (6.12)for each w ∈ X . It is clear that each A ( w ) ⊆ el ( M w ) and ∣ A ( w )∣ ≤ κ .Since M is cartesian, we know that α ∗ M y = M z . Since α ∶ ( R y , C, ψ y ) Ð→ ( R z , C, ψ z ) is flat in E nt C , we use Lemma6.6 with A ( y ) ⊆ el ( M y ) and A ( z ) ⊆ el ( α ∗ M y ) = el ( M z ) to obtain A ( y ) ↪ M y in M C R y ( ψ y ) such that ∣ A ( y )∣ ≤ κ ∣ α ∗ A ( y )∣ ≤ κ A ( y ) ⊆ el ( A ( y )) A ( z ) ⊆ el ( α ∗ A ( y )) (6.13)18e now set A ( z ) ∶ = α ∗ A ( y ) . Then, (6.13) can be rewritten as ∣ A ( y )∣ ≤ κ ∣ A ( z )∣ ≤ κ A ( y ) ⊆ el ( A ( y )) A ( z ) ⊆ el ( A ( z )) (6.14)We observe here that since X is a poset, then y = z implies α ∶ y Ð→ z is the identity and hence A ( y ) = A ( z ) . Forany w ≠ y, z in X , we set A ( w ) = A ( w ) . Combining with (6.14), we have A ( w ) ⊆ A ( w ) for every w ∈ X andeach ∣ A ( w )∣ ≤ κ . Lemma 6.7.
Let B ⊆ el X ( M ) with ∣ B ∣ ≤ κ . Then, there is a submodule Q ↪ M in M od C − R such that B ⊆ el X ( Q ) and ∣ Q ∣ ≤ κ .Proof. For any m ∈ B ⊆ el X ( M ) we can choose, as in the proof of Theorem 4.9, a subobject Q m ⊆ M such that m ∈ el X ( Q m ) and ∣ Q m ∣ ≤ κ . Then, we set Q ∶ = ∑ m ∈ B Q m . In particular, Q is a quotient of ⊕ m ∈ B Q m . Since ∣ B ∣ ≤ κ , theresult follows.Using Lemma 6.7, we now choose a submodule Q ( n, α ) ↪ M in M od C − R such that ⋃ w ∈ X A ( w ) ⊆ el X ( Q ( n, α )) and ∣ Q ( n, α )∣ ≤ κ . In particular, A ( w ) ⊆ Q ( n, α ) w for each w ∈ X .We now iterate this construction. Suppose we have constructed a submodule Q l ( n, α ) ↪ M for every l ≤ m suchthat ⋃ w ∈ X A l ( w ) ⊆ el X ( Q l ( n, α )) and ∣ Q l ( n, α )∣ ≤ κ . Then, we set A m + ( w ) ∶ = Q m ( n, α ) w for each w ∈ X . We thenuse Lemma 6.6 with A m + ( y ) ⊆ el ( M y ) and A m + ( z ) ⊆ el ( α ∗ M y ) = el ( M z ) to obtain A m + ( y ) ↪ M y in M C R y ( ψ y ) such that ∣ A m + ( y )∣ ≤ κ ∣ α ∗ A m + ( y )∣ ≤ κ A m + ( y ) ⊆ el ( A m + ( y )) A m + ( z ) ⊆ el ( α ∗ A m + ( y )) (6.15)We now set A m + ( z ) ∶ = α ∗ A m + ( y ) . Then, (6.15) can be rewritten as ∣ A m + ( y )∣ ≤ κ ∣ A m + ( z )∣ ≤ κ A m + ( y ) ⊆ el ( A m + ( y )) A m + ( z ) ⊆ el ( A m + ( z )) (6.16)For any w ≠ y, z in X , we set A m + ( w ) = A m + ( w ) . Combining with (6.16), we have A m + ( w ) ⊆ A m + ( w ) for every w ∈ X and each ∣ A m + ( w )∣ ≤ κ .Using Lemma 6.7, we now choose a submodule Q m + ( n, α ) ↪ M in M od C − R such that ⋃ w ∈ X A m + ( w ) ⊆ el X ( Q m + ( n, α )) and ∣ Q m + ( n, α )∣ ≤ κ . In particular, A m + ( w ) ⊆ Q m + ( n, α ) w for each w ∈ X .Finally, we set P ( n, α ) ∶ = lim Ð→ m ≥ Q m ( n, α ) (6.17)in M od C − R . Lemma 6.8.
The family { P ( n, α ) ∣ ( n, α ) ∈ N × M or ( X )} satisfies the following conditions.(1) m ∈ el X ( P ( , α )) , where α is the least element of M or ( X ) .(2) P ( n, α ) ⊆ P ( m, β ) , whenever ( n, α ) ≤ ( m, β ) in N × M or ( X ) (3) For each ( n, α ∶ y Ð→ z ) ∈ N × M or ( X ) , the morphism P ( n, α ) α ∶ α ∗ P ( n, α ) y Ð→ P ( n, α ) z is an isomorphismin M C R z ( ψ z ) .(4) ∣ P ( n, α )∣ ≤ κ .Proof. The conditions (1) and (2) are immediate from the definition in (6.12). The condition (4) follows from (6.17)and the fact that each ∣ Q m + ( n, α )∣ ≤ κ . 19o prove (3), we notice that P ( n, α ) y may be expressed as the filtered union A ( y ) ↪ Q ( n, α ) y ↪ A ( y ) ↪ Q ( n, α ) y ↪ ⋅ ⋅ ⋅ ↪ A m + ( y ) ↪ Q m + ( n, α ) y ↪ ... (6.18)of objects in M C R y ( ψ y ) . Since α ∗ is exact and a left adjoint, we can express α ∗ P ( n, α ) y as the filtered union α ∗ A ( y ) ↪ α ∗ Q ( n, α ) y ↪ α ∗ A ( y ) ↪ α ∗ Q ( n, α ) y ↪ ⋅ ⋅ ⋅ ↪ α ∗ A m + ( y ) ↪ α ∗ Q m + ( n, α ) y ↪ ... (6.19)in M C R z ( ψ z ) . Similarly, P ( n, α ) z may be expressed as the filtered union A ( z ) ↪ Q ( n, α ) z ↪ A ( z ) ↪ Q ( n, α ) z ↪ ⋅ ⋅ ⋅ ↪ A m + ( z ) ↪ Q m + ( n, α ) z ↪ ... (6.20)in M C R z ( ψ z ) . By definition, we know that A m ( z ) = α ∗ A m ( y ) for each m ≥
0. From (6.19) and (6.20), it is clear that thefiltered colimit of the isomorphisms α ∗ A m ( y ) = A m ( z ) induces an isomorphism P ( n, α ) α ∶ α ∗ P ( n, α ) y Ð→ P ( n, α ) z . Lemma 6.9.
Let M be a cartesian module over a flat representation R ∶ X Ð→ E nt C . Choose m ∈ el X ( M ) .Let κ = max {∣ N ∣ , ∣ C ∣ , ∣ K ∣ , ∣ M or ( X )∣ , ∣ M or ( R x )∣ , x ∈ X } . Then, there is a cartesian submodule P ⊆ M with m ∈ el X ( P ) such that ∣ P ∣ ≤ κ .Proof. It is clear that N × M or ( X ) with the lexicographic ordering is filtered. We set P ∶ = ⋃ ( n,α )∈ N × Mor ( X ) P ( n, α ) ⊆ M (6.21)in M od C − R . It is immediate that m ∈ el X ( P ) . Since each ∣ P ( n, α )∣ ≤ κ , it is clear that ∣ P ∣ ≤ κ .We now consider a morphism β ∶ z Ð→ w in X . Then, the family {( m, β )} m ≥ is cofinal in N × M or ( X ) and henceit follows that P ∶ = lim Ð→ m ≥ P ( m, β ) (6.22)Since each P ( m, β ) β ∶ β ∗ P ( m, β ) z Ð→ P ( m, β ) w is an isomorphism, the filtered colimit P β ∶ β ∗ P z Ð→ P w is anisomorphism. Theorem 6.10.
Let C be a right semiperfect coalgebra over a field K . Let X be a poset and let R ∶ X Ð→ E nt C be an entwined C -representation of X . Suppose that R is flat. Then, Cart C − R is a Grothendieck category.Proof. It is already clear that
Cart C − R satisfies the (AB5) condition. From Lemma 6.9, it is clear that any M ∈ Cart C − R can be expressed as a sum of a family { P m } m ∈ el X ( M ) of cartesian submodules such that each ∣ P m ∣ ≤ κ . As such, isomorphism classes of cartesian modules P with ∣ P ∣ ≤ κ form a family of generators for Cart C − R . Let ( R , C, ψ ) be an entwining structure. We consider the forgetful functor F ∶ M C R ( ψ ) Ð→ M R . By [4, Lemma2.4 & Lemma 3.1], we know that F has a right adjoint G ∶ M R Ð→ M C R ( ψ ) given by setting G ( N ) ∶ = N ⊗ C , i.e. G ( N )( r ) ∶ = N ( r ) ⊗ C for each r ∈ R . The right R -module structure on G ( N ) is given by ( n ⊗ c ) ⋅ f ∶ = nf ψ ⊗ c ψ for f ∈ R ( r ′ , r ) , n ∈ N ( r ) and c ∈ C .We continue with X being a poset, C being a right semiperfect coalgebra and let R ∶ X Ð→ E nt C be an entwined C -representation. We denote by L in the category of small K -linear categories. Then, for each x ∈ X , we may replace20he entwining structure ( R x , C, ψ x ) by the K -linear category R x to obtain a functor that we continue to denote by R ∶ X Ð→ L in . We consider modules over R ∶ X Ð→ L in in the sense of Estrada and Virili [19, Definition 3.6]and denote their category by M od − R . Explicitly, an object N in M od − R consists of a module N x ∈ M R x for each x ∈ X as well as compatible morphisms N α ∶ N x Ð→ α ∗ N y (equivalently N α ∶ α ∗ N x Ð→ N y ) for each α ∶ x Ð→ y in X . The module N is said to be cartesian if each N α ∶ α ∗ N x Ð→ N y is an isomorphism. We denote by Cart − R the full subcategory of cartesian modules on R .For each x ∈ X , we have a forgetful functor F x ∶ M C R x ( ψ x ) Ð→ M R x having right adjoint G x ∶ M R x Ð→ M C R x ( ψ x ) .From the proofs of Propositions 2.3 and 2.4, it is clear that we have commutative diagrams M C R y ( ψ y ) α ∗ ÐÐÐÐ→ M C R x ( ψ x ) F y ×××Ö ×××Ö F x M R y α ∗ ÐÐÐÐ→ M R x M C R x ( ψ x ) α ∗ ÐÐÐÐ→ M C R y ( ψ y ) F x ×××Ö ×××Ö F y M R x α ∗ ÐÐÐÐ→ M R y M R y α ∗ ÐÐÐÐ→ M R x G y ×××Ö ×××Ö G x M C R y ( ψ y ) α ∗ ÐÐÐÐ→ M C R x ( ψ x ) (7.1)for each α ∶ x Ð→ y in X . Proposition 7.1.
Let R ∶ X Ð→ E nt C be an entwined C -representation. Then, the collection { F x ∶ M C R x ( ψ x ) Ð→ M R x } x ∈ X (resp. the collection { G x ∶ M R x Ð→ M C R x ( ψ x )} x ∈ X ) together defines a functor F ∶ M od C − R Ð→ M od − R (resp. a functor G ∶ M od − R Ð→ M od C − R ).Proof. We consider M ∈ M od C − R and set F ( M ) x ∶ = F x ( M x ) ∈ M R x . For a morphism α ∶ x Ð→ y , we obtainfrom (7.1) a morphism F ( M ) α ∶ = F x ( M α ) ∶ F x ( M x ) Ð→ F x ( α ∗ M y ) = α ∗ F y ( M y ) . This shows that F ( M ) is anobject of M od − R . Similarly, it follows from (7.1) that for any N ∈ M od − R , we have G ( N ) ∈ M od C − R obtainedby setting G ( N ) x ∶ = G x ( N x ) = N x ⊗ C . Proposition 7.2.
Let R ∶ X Ð→ E nt C be an entwined C -representation. Then, the functor F ∶ M od C − R Ð→ M od − R has a right adjoint, given by G ∶ M od − R Ð→ M od C − R .Proof. We consider M ∈ M od C − R and N ∈ M od − R along with a morphism η ∶ F ( M ) Ð→ N in M od − R . Wewill show how to construct a morphism ζ ∶ M Ð→ G ( N ) in M od C − R corresponding to η .For each x ∈ X , we consider η x ∶ F ( M ) x = F x ( M x ) Ð→ N x in M R x . By [4, Lemma 3.1], we already know that ( F x , G x ) is a pair of adjoint functors, which gives us M R x ( F x ( M x ) , N x ) ≅ M C R x ( M x , G x ( N x )) . Accordingly, wedefine ζ x ∶ M x Ð→ G x ( N x ) = N x ⊗ C by setting ζ x ( m ′ ) ∶ = η x ( r )( m ′ ) ⊗ m ′ for m ′ ∈ M x ( r ) , r ∈ R x . We now considerthe diagrams F x ( M x ) η x ÐÐÐÐ→ N x F x ( M α ) ×××Ö ×××Ö N α α ∗ F y ( M y ) α ∗ ( η y ) ÐÐÐÐ→ α ∗ N y ⇒ M x ζ x ÐÐÐÐ→ G x ( N x ) M α ×××Ö ×××Ö G x ( N α ) α ∗ M y α ∗ ( ζ y ) ÐÐÐÐ→ α ∗ G y ( N y ) (7.2)The left hand side diagram in (7.2) is commutative because η ∶ F ( M ) Ð→ N is a morphism in M od − R . In order toprove that we have a morphism ζ ∶ M Ð→ G ( N ) in M od C − R , it suffices to show that this implies the commutativityof the right hand side diagram in (7.2).We consider m ∈ el ( M x ) . Then, we have G x ( N α )( ζ x ( m )) = N α ( η x ( m )) ⊗ m . On the other hand, we have α ∗ ( ζ y )( M α ( m )) = η y (( M α ( m )) ) ⊗ ( M α ( m )) . Since M α is C -colinear, we have ( M α ( m )) ⊗ ( M α ( m )) = M α ( m ) ⊗ m . It follows that α ∗ ( ζ y )( M α ( m )) = η y ( M α ( m )) ⊗ m . From the left hand side commutative diagramin (7.2), we get η y ( M α ( m )) = N α ( η x ( m )) , which shows that the right hand diagram in (7.2) is commutative.Similarly, we may show that a morphism ζ ′ ∶ M Ð→ G ( N ) in M od C − R induces a morphism η ′ ∶ F ( M ) Ð→ N in M od − R and that these two associations are inverse to each other. This proves the result.21e now recall that a functor F ∶ A Ð→ B is said to be separable if the natural transformation A ( , ) Ð→ B ( F ( ) , F ( )) is a split monomorphism (see [26], [27]). If F has a right adjoint G ∶ B Ð→ A , then F is sepa-rable if and only if there exists a natural transformation υ ∈ N at ( GF, A ) satisfying υ ○ µ = A , where µ is the unitof the adjunction (see [27, Theorem 1.2]).We now consider the forgetful functor F ∶ M od C − R Ð→ M od − R as well as its right adjoint G ∶ M od − R Ð→ M od C − R constructed in Proposition 7.2. We will need an alternate description for the natural transformations G F Ð→ Mod C − R . Proposition 7.3.
A natural transformation υ ∈ N at ( G F , Mod C − R ) corresponds to a collection of natural transfor-mations { υ x ∈ N at ( G x F x , M C R x ( ψ x ) )} x ∈ X such that for any α ∶ x Ð→ y in X and object M ∈ M od C − R , we have acommutative diagram G x F x ( M x ) υ x ( M x ) ÐÐÐÐ→ M x G x F x ( M α ) ×××Ö ×××Ö M α α ∗ G y F y ( M y ) α ∗ υ y ( M y ) ÐÐÐÐÐÐ→ α ∗ M y (7.3) in M C R x ( ψ x ) .Proof. We consider υ ∈ N at ( G F , Mod C − R ) . For x ∈ X , we define the natural transformation υ x ∈ N at ( G x F x , M C R x ( ψ x ) ) by setting υ x ( M ) ∶ = υ ( ex Cx ( M )) x ∶ G x F x ( M ) = G x F x (( ex Cx ( M )) x ) Ð→ ( ex Cx ( M )) x = M (7.4)for M ∈ M C R x ( ψ x ) . We now consider M ∈ M od C − R . For α ∶ x Ð→ y in X , the morphism υ ( M ) ∶ G F ( M ) Ð→ M in M od C − R leads to a commutative diagram ( G F ( M )) x = G x F x ( M x ) υ ( M ) x ÐÐÐÐ→ M x G x F x ( M α ) ×××Ö ×××Ö M α α ∗ ( G F ( M )) y = α ∗ G y F y ( M y ) α ∗ ( υ ( M ) y ) ÐÐÐÐÐÐ→ α ∗ M y (7.5)We now claim that υ ( M ) x = ( υ ( ex Cx ( M x ))) x = υ x ( M x ) for each x ∈ X . For this, we consider the canonical morphism ζ ∶ ex Cx ( M x ) = ex Cx ( ev Cx ( M )) Ð→ M in M od C − R corresponding to the adjoint pair ( ex Cx , ev Cx ) in Proposition 5.3.It is clear that ev Cx ( ζ ) = id . Then, we have commutative diagrams G F ( ex Cx ( M x )) υ ( ex Cx ( M x )) ÐÐÐÐÐÐÐ→ ex Cx ( M x ) G F ( ζ ) ×××Ö ×××Ö ζ G F ( M ) υ ( M ) ÐÐÐÐ→ M ⇒ G x F x ( M x ) ( υ ( ex Cx ( M x ))) x ÐÐÐÐÐÐÐÐÐ→ M xid ×××Ö ×××Ö id G x F x ( M x ) υ ( M ) x ÐÐÐÐ→ M x (7.6)This proves that υ ( M ) x = ( υ ( ex Cx ( M x ))) x = υ x ( M x ) for each x ∈ X . The commutativity of the diagram (7.3) nowfollows from (7.5).Conversely, given a collection of natural transformations { υ x ∈ N at ( G x F x , M C R x ( ψ x ) )} x ∈ X satisfying (7.3) for each M ∈ M od C − R , we get υ ( M ) ∶ G F ( M ) Ð→ M in M od C − R by setting υ ( M ) x = υ x ( M x ) for each x ∈ X . From(7.3), it is clear that υ ∈ N at ( G F , Mod C − R ) . 22ore explicitly, the diagram in (7.3) shows that for each α ∶ x Ð→ y in X and r ∈ R x , we have a commutativediagram M x ( r ) ⊗ C = ( G x F x ( M x ))( r ) ( υ x ( M x ))( r ) ÐÐÐÐÐÐÐ→ M x ( r ) ( G x F x ( M α ))( r ) ×××Ö ×××Ö M α ( r ) M y ( α ( r )) ⊗ C = ( G y F y ( M y ))( α ( r )) = ( α ∗ G y F y ( M y ))( r ) ( α ∗ υ y ( M y ))( r ) ÐÐÐÐÐÐÐÐÐÐ→ =( υ y ( M y ))( α ( r )) ( α ∗ M y )( r ) = M y ( α ( r )) (7.7)We note that all morphisms in (7.7) are C -colinear. We now give another interpretation of the space N at ( G F , Mod C − R ) .For this, we consider a collection θ ∶ = { θ x ( r ) ∶ C ⊗ C Ð→ R x ( r, r )} x ∈ X ,r ∈ R x of K -linear maps satisfying the followingconditions.(1) Fix x ∈ X and r ∈ R x . Then, for c , d ∈ C , we have θ x ( r )( c ⊗ d ) ⊗ d = ( θ x ( r )( c ⊗ d )) ψ x ⊗ c ψ x (7.8)(2) Fix x ∈ X and c , d ∈ C . Then, for f ∶ s Ð→ r in R x , we have ( θ x ( r )( c ⊗ d )) ○ f = f ψ xψx ○ ( θ x ( s )( c ψ x ⊗ d ψ x )) (7.9)(3) Fix c , d ∈ C . Then, for any α ∶ x Ð→ y in X and r ∈ R x , we have α ( θ x ( r )( c ⊗ d )) = θ y ( α ( r ))( c ⊗ d ) (7.10)The space of all such θ will be denoted by V . Proposition 7.4.
Let θ ∈ V . Then, θ induces a natural transformation υ ∈ N at ( G F , Mod C − R ) , such that for each x ∈ X , υ x ∈ N at ( G x F x , M C R x ( ψ x ) ) is given by υ x ( M ) ∶ M ⊗ C Ð→ M ( m ⊗ c ) ↦ M ( θ x ( r )( m ⊗ c ))( m ) (7.11) for any M ∈ M C R x ( ψ x ) , r ∈ R x , m ∈ M ( r ) and c ∈ C .Proof. From [4, Proposition 3.6], it follows that each υ x as defined in (7.11) by the collection θ x ∶ = { θ x ( r ) ∶ C ⊗ C Ð→ R x ( r, r )} r ∈ R x gives a natural transformation υ x ∈ N at ( G x F x , M C R x ( ψ x ) ) . To prove the result, it therefore suffices toshow the commutativity of the diagram (7.7) for any M ∈ M od C − R . Accordingly, for α ∶ x Ð→ y in X and r ∈ R x ,we have (( M α ( r )) ○ ( υ x ( M x ))( r ))( m ⊗ c ) = ( M α ( r ))( M x ( θ x ( r )( m ⊗ c ))( m )) (7.12)for m ⊗ c ∈ M x ( r ) ⊗ C . On the other hand, we have ((( υ y ( M y ))( α ( r ))) ○ (( G x F x ( M α ))( r )))( m ⊗ c ) = M y ( θ y ( α ( r ))( M α ( m ) ⊗ c ))( M α ( r )( m )) = M y ( θ y ( α ( r ))( m ⊗ c ))( M α ( r )( m ))= M y ( α ( θ x ( r )( m ⊗ c )))( M α ( r )( m )) (7.13)The second equality in (7.13) follows from the C -colinearity of M α ( r ) and the third equality follows by applyingcondition (7.10). We now notice that for any f ∈ R x ( r, r ) , we have a commutative diagram M x ( r ) M α ( r ) ÐÐÐÐ→ M y ( α ( r )) M x ( f ) ×××Ö ×××Ö M y ( α ( f )) M x ( r ) M α ( r ) ÐÐÐÐ→ M y ( α ( r )) (7.14)23pplying (7.14) to f = θ x ( r )( m ⊗ c ) ∈ R x ( r, r ) , we obtain from (7.13) that ((( υ y ( M y ))( α ( r ))) ○ (( G x F x ( M α ))( r )))( m ⊗ c ) = ( M α ( r ))( M x ( θ x ( r )( m ⊗ c ))( m )) (7.15)This proves the result.Fix x ∈ X and r ∈ R x . We now set H ( x,r ) y ∶ = { R y ( , α ( r )) ⊗ C if α ∶ x Ð→ y x /≤ y (7.16)for each y ∈ X . Lemma 7.5.
For each x ∈ X and r ∈ R x , the collection H ( x,r ) ∶ = { H ( x,r ) y } y ∈ X determines an object of M od C − R .Proof. For each y ∈ X , it follows by [4, Lemma 2.4] that H ( x,r ) y is an object of M C R y ( ψ y ) . We consider β ∶ y Ð→ z in X and suppose we have α ∶ x Ð→ y , i.e., x ≤ y . Then, for r ′ ∈ R y , we have an obvious morphism β ( ) ⊗ C ∶ H ( x,r ) y ( r ′ ) = R y ( r ′ , α ( r )) ⊗ C Ð→ β ∗ ( R z ( , βα ( r )) ⊗ C )( r ′ ) = R z ( β ( r ′ ) , βα ( r )) ⊗ C (7.17)which is C -colinear. To prove that H ( x,r ) y Ð→ β ∗ H ( x,r ) z is a morphism in M C R y ( ψ y ) , it remains to show that for any g ∶ r ′′ Ð→ r ′ in R y , the following diagram commutes R y ( r ′ , α ( r )) ⊗ C ⋅ g ÐÐÐÐ→ R y ( r ′′ , α ( r )) ⊗ C β ( ) ⊗ C ×××Ö ×××Ö β ( ) ⊗ C R z ( β ( r ′ ) , βα ( r )) ⊗ C ⋅ β ( g ) ÐÐÐÐ→ R z ( β ( r ′′ ) , βα ( r )) ⊗ C (7.18)For f ⊗ c ∈ R y ( r ′ , α ( r )) ⊗ C , we have ( β ( ) ⊗ C )(( f ⊗ c ) ⋅ g ) = ( β ( ) ⊗ C )( f g ψ y ⊗ c ψ y ) = β ( f ) β ( g ψ y ) ⊗ c ψ y = β ( f ) β ( g ) ψ z ⊗ c ψ z = ( β ( f ) ⊗ c ) ⋅ β ( g ) This shows that (7.18) is commutative. Finally, if x /≤ y , then 0 = H ( x,r ) y Ð→ β ∗ H ( x,r ) z is obviously a morphism in M C R y ( ψ y ) . This proves the result. Proposition 7.6.
Let υ ∈ N at ( G F , Mod C − R ) . For each x ∈ X and r ∈ R x , define θ x ( r ) ∶ C ⊗ C Ð→ R x ( r, r ) bysetting θ x ( r )( c ⊗ d ) ∶ = (( id ⊗ ε C ) ○ ( υ x ( H ( x,r ) x )( r )))( id r ⊗ c ⊗ d ) (7.19) for c , d ∈ C . Then, the collection θ ∶ = { θ x ( r ) ∶ C ⊗ C Ð→ R x ( r, r )} x ∈ X ,r ∈ R x is an element of V .Proof. From the definition in (7.19), we have explicitly that θ x ( r )( c ⊗ d ) = (( id ⊗ ε C ) ○ ( υ x ( R x ( , r ) ⊗ C )( r )))( id r ⊗ c ⊗ d ) (7.20)Then, it follows from [4, Proposition 3.5] that θ x ( r ) satisfies the conditions in (7.8) and (7.9). It remains to verifythe condition (7.10). For this we take α ∶ x Ð→ y in X and consider the commutative diagram R x ( r ′ , r ) ⊗ C ⊗ C υ x ( H ( x,r ) x )( r ′ ) ÐÐÐÐÐÐÐÐ→ R x ( r ′ , r ) ⊗ C id ⊗ ε C ÐÐÐÐ→ R x ( r ′ , r ) α ( ) ⊗ C ⊗ C ×××Ö ×××Ö α ( ) ⊗ C ×××Ö α ( ) R y ( α ( r ′ ) , α ( r )) ⊗ C ⊗ C υ y ( H ( x,r ) y )( α ( r ′ )) ÐÐÐÐÐÐÐÐÐÐ→ R y ( α ( r ′ ) , α ( r )) ⊗ C id ⊗ ε C ÐÐÐÐ→ R y ( α ( r ′ ) , α ( r )) (7.21)24or any r, r ′ ∈ R x . Since υ ∈ N at ( G F , Mod C − R ) , the commutativity of the left hand side square in (7.21) followsfrom (7.7). It is clear that the right hand square in (7.21) is commutative.We notice that H ( y,α ( r )) y = H ( x,r ) y in M C R y ( ψ y ) . Applying (7.21) with r ′ = r ∈ R x and id r ⊗ c ⊗ d ∈ R x ( r, r ) ⊗ C ⊗ C ,it follows from (7.20) that α ( θ x ( r )( c ⊗ d )) = θ y ( α ( r ))( c ⊗ d ) . This proves (7.10). Proposition 7.7.
N at ( G F , Mod C − R ) is isomorphic to V .Proof. From Proposition 7.4 and Proposition 7.6, we see that we have maps ψ ∶ V Ð→ N at ( G F , Mod C − R ) and φ ∶ N at ( G F , Mod C − R ) Ð→ V in opposite directions.We consider υ ∈ N at ( G F , Mod C − R ) . By Proposition 7.6, υ induces an element θ ∈ V . Applying Proposition 7.4, θ induces an element in N at ( G F , Mod C − R ) , which we denote by υ ′ . Then, υ and υ ′ are determined respectively bynatural transformations { υ x ∈ N at ( G x F x , M C R x ( ψ x ) )} x ∈ X and { υ ′ x ∈ N at ( G x F x , M C R x ( ψ x ) )} x ∈ X satisfying compat-ibility conditions as in (7.3). From [4, Proposition 3.7], it follows that υ ′ x = υ x for each x ∈ X . Hence, υ ′ = υ and ψ ○ φ = id . Similarly, we can show that φ ○ ψ = id . Theorem 7.8.
Let X be a partially ordered set. Let C be a right semiperfect K -coalgebra and let R ∶ X Ð→ E nt C be an entwined C -representation. Then, the functor F ∶ M od C − R Ð→ M od − R is separable if and only if thereexists θ ∈ V such that θ x ( r )( c ⊗ c ) = ε C ( c ) ⋅ id r (7.22) for every x ∈ X , r ∈ R x and c ∈ C .Proof. We suppose that F ∶ M od C − R Ð→ M od − R is separable. As mentioned before, this implies that there exists υ ∈ N at ( G F , Mod C − R ) such that υ ○ µ = Mod C − R , where µ is the unit of the adjunction ( F , G ) . We set θ = φ ( υ ) ,where φ ∶ N at ( G F , Mod C − R ) Ð→ V is the isomorphism described in the proof of Proposition 7.7. In particular, forevery x ∈ X , r ∈ R x , we have υ ( H ( x,r ) ) ○ µ ( H ( x,r ) ) = id . From (7.19), it now follows that for every c ∈ C , we have θ x ( r )( c ⊗ c ) = (( id ⊗ ε C ) ○ ( υ x ( H ( x,r ) x )( r )))( id r ⊗ c ⊗ c )= (( id ⊗ ε C ) ○ ( υ x ( H ( x,r ) x )( r )) ○ µ ( H ( x,r ) ) x ( r ))( id r ⊗ c )= ( id ⊗ ε C )( id r ⊗ c ) = ε C ( c ) ⋅ id r (7.23)Conversely, suppose that there exists θ ∈ V satisfying the condition in (7.22). We set υ ∶ = ψ ( θ ) , where ψ ∶ V Ð→ N at ( G F , Mod C − R ) is the other isomorphism described in the proof of Proposition 7.7. We consider M ∈ M od C − R .By (7.11), we know that υ x ( M x ) ∶ M x ⊗ C Ð→ M x ( m ⊗ c ) ↦ M x ( θ x ( r )( m ⊗ c ))( m ) (7.24)for any x ∈ X , r ∈ R x , m ∈ M x ( r ) and c ∈ C . We claim that υ ○ µ = Mod C − R . For this, we see that (( υ ( M ) ○ µ ( M )) x ( r ))( m ) = ( υ x ( M x )( r ))( m ⊗ m )= M x ( θ x ( r )( m ⊗ m ))( m )= M x ( θ x ( r )( m ⊗ m ))( m )= ε C ( m ) m = m (7.25)This proves the result.We now turn to cartesian modules over entwined C -representations. For this, we assume additionally that R ∶ X Ð→ E nt C is flat. Then, it follows from Theorem 6.10 that Cart C − R is a Grothendieck category. In particular, by taking C = K , we note that Cart − R is also a Grothendieck category.25 roposition 7.9. Let X be a poset, C be a right semiperfect K -coalgebra and R ∶ X Ð→ E nt C be an entwined C -representation that is also flat. Then, the functor F ∶ M od C − R Ð→ M od − R restricts to a functor F c ∶ Cart C − R Ð→ Cart − R . Additionally, F c has a right adjoint G c ∶ Cart − R Ð→ Cart C − R .Proof. We consider M ∈ Cart C − R . We claim that F ( M ) ∈ M od − R actually lies in the subcategory Cart − R .Indeed, for α ∶ x Ð→ y in X , we have F ( M ) α ∶ F x ( M x ) = M x Ð→ α ∗ M y = α ∗ F y ( M y ) in M R x . By adjunction, thiscorresponds to a morphism α ∗ M x Ð→ M y in M R y . But since M ∈ Cart C − R , we already know that α ∗ M x Ð→ M y is an isomorphism. Hence, F c ( M ) ∶ = F ( M ) ∈ Cart − R .We also notice that Cart C − R is closed under taking colimits in M od C − R . Then F c ∶ Cart C − R Ð→ Cart − R preserves colimits and we know from Theorem 6.10 that both Cart C − R and Cart − R are Grothendieck categories.It now follows from [23, Proposition 8.3.27] that F c has a right adjoint. Proposition 7.10.
Let X be a poset, C be a right semiperfect K -coalgebra and R ∶ X Ð→ E nt C be an entwined C -representation that is also flat. Suppose there exists θ ∈ V such that θ x ( r )( c ⊗ c ) = ε C ( c ) ⋅ id r (7.26) for every x ∈ X , r ∈ R x and c ∈ C . Then, F c ∶ Cart C − R Ð→ Cart − R is separable.Proof. From Theorem 7.8, it follows that F ∶ M od C − R Ð→ M od − R is separable. In other words, for any M , N ∈ M od C − R , the canonical morphism M od C − R ( M , N ) Ð→ M od − R ( F ( M ) , F ( N )) is a split monomorphism.Since Cart C − R and Cart − R are full subcategories of M od C − R and M od − R respectively and F c is a restrictionof F , the result follows. G ∶ M od − R Ð→ M od C − R We continue with X being a poset, C being a right semiperfect coalgebra and let R ∶ X Ð→ E nt C be an entwined C -representation. In this section, we will give conditions for the right adjoint G ∶ M od − R Ð→ M od C − R to beseparable.Putting C = K in Proposition 5.3, we see that for each x ∈ X , there is a functor ex x ∶ M R x Ð→ M od − R having rightadjoint ev x ∶ M od − R Ð→ M R x . In a manner similar to Proposition 7.3, we now can show that a natural transfor-mation ω ∈ N at ( Mod − R , F G ) consists of a collection of natural transformations { ω x ∈ N at ( M R x , F x G x )} x ∈ X suchthat for any α ∶ x Ð→ y in X and any N ∈ M od − R , we have the following commutative diagram N x ω x ( N x ) ÐÐÐÐ→ F x G x ( N x ) N α ×××Ö ×××Ö F x G x ( N α ) α ∗ N y α ∗ ω y ( N y ) ÐÐÐÐÐÐ→ α ∗ F y G y ( N y ) (8.1)Here, ω x ∈ N at ( M R x , F x G x ) is determined by setting ω x ( N ) ∶ = ω ( ex x ( N )) x ∶ ( ex x ( N )) x = N Ð→ F x G x ( N ) = F x G x (( ex x ( N )) x ) (8.2)for N ∈ M R x . As in the proof of Proposition 7.3, we can also show that ω x ( N x ) = ω ( ex x ( N x )) x = ω ( N ) x (8.3)26or any N ∈ M od − R and x ∈ X . More explicitly, for each x ∈ X and r ∈ R x , we have a commutative diagram N x ( r ) ( ω x ( N x ))( r ) ÐÐÐÐÐÐÐ→ ( F x G x ( N x ))( r ) = N x ( r ) ⊗ C N α ( r ) ×××Ö ×××Ö ( F x G x ( N α ))( r ) N y ( α ( r )) = ( α ∗ N y )( r ) ( α ∗ ω y ( N y ))( r ) ÐÐÐÐÐÐÐÐ→ = ω y ( N y )( α ( r )) ( α ∗ F y G y ( N y ))( r ) = N y ( α ( r )) ⊗ C (8.4)We will now give another interpretation for the space N at ( Mod − R , F G ) . For this, we consider a collection η ={ η x ( s, r ) ∶ H xr ( s ) = R x ( s, r ) Ð→ H xr ( s ) ⊗ C = R x ( s, r ) ⊗ C ∶ f ↦ ˆ f ⊗ c f } x ∈ X ,r,s ∈ R x of K -linear maps satisfying thefollowing conditions:(1) Fix x ∈ X . Then, for s ′ h Ð→ s f Ð→ r g Ð→ r ′ in R x , we have η x ( s ′ , r ′ )( gf h ) = ∑ ̂ gf h ⊗ c gfh = g ˆ f h ψ x ⊗ c ψ x f ∈ R x ( s ′ , r ′ ) ⊗ C (8.5)(2) For α ∶ x Ð→ y in X and f ∈ R x ( s, r ) we have α ( ˆ f ) ⊗ c f = ̂ α ( f ) ⊗ c α ( f ) ∈ R y ( α ( s ) , α ( r )) ⊗ C (8.6)The space of all such η will be denoted by W . We note that condition (1) is equivalent to saying that for each x ∈ X ,the element η x = { η x ( s, r ) ∶ R x ( s, r ) Ð→ R x ( s, r ) ⊗ C ∶ f ↦ ˆ f ⊗ c f } r,s ∈ R x ∈ N at ( H x , H x ⊗ C ) , i.e., η x is a morphismin the category of R x -bimodules (functors R opx ⊗ R x Ð→ V ect K ). Here H x is the canonical R x -bimodule that takesa pair of objects ( s, r ) ∈ Ob ( R opx ⊗ R x ) to R x ( s, r ) . Further, H x ⊗ C is the R x -bimodule defined by setting ( H x ⊗ C )( s, r ) = R x ( s, r ) ⊗ C ( H x ⊗ C )( h, g )( f ⊗ c ) = gf h ψ x ⊗ c ψ x (8.7)for s ′ h Ð→ s f Ð→ r g Ð→ r ′ in R x and c ∈ C . Lemma 8.1.
There is a canonical morphism
N at ( Mod − R , F G ) Ð→ W .Proof. As mentioned above, any ω ∈ N at ( Mod − R , F G ) corresponds to a collection of natural transformations { ω x ∈ N at ( M R x , F x G x )} x ∈ X satisfying (8.1). From the proof of [4, Proposition 3.10], we know that each ω x ∈ N at ( M R x , F x G x ) corresponds to η x ∈ N at ( H x , H x ⊗ C ) determined by setting η x ( s, r ) ∶ H xr ( s ) = R x ( s, r ) Ð→ H xr ( s ) ⊗ C = R x ( s, r ) ⊗ C η x ( s, r ) ∶ = ω x ( H xr )( s ) (8.8)for r , s ∈ R x . Here, H xr is the right R x -module H xr ∶ = R x ( , r ) ∶ R opx Ð→ V ect K . We now consider α ∶ x Ð→ y in X and some f ∈ R x ( s, r ) . By applying Lemma 5.1 with C = K , we have ex x ( H xr ) ∈ M od − R which satisfies ( ex x ( H xr )) y = α ∗ H xr = H yα ( r ) . Setting N = ex x ( H xr ) in (8.4), we have N x ( s ) = H xr ( s ) ( ω x ( H xr ))( s ) ÐÐÐÐÐÐÐ→ = η x ( s,r ) ( F x G x ( N x ))( s ) = H xr ( s ) ⊗ C N α ( s ) ×××Ö ×××Ö ( F x G x ( N α ))( s ) N y ( α ( s )) = H yα ( r ) ( α ( s )) η y ( α ( s ) ,α ( r )) ÐÐÐÐÐÐÐÐÐÐ→ = ω y ( H yα ( r ) )( α ( s )) N y ( α ( s )) ⊗ C = H yα ( r ) ( α ( s )) ⊗ C (8.9)It follows that that the collection η x ( s, r ) satisfies condition (8.6). This proves the result.27 roposition 8.2. The spaces
N at ( Mod − R , F G ) and W are isomorphic.Proof. We consider an element η ∈ W . As mentioned before, this gives a collection { η x ∈ N at ( H x , H x ⊗ C )} x ∈ X satisfying the compatibility condition in (8.6). From the proof of [4, Proposition 3.10], it follows that each η x corresponds to a natural transformation ω x ∈ N at ( M R x , F x G x ) which satisfies ω x ( H xr )( s ) = η x ( s, r ) for r , s ∈ R x .We claim that the collection { ω x } x ∈ X satisfies the compatibility condition in (8.1) for each N ∈ M od − R , thusdetermining an element ω ∈ N at ( Mod − R , F G ) .We start with N = ex x ( H xr ) for some x ∈ X and r ∈ R x . We consider a morphism α ∶ y Ð→ z in X . If x /≤ y , then N y = β ∶ x Ð→ y in X and set s = β ( r ) . In particular, N y = β ∗ H xr = H yβ ( r ) = H ys and N z = H zαβ ( r ) = H zα ( s ) . Applying the condition (8.6), we see that the following diagramis commutative for any s ′ ∈ R y : R y ( s ′ , s ) = N y ( s ′ ) η y ( s ′ ,s ) ÐÐÐÐÐÐÐ→ = ω y ( N y )( s ′ ) N y ( s ′ ) ⊗ C = R y ( s ′ , s ) ⊗ C N α ( s ′ ) ×××Ö ×××Ö ( F y G y ( N α ))( s ′ ) R z ( α ( s ′ ) , α ( s )) = N z ( α ( s ′ )) η z ( α ( s ′ ) ,α ( s )) ÐÐÐÐÐÐÐÐÐ→ = ω z ( N z )( α ( s ′ )) N z ( α ( s ′ )) ⊗ C = R z ( α ( s ′ ) , α ( s )) ⊗ C (8.10)In other words, the condition in (8.1) is satisfied for N = ex x ( H xr ) . From Theorem 5.5, we know that the collection { ex x ( H xr ) ∣ x ∈ X , r ∈ R x } (8.11)is a set of generators for M od − R . Accordingly, for any N ′ ∈ M od − R , we can choose an epimorphism φ ∶ N Ð→ N ′ where N is a direct sum of copies of objects in (8.11). Then, N satisfies (8.1) and we have commutative diagrams N y N α ÐÐÐÐ→ α ∗ N z α ∗ ω z ( N z ) ÐÐÐÐÐÐ→ α ∗ F z G z ( N z ) φ y ×××Ö α ∗ φ z ×××Ö ×××Ö α ∗ F z G z ( φ z ) N ′ y N ′ α ÐÐÐÐ→ α ∗ N ′ z α ∗ ω z ( N ′ z ) ÐÐÐÐÐÐ→ α ∗ F z G z ( N ′ z ) (8.12) N y ω y ( N y ) ÐÐÐÐ→ F y G y ( N y ) F y G y ( N α ) ÐÐÐÐÐÐ→ α ∗ F z G z ( N z ) φ y ×××Ö F y G y ( φ y ) ×××Ö ×××Ö α ∗ F z G z ( φ z ) N ′ y ω y ( N ′ y ) ÐÐÐÐ→ F y G y ( N ′ y ) F y G y ( N ′ α ) ÐÐÐÐÐÐ→ α ∗ F z G z ( N ′ z ) N y ω y ( N y ) ÐÐÐÐ→ F y G y ( N y ) N α ×××Ö ×××Ö F y G y ( N α ) α ∗ N z α ∗ ω z ( N z ) ÐÐÐÐÐÐ→ α ∗ F z G z ( N z ) (8.13)for any α ∶ y Ð→ z in X . Since φ y ∶ N y Ð→ N ′ y is an epimorphism, it follows that N ′ also satisfies the conditionin (8.1). This gives a morphism W Ð→ N at ( Mod − R , F G ) . It may be verified that this is inverse to the morphism N at ( Mod − R , F G ) Ð→ W in Lemma 8.1, which proves the result.We will now give conditions for the functor G ∶ M od − R Ð→ M od C − R to be separable. Since G has a leftadjoint, it follows (see [27, Theorem 1.2]) that G is separable if and only if there exists a natural transformation ω ∈ N at ( Mod − R , F G ) such that ν ○ ω = Mod − R , where ν is the counit of the adjunction. Theorem 8.3.
Let X be a partially ordered set, C be a right semiperfect K -coalgebra and let R ∶ X Ð→ E nt C bean entwined C -representation. Then, the functor G ∶ M od − R Ð→ M od C − R is separable if and only if there exists η ∈ W such that id = ( id ⊗ ε C ) ○ η x ( s, r ) ∶ R x ( s, r ) η x ( s,r ) ÐÐÐÐ→ R x ( s, r ) ⊗ C ( id ⊗ ε C ) ÐÐÐÐ→ R x ( s, r ) (8.14) for each x ∈ X and s , r ∈ R x . roof. First, we suppose that G is separable, i.e., there exists a natural transformation ω ∈ N at ( Mod − R , F G ) suchthat ν ○ ω = Mod − R . Using Proposition 8.2, we consider η ∈ W corresponding to ω .By definition, the counit ν of the adjunction ( F , G ) is described as follows: for any N ∈ M od − R , we have ν ( N ) x ( s ) ∶ N x ( s ) ⊗ C Ð→ N x ( s ) n ⊗ c ↦ nε C ( c ) (8.15)for each x ∈ X , s ∈ R x . We choose x ∈ X , r ∈ R x and set N = ex x ( H xr ) . Since ν ○ ω = Mod − R , it now follows from(8.8) that id = ν ( ex x ( H xr )) x ( s ) ○ ω ( ex x ( H xr )) x ( s ) = ( id ⊗ ε C ) ○ ω x ( H xr )( s ) = ( id ⊗ ε C ) ○ η x ( s, r ) (8.16)Conversely, suppose that we have η ∈ W such that the condition in (8.14) is satisfied. Using the isomorphism inProposition 8.2, we obtain the natural transformation ω ∈ N at ( Mod − R , F G ) corresponding to η . Then, it is clearfrom (8.16) that ν ( N ) ○ ω ( N ) = id for N = ex x ( H xr ) . Since { ex x ( H xr ) ∣ x ∈ X , r ∈ R x } is a set of generators for M od − R , it follows that for any N ′ ∈ M od − R , there is an epimorphism φ ∶ N Ð→ N ′ such that ν ( N ) ○ ω ( N ) = id .We now consider the commutative diagram N ω ( N ) ÐÐÐÐ→
F G ( N ) ν ( N ) ÐÐÐÐ→ N φ ×××Ö FG ( φ ) ×××Ö ×××Ö φ N ′ ω ( N ′ ) ÐÐÐÐ→
F G ( N ′ ) ν ( N ′ ) ÐÐÐÐ→ N ′ (8.17)Since the upper horizontal composition in (8.17) is the identity and φ is an epimorphism, it follows that ν ( N ′ ) ○ ω ( N ′ ) = id . This proves the result. ( F , G ) as a Frobenius pair In Sections 7 and 8, we have given conditions for the functor F ∶ M od C − R Ð→ M od − R and its right adjoint G ∶ M od − R Ð→ M od C − R to be separable. In this section, we will give necessary and sufficient conditions for ( F , G ) to be a Frobenius pair, i.e., G is both a right and a left adjoint of F . First, we note that it follows from thecharacterization of Frobenius pairs (see for instance, [10, § ( F , G ) is a Frobenius pair if and only if thereexist υ ∈ N at ( G F , Mod C − R ) and ω ∈ N at ( Mod − R , F G ) such that F ( υ ( M )) ○ ω ( F ( M )) = id F ( M ) υ ( G ( N )) ○ G ( ω ( N )) = id G ( N ) (9.1)for any M ∈ M od C − R and N ∈ M od − R . Equivalently, for each x ∈ X , we must have ( F ( υ ( M ))) x ○ ω ( F ( M )) x = F x ( υ x ( M x )) ○ ω x ( F x ( M x )) = id F x ( M x ) υ ( G ( N )) x ○ G ( ω ( N )) x = υ x ( G x ( N x )) ○ G x ( ω x ( N x )) = id G x ( N x ) (9.2)for any M ∈ M od C − R and N ∈ M od − R . Theorem 9.1.
Let X be a partially ordered set, C be a right semiperfect K -coalgebra and let R ∶ X Ð→ E nt C be anentwined C -representation. Let F ∶ M od C − R Ð→ M od − R be the forgetful functor and G ∶ M od − R Ð→ M od C − R its right adjoint. Then, ( F , G ) is a Frobenius pair if and only if there exist θ ∈ V and η ∈ W such that ε C ( d ) f = ∑ ̂ f ○ θ x ( r )( c f ⊗ d ) ε C ( d ) f = ∑ ̂ f ψ x ○ θ x ( r )( d ψ x ⊗ c f ) (9.3) for every x ∈ X , r ∈ R x , f ∈ R x ( r, s ) and d ∈ C , where η x ( r, s )( f ) = ̂ f ⊗ c f . roof. We suppose there exist θ ∈ V and η ∈ W satisfying (9.3) and consider M ∈ M od C − R , N ∈ M od − R . Using theisomorphisms in Proposition 7.7 and Proposition 8.2, we obtain υ ∈ N at ( G F , Mod C − R ) and ω ∈ N at ( Mod − R , F G ) corresponding to θ and η respectively.For fixed x ∈ X , it follows that θ x = { θ x ( r ) ∶ C ⊗ C Ð→ R x ( r, r )} r ∈ R x and the R x -bimodule morphism η x ∈ N at ( H x , H x ⊗ C ) satisfy the conditions in [4, Theorem 3.14]. Hence, we have F x ( υ x ( M )) ○ ω x ( F x ( M )) = id F x (M) υ x ( G x ( N )) ○ G x ( ω x ( N )) = id G x (N ) (9.4)for any M ∈ M C R x ( ψ x ) and N ∈ M R x . In particular, (9.2) holds for M x ∈ M C R x ( ψ x ) and N x ∈ M R x .Conversely, suppose that ( F , G ) is a Frobenius pair. Then, there exist υ ∈ N at ( G F , Mod C − R ) and ω ∈ N at ( Mod − R , F G ) satisfying (9.2) for each x ∈ X . Again using the isomorphisms in Proposition 7.7 and Proposition 8.2, we obtaincorresponding θ ∈ V and η ∈ W .We now consider M ∈ M C R x ( ψ x ) and N ∈ M R x . Applying (9.2) with M = ex Cx ( M ) and N = ex x ( N ) , we have F x ( υ x ( M )) ○ ω x ( F x ( M )) = id F x (M) υ x ( G x ( N )) ○ G x ( ω x ( N )) = id G x (N ) (9.5)It now follows from [4, Theorem 3.14] that θ x = { θ x ( r ) ∶ C ⊗ C Ð→ R x ( r, r )} r ∈ R x and the R x -bimodule morphism η x ∈ N at ( H x , H x ⊗ C ) satisfy (9.3). This proves the result. Corollary 9.2.
Let ( F , G ) be a Frobenius pair. Then, for each x ∈ X , ( F x , G x ) is a Frobenius pair of adjointfunctors.Proof. This is immediate from (9.4).We consider α ∶ x Ð→ y in X . In (7.1), we observed directly that the functors { F x ∶ M C R x ( ψ x ) Ð→ M R x } x ∈ X commute with both α ∗ and α ∗ , while the functors { G x ∶ M R x Ð→ M C R x ( ψ x )} x ∈ X commute only with α ∗ . We willnow give a sufficient condition for the functors { G x ∶ M R x Ð→ M C R x ( ψ x )} x ∈ X to commute with α ∗ . Lemma 9.3.
Let ( F , G ) be a Frobenius pair. Then, for any α ∶ x Ð→ y in X , we have a commutative diagram M R x α ∗ ÐÐÐÐ→ M R y G x ×××Ö ×××Ö G y M C R x ( ψ x ) α ∗ ÐÐÐÐ→ M C R y ( ψ y ) (9.6) Proof.
For M ∈ M R x , we will show that G y α ∗ ( M ) = α ∗ G x ( M ) ∈ M C R y ( ψ y ) . From Corollary 9.2 we know that each ( F x , G x ) is a Frobenius pair of adjoint functors. Using this fact and the commutative diagrams in (7.1), we nowhave that for any N ∈ M C R y ( ψ y ) : M C R y ( ψ y )( G y α ∗ ( M ) , N ) = M R x ( M , α ∗ F y ( N )) = M R x ( M , F x α ∗ ( N )) = M C R y ( ψ y )( α ∗ G x ( M ) , N ) (9.7) Proposition 9.4.
Let ( F , G ) be a Frobenius pair. Suppose that R ∶ X Ð→ E nt C is flat. Then, G ∶ M od − R Ð→ M od C − R restricts to a functor G c ∶ Cart − R Ð→ Cart C − R . roof. For any N ∈ Cart − R , we claim that G ( N ) ∈ M od C − R actually lies in Cart C − R . By definition of G ,we have for any α ∶ x Ð→ y , a morphism G ( N ) α = G x ( N α ) ∶ G x ( N x ) Ð→ G x ( α ∗ ( N y )) = α ∗ ( G y ( N y )) in M C R x ( ψ x ) which corresponds to a morphism G ( N ) α ∶ α ∗ ( G x ( N x )) Ð→ G y ( N y ) in M C R y ( ψ y ) . Since ( F , G ) is a Frobenius pair,it follows from Lemma 9.3 that G y α ∗ ( N x ) = α ∗ G x ( N x ) ∈ M C R y ( ψ y ) . Since N is cartesian, we know that α ∗ N x isisomorphic to N y and hence G ( N ) α = G y ( N α ) is an isomorphism. Corollary 9.5.
Let ( F , G ) be a Frobenius pair. Suppose that R ∶ X Ð→ E nt C is flat. Then, ( F c , G c ) is a Frobeniuspair of adjoint functors between Cart C − R and Cart − R .Proof. From Proposition 7.9, we know that F ∶ M od C − R Ð→ M od − R restricts to a functor F c ∶ Cart C − R Ð→ Cart − R . From Proposition 9.4, we know that G ∶ M od − R Ð→ M od C − R restricts to a functor G c ∶ Cart − R Ð→ Cart C − R on the full subcategories of cartesian modules. Since G is both right and left adjoint to F , it is clear that G c is both right and left adjoint to F c .
10 Constructing entwined representations
In this final section, we will give examples of how to construct entwined representations and describe modules overthem. Let ( R , C, ψ ) be an entwining structure. Then, we consider the K -linear category ( C, R ) ψ defined as follows Ob (( C, R ) ψ ) = Ob ( R ) ( C, R ) ψ ( s, r ) ∶ = Hom K ( C, R ( s, r )) (10.1)for s , r ∈ R . The composition in ( C, R ) ψ is as follows: given φ ∶ C Ð→ R ( s, r ) and φ ′ ∶ C Ð→ R ( t, s ) respectively in ( C, R ) ψ ( s, r ) and ( C, R ) ψ ( t, s ) , we set φ ∗ φ ′ ∶ C Ð→ R ( t, r ) c ↦ ∑ φ ( c ) ψ ○ φ ′ ( c ψ ) (10.2) Lemma 10.1.
Let ( R , C, ψ ) be an entwining structure. Then, there is a canonical functor P ψ ∶ M C R ( ψ ) Ð→ M ( C, R) ψ .Proof. We consider M ∈ M C R ( ψ ) . We will define N = P ψ ( M ) ∈ M ( C, R) ψ by setting N ( r ) ∶ = M ( r ) for each r ∈ ( C, R ) .Given φ ∶ C Ð→ R ( s, r ) in ( C, R ) ψ ( s, r ) , we define m ∗ φ ∈ N ( s ) = M ( s ) by setting m ∗ φ = ∑ m φ ( m ) . Here, ρ M( r ) ( m ) = ∑ m ⊗ m is the right C -comodule structure on M ( r ) .For φ ′ ∶ C Ð→ R ( t, s ) in ( C, R ) ψ ( t, s ) , we now have m ∗ ( φ ∗ φ ′ ) = ∑ m ( φ ∗ φ ′ )( m ) = ∑ m φ ( m ) ψ φ ′ ( m ψ ) = ∑ m φ ( m ) ψ φ ′ ( m ψ )( m ∗ φ ) ∗ φ ′ = ∑ ( m ∗ φ ) φ ′ (( m ∗ φ ) ) = ∑ ( m φ ( m )) φ ′ (( m φ ( m )) )= ∑ ( m φ ( m ) ψ ) φ ′ ( m ψ ) = ∑ m φ ( m ) ψ φ ′ ( m ψ ) (10.3)This proves the result. Lemma 10.2.
Let ( α, id ) ∶ ( R , C, ψ ) Ð→ ( S , C, ψ ′ ) be a morphism of entwining structures. Then, P ψ ○ ( α, id ) ∗ = α ∗ ○ P ψ ′ ∶ M C S ( ψ ′ ) Ð→ M ( C, R) .Proof. We begin with N ∈ M C S ( ψ ′ ) . From the construction in Lemma 10.1, it is clear that for any r ∈ ( C, R ) ψ , wehave ( P ψ ○ ( α, id ) ∗ )( N )( r ) = ( α ∗ ○ P ψ ′ )( N )( r ) = N ( α ( r )) . We set N ∶ = ( P ψ ○ ( α, id ) ∗ )( N ) and N ∶ = ( α ∗ ○ P ψ ′ )( N ) and consider n ∈ N ( r ) = N ( r ) as well as φ ∶ C Ð→ R ( s, r ) in ( C, R ) ψ ( s, r ) . Then, in both N ( s ) and N ( s ) , wehave n ∗ φ = ∑ n α ( φ ( n )) . This proves the result. 31ow let X be a small category and R ∶ X Ð→ E nt C an entwined C -representation. By replacing each entwiningstructure ( R x , C, ψ x ) with the category ( C, R x ) ψ x , we obtain an induced representation ( C, R ) ψ ∶ X Ð→ L in (werecall that L in is the category of small K -linear categories). Proposition 10.3.
There is a canonical functor
M od C − R Ð→ M od − ( C, R ) ψ .Proof. By definition, an object M ∈ M od C − R consists of a collection { M x ∈ M C R x ( ψ x )} x ∈ X and for each α ∶ x Ð→ y in X , a morphism M α ∶ M x Ð→ α ∗ M y in M C R x ( ψ x ) . Applying the functors P ψ x ∶ M C R x ( ψ x ) Ð→ M ( C, R x ) ψx for x ∈ X and using Lemma 10.2, the result is now clear.Now let C be finitely generated as a K -vector space and let C ∗ denote its K -linear dual. Then, the canonical map C ∗ ⊗ V Ð→ Hom K ( C, V ) is an isomorphism for any vector space V . For an entwining structure ( R , C, ψ ) , the category ( C, R ) ψ can now be rewritten as ( C ∗ ⊗ R ) ψ where ( C ∗ ⊗ R ) ψ ( s, r ) = C ∗ ⊗ R ( s, r ) for s , r ∈ Ob (( C ∗ ⊗ R ) ψ ) = Ob ( R ) .Given c ∗ ⊗ f ∈ C ∗ ⊗ R ( s, r ) and d ∗ ⊗ g ∈ C ∗ ⊗ R ( t, s ) , the composition in ( C ∗ ⊗ R ) ψ is expressed as ( c ∗ ⊗ f ) ○ ( d ∗ ⊗ g ) ∶ C Ð→ R ( t, r ) x ↦ ∑ c ∗ ( x ) d ∗ ( x ψ )( f ψ ○ g ) (10.4)for x ∈ C . It is important to note that when f and g are identity maps, the composition in (10.4) simplifies to ( c ∗ ⊗ id r ) ○ ( d ∗ ⊗ id r ) ∶ C Ð→ R ( t, r ) x ↦ ∑ c ∗ ( x ) d ∗ ( x ) id r (10.5)In other words, for the canonical morphism C ∗ Ð→ C ∗ ⊗ R ( r, r ) given by c ∗ ↦ c ∗ ⊗ id r to be a morphism of algebras,we must use the opposite of the usual convolution product on C ∗ .Similarly, given an entwined C -representation R ∶ X Ð→ E nt C with C finitely generated as a K -vector space, wecan replace the induced representation ( C, R ) ψ ∶ X Ð→ L in by ( C ∗ ⊗ R ) ψ . Then, M od − ( C, R ) ψ may be replacedby M od − ( C ∗ ⊗ R ) ψ . Proposition 10.4.
Let X be a small category and R ∶ X Ð→ E nt C an entwined C -representation. Suppose that C is finitely generated as a K -vector space. Then, the categories M od C − R and M od − ( C ∗ ⊗ R ) ψ are equivalent.Proof. By Proposition 10.3, we already know that any object in
M od C − R may be equipped with a ( C ∗ ⊗ R ) ψ -modulestructure. For the converse, we consider some M ∈ M od − ( C ∗ ⊗ R ) ψ and choose some x ∈ X .We make M x into an R x -module as follows: for f ∈ R x ( s, r ) and m ∈ M x ( r ) , we set mf ∈ M x ( s ) to be mf ∶ = m ( ε C ⊗ f ) . By considering the canonical morphism C ∗ Ð→ C ∗ ⊗ R x ( r, r ) , it follows that the right ( C ∗ ⊗ R x ) ψ x ( r, r ) module M x ( r ) carries a right C ∗ -module structure. As observed in (10.5), here the product on C ∗ happens to be theopposite of the usual convolution product. Hence, the right C ∗ -module structure on M x ( r ) leads to a left C ∗ -modulestructure on M x ( r ) when C ∗ is equipped with the usual product. Since C is finite dimensional, it is well known(see, for instance, [18, § C -comodule structure on M x ( r ) . It may be verified bydirect computation that M x ∈ M C R x ( ψ x ) . Finally, for a morphism α ∶ x Ð→ y in X , the map M α ∶ M x Ð→ α ∗ M y in M ( C ∗ ⊗ R x ) ψx induces a morphism in M C R x ( ψ x ) . Hence, M ∈ M od − ( C ∗ ⊗ R ) ψ may be treated as an object of M od C − R . It may be directly verified that this structure is the inverse of the one defined by Propostion 10.3.Finally, we will give an example of constructing entwined representations starting from B -comodule categories, where B is a bialgebra. So let B be a bialgebra over K , having multiplication µ B , unit map u B as well as comultiplication∆ B and counit map ε B . Then, the notion of a “ B -comodule category,” which behaves like a B -comodule algebrawith many objects, is implicit in the literature. 32 efinition 10.5. Let B be a K -bialgebra. We will say that a small K -linear category R is a right B -comodulecategory if it satisfies the following conditions:(i) For any r , s ∈ R , there is a coaction ρ = ρ ( r, s ) ∶ R ( r, s ) Ð→ R ( r, s ) ⊗ B , f ↦ ∑ f ⊗ f , making R ( r, s ) a right B -comodule. Further, ρ ( id r ) = id r ⊗ B for each r ∈ R .(ii) For f ∈ R ( r, s ) and g ∈ R ( s, t ) , we have ρ ( g ○ f ) = ( g ○ f ) ⊗ ( g ○ f ) = ( g ○ f ) ⊗ ( g f ) (10.6) We have suppressed the summation signs in (10.6) . We will always refer to a right B -comodule category as a co- B -category. We will only consider those K -linear functors between co- B -categories whose action on morphisms is B -colinear. Together, the co- B -categories form a new category, which we will denote by Cat B . Lemma 10.6.
Let B be a bialgebra over K . Let R be a co- B -category and let C be a right B -module coalgebra. Thecollection ψ ∶ = ψ R = { ψ rs ∶ C ⊗ R ( r, s ) Ð→ R ( r, s ) ⊗ C } r,s ∈R defined by setting ψ rs ( c ⊗ f ) = f ψ ⊗ c ψ = f ⊗ cf f ∈ R ( r, s ) , c ∈ C (10.7) makes ( R , C, ψ ) an entwining structure.Proof. We consider morphisms f , g in R so that gf is defined. Then, for c ∈ C , we see that ( gf ) ψ ⊗ c ψ = ( gf ) ⊗ c ( gf ) = ( g f ) ⊗ c ( g f ) = g ψ f ψ ⊗ c ψψ f ψ ⊗ ∆ C ( c ψ ) = f ⊗ ∆ C ( cf ) = f ⊗ c f ⊗ c f = f ⊗ c f ⊗ c f = f ψψ ⊗ c ψ ⊗ c ψ ε C ( c ψ ) f ψ = ε C ( c ) ε B ( f ) f = ε C ( c ) f ψ ( c ⊗ id r ) = id r ⊗ c B (10.8)This proves the result. Proposition 10.7.
Let B be a K -bialgebra and let C be a right B -module coalgebra. If X is a small category, afunctor R ′ ∶ X Ð→ Cat B induces an entwined C -representation of XR ∶ X Ð→ E nt C x ↦ ( R x , C, ψ x ) ∶ = ( R ′ x , C, ψ R ′ x ) (10.9) Proof.
It may be easily verified that the entwining structures constructed in Lemma 10.6 are functorial with respectto B -colinear functors between B -comodule categories. This proves the result.We now consider a representation R ′ ∶ X Ð→ Cat B as in Proposition 10.7 and the corresponding entwined C -representation R ∶ X Ð→ E nt C . By considering the underlying K -linear category of any co- B -category, we obtainan induced representation that we continue to denote by R ′ ∶ X Ð→ Cat B Ð→ L in . We conclude by showing howentwined modules over R are related to modules over R ′ in the sense of Estrada and Virili [19]. Proposition 10.8.
Let B be a K -bialgebra and let C be a right B -module coalgebra. Let X be a small category, R ′ ∶ X Ð→ Cat B a functor and let R ∶ X Ð→ E nt C be the corresponding entwined C -representation. Then, amodule M over R consists of the following data:(1) A module M over the induced representation R ′ ∶ X Ð→ Cat B Ð→ L in .(2) For each x ∈ X and r ∈ R x a right C -comodule structure ρ xr ∶ M x ( r ) Ð→ M x ( r ) ⊗ C such that ρ xs ( mf ) = ( mf ) ⊗ ( mf ) = m f ⊗ m f for every f ∈ R x ( s, r ) and m ∈ M x ( r ) . (3) For each morphism α ∶ x Ð→ y in X , the morphism M α ( r ) ∶ M x ( r ) Ð→ ( α ∗ M y )( r ) is C -colinear for each r ∈ R x . roof. We consider a datum as described by the three conditions above. The conditions (1) and (2) ensure that each M x ∈ M C R x ( ψ x ) . For each x ∈ X , there is a forgetful functor F x ∶ M C R x ( ψ x ) Ð→ M R x . Let α ∶ x Ð→ y be a morphismin X . From (7.1), we know that ( α, id ) ∗ ∶ M C R y ( ψ y ) Ð→ M C R x ( ψ x ) and α ∗ ∶ M R y Ð→ M R x are well behaved withrespect to these forgetful functors. For each r ∈ R x , if M α ( r ) ∶ M x ( r ) Ð→ ( α ∗ M y )( r ) is also C -colinear, it followsthat M α is a morphism in M C R x ( ψ x ) . The result is now clear. References [1] J. Y. Abuhlail,
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