Categories of Modules, Comodules and Contramodules Over

arXiv:2106.12237v1 [math.RA] 23 Jun 2021 ainlpiig,trincass ucosbtencom between functors classes, torsion pairings, Rational represent 7 coalgebra over trans-contramodules Cartesian 6 representations coalgebra over Contramodules 5 representations coalgebra over comodules Cartesian 4 gener projective and poset a of representations Coalgebra 3 representations coalgebra over Comodules 2 Introduction 1 Contents objec cartesian contramodules, comodules, Modules, : Keywords 18E10 16T15, Classiﬁcation: Subject MSC(2020) aeoiso oue,cmdlsadcnrmdlsoe re over contramodules and comodules modules, of Categories ¶ ∗ § ‡ † . oue vragbarpeettos...... comodule between . functors . classes, torsion . modules, . Rational representations algebra over 7.2 Modules 7.1 ...... representations . . . coalgebra . over . . trans-comodules . representations . Cartesian coalgebra . . over . cis-comodules . 4.2 Cartesian . . . . 4.1 ...... trans-comodules . . for . . generators . . Projective . cis-comodules . for . . generators 3.2 . Projective . . . . 3.1 ...... representations coalgebra . over Trans-comodules representations coalgebra over 2.2 Cis-comodules 2.1 eateto ahmtc,Ida nttt fSineEdu Science of Institute Indian Mathematics, of Department Bwsprilyspotdb EBMtisflosi MTR/2 fellowship Matrics SERB by supported partially was AB Rwsprilyspotdb EBNtoa ototrlFe Postdoctoral National SERB by supported partially was SR eateto ahmtc,Ida nttt fSine Ba Science, of Institute Indian Mathematics, of Department eateto ahmtc,Ida nttt fSine Ba Science, of Institute Indian Mathematics, of Department rtedekctgre,bcuetelte a etreated be may latter arguments the counterpa cardinality because as categories, using viewed Grothendieck be properties may categorical which natural categories, their arise these morphisms of their each in and coalgebras betw functors of adjoint module properties the to all similar incorporates manner which a framework in (co)algebras, in values taking esuyadrlt aeoiso oue,cmdlsadcon and comodules modules, of categories relate and study We at Balodi Mamta ∗ bihkBanerjee Abhishek glr 602 ni.Eal abhishekbanerjee1313@gmai Email: India. 560012, ngalore glr 602 ni.Eal [email protected] Email: India. 560012, ngalore ainadRsac,Pn 108 ni.Eal ray.samarp Email: India. 411008, Pune Research, and cation lwhpPDF/2020/000670 llowship Abstract 017/000112 u ou so eeaosfrteectgre n on and categories these for generators on is focus Our . 1 e hs aeoisi aua anr aiu classical Various manner. natural a in categories these een ywti hster.W locnie atsa objects cartesian consider also We theory. this within ly vrarne pc.A eut eoti categorical a obtain we result, a As space. ringed a over s srpaeet o ocmuaiespaces. noncommutative for replacements as t fqaichrn hae vrashm.W study We scheme. a over sheaves quasi-coherent of rts s rtedekcategories Grothendieck ts, rmdlsoe ersnaino ml category small a of representation a over tramodules ations dlsadcnrmdls30 contramodules and odules †‡ 17 ...... tr o ooue 11 comodules for ators n otaoue 31 ...... contramodules and s 14 ...... 9 ...... 12 ...... 11 ...... 5 ...... aapt Ray Samarpita 30 ...... §¶ presentations l.com [email protected] 25 20 14 5 2 1 Introduction

The purpose of this paper is to obtain an algebraic geometry like categorical framework that studies modules, comodules and contramodules over a representation of a small category taking values in (co)algebras. In classical algebraic geometry, one usually has a ringed site, or more generally a ringed category (X, O) consisting of a small category X and a presheaf O of commutative rings on X. Accordingly, a module M over (X, O) corresponds to a family {Mx}x∈X , where each Mx is an Ox module, along with compatible morphisms. In more abstract settings, the idea of studying schemes by means of module categories linked with adjoint pairs given by extension and restriction of scalars is well developed in the literature. This appears for instance in the relative algebraic geometry over symmetric monoidal categories (see Deligne [12], Toën and Vaquié[43]), in derived algebraic geometry (see Lurie [23]) and in homotopical algebraic geometry (see Toën and Vezzosi [41], [42]). In [15], Estrada and Virili considered a representation A : X −→ Add of a small category X taking values in the category Add of small preadditive categories. Following the philosophy of Mitchell [24], the small preadditive categories play the role of “algebras with several objects.” An object M in the category Mod-A of A-modules consists of the data of an Ax-module Mx for each x ∈ Ob(X ), along with compatible morphisms corresponding to extension or restriction of scalars. The authors in [15] then establish a number of categorical properties of A-modules, as also those of cartesian objects in Mod-A, the latter being similar to quasi-coherent modules over a scheme. As such, the study in [15] not only takes the philosophy of Mitchell one step further, but also provides a categorical framework for studying modules over ringed categories where the algebras are not necessarily commutative. The work of [15] is our starting point. For a small category X , we consider either a representation C : X −→ Coalg taking values in coalgebras or a representation A : X −→ Alg taking values in algebras. In place of modules, we consider three different “module like” categories; those of modules, comodules and contramodules as well as incorporate all the adjoint functors between them into our theory. In doing so, we have two objectives. First of all, in each of these contexts, we also work with cartesian objects, which play a role similar to quasi-coherent sheaves over a scheme. By a classical result of Gabriel [16] (see also Rosenberg [34], [35], [36]) we know that under certain conditions, a scheme can be reconstructed from its category of quasi-coherent sheaves. As such, the categories of cartesian comodules or cartesian contramodules may be viewed as a step towards constructing a scheme like object related to comodules or contramodules over coalgebras. Our focus is on Grothendieck categories appearing in these contexts and more generally on generators of these categories. This is because Grothendieck categories may be treated as a replacement for noncommutative spaces as noted in [22]. The latter is motivated by the work of [4], [5], [39] as well as the observation in [21] that the Gabriel-Popescu theorem for Grothendieck categories may be viewed as an additive version of Giraud’s theorem. Secondly, our methods enable us to explore the richness of the theory of comodules and coalgebras with the flavor of algebraic geometry. In this framework, it becomes natural to include the theory of contramodules, which have been somewhat neglected in the literature. Formally, the notion of a contramodule is also dual to that of a module, if we write the structure map of a module P over an algebra A as a morphism P −→ Hom(A, P ) instead of the usual P ⊗ A −→ P . Accordingly, a contramodule M over a coalgebra C consists of a space M as well as a morphism Hom(C,M) −→ M satisfying certain coassociativity and counit conditions (see Section 5). While contramodules were introduced much earlier by Eilenberg and Moore [14, § IV.5], the subject has seen a lot of interest in recent years (see, for instance, [7], [8], [10], [26], [27], [28], [29], [30], [31], [32], [37], [44]). One important aspect of our paper is that for comodules over a coalgebra representation C : X −→ Coalg or modules over an algebra representation A : X −→ Alg, it becomes necessary to work with objects of two different orientations, which we refer to as “cis-objects” and “trans-objects.” We shall see that cis-comodules over a coalgebra representation are related to trans-modules over its dual algebra representation and vice-versa. For a coalgebra C over a field K, let MC denote the category of right C-comodules. Given a morphism α : C −→ D of coalgebras, we consider a system of three different functors between comodule categories ! D C ∗ C D D C α : M −→ M α : M −→ M α∗ : M −→ M (1.1) ∗ C D Here, α : M −→ M is the corestriction of scalars and its right adjoint is given by the cotensor product α∗ = D C DC : M −→ M . In addition, if α : C −→ D makes C quasi-finite as a right D-comodule (see Section 2.2), then

2 the left adjoint α! : MD −→ MC of the corestriction functor α∗ also exists. In order to define a cis-comodule M over a coalgebra representation C : X −→ Coalg, we need a collection {Mx}x∈Ob(X ) where each Mx is a Cx-comodule, along α ∗ with compatible morphisms M : α Mx −→ My of Cy-comodules for α ∈ X (x, y) (see Definition 2.2). Equivalently, we have morphisms Mα : Mx −→ α∗My of Cx-comodules for each α ∈ X (x, y). By combining techniques on comodules with adapting the cardinality arguments of [15], we study the category Comcs-C of cis-comodules and give conditions for it to be a Grothendieck category. It turns out that the relevant criterion is for the representation C : X −→ Coalg to take values in semiperfect coalgebras, i.e., those for which the category of comodules has enough projectives. The semiperfect coalgebras also return in the last section, where they make an interesting appearance with respect to torsion theories. ∗ On the other hand, a trans-comodule M over C : X −→ Coalg consists of morphisms αM : My −→ α Mx for each α ∈ X (x, y). Whenever the coalgebra representation is quasi-finite, i.e., each morphism Cx −→ Cy of coalgebras induced α ! by C : X −→ Coalg makes Cx quasi-finite as a Cy-comodule, this is equivalent to having morphisms M : α My −→ Mx for each α ∈ X (x, y). We study the category Comtr-C of trans-comodules in a manner similar to Comcs-C. When the small category X is a poset, we show that the evaluation functor at each x ∈ Ob(X ) has both a left and a right adjoint. This enables us to construct explicit projective generators for Comcs-C and Comtr-C, by making use of the projective generators in the category of comodules over each of the semiperfect coalgebras {Cx}x∈Ob(X ). A cartesian object in the category of cis-comodules consists of M ∈ Comcs-C such that for each α ∈ X (x, y), the morphism Mα : Mx −→ α∗My is an isomorphism. In order to study these objects, we will suppose that the repre- X sentation C : −→ Coalg is coflat, i.e., the cotensor products Cy Cx corresponding to any morphism Cx −→ Cy of coalgebras induced by C are exact. By using a transfinite induction argument adapted from [15], we will show that for any coflat and semiperfect representation C : X −→ Coalg of a poset, there exists a cardinal κ′ such that any cartesian cis-comodule M over C can be expressed as a filtered union of cartesian subcomodules each of cardinality ≤ κ′ (see cs Theorem 4.8). It follows in particular that the category Comc -C of cartesian cis-comodules over such a representation is a Grothendieck category. Here, we also refer the reader to the classical result of Gabber (see, for instance, [38, Tag 077K]), which shows that the category of quasi-coherent sheaves over a scheme is a Grothendieck category. We also cs cs obtain a right adjoint of the inclusion functor Comc -C ֒→ Com -C, which may be viewed as a coalgebraic counterpart of the classical quasi-coherator construction (see Illusie [18, Lemme 3.2]). We will say that a quasi-finite morphism α : C −→ D of coalgebras is right Σ-injective if the direct sum C(Λ) is injective as a right D-comodule for any indexing set Λ. This is equivalent (see [3, Corollary 3.10]) to the functor α! : MD −→ MC tr α ! being exact. An object M ∈ Com -C is cartesian if the morphism M : α My −→ Mx is an isomorphism for each X tr α ∈ (x, y). We then study the category Comc -C of cartesian trans-contramodules over a Σ-injective and semiperfect cs coalgebra representation in a manner similar to Comc -C.

If C is a coalgebra over a ﬁeld K, we denote by M[C, ] the category of (right) C-contramodules. If α : C −→ D is a morphism of coalgebras, we have a pair of adjoint functors

• α : M[D, ] −→ M[C, ] α• : M[C, ] −→ M[D, ] (1.2) between corresponding categories of contramodules. Here, the functor α• is the contrarestriction of scalars, obtained C by treating a C-contramodule (M, πM : HomK (C,M) −→ M) as a D-contramodule with induced structure map C πM HomK (D,M) −→ HomK (C,M) −−→ M. Let C : X −→ Coalg be a coalgebra representation. A trans-contramodule M over C consists of a Cx-contramodule Mx for each x ∈ Ob(X ) along with compatible morphisms αM : My −→ α•Mx α • for α ∈ X (x, y). Equivalently, we have a morphism M : α My −→ Mx for α ∈ X (x, y). We note that we have only a single pair of adjoint functors in (1.2), unlike the system of three adjoint functors for comodule categories in (1.1). As such, we consider only the category Conttr-C of trans-contramodules over a representation C : X −→ Coalg but no cis-contramodules. In comparison to comodule categories, working with contramodules presents certain diﬃculties. The ﬁrst among these is the fact that the category M[C, ] of contramodules over a K-coalgebra C is not usually a Grothendieck category. Further, direct sums in the category of contramodules do not correspond in general to the direct sums of their underlying

3 vector spaces. As a result, we work by considering morphisms from presentable generators in contramodule categories, using an adjunction between vector spaces and contramodules. Interestingly, the category of contramodules M[C, ] does contain enough projectives. For a coalgebra representation C : X −→ Coalg we show that Conttr-C has a set of generators. When the small category X is a poset, we show that Conttr-C is in fact locally presentable and we construct a set of projective generators for Conttr-C. • When α : C −→ D is a coflat morphism of coalgebras, one observes that the contraextension functor α : M[D, ] −→ M[C, ] is exact. Accordingly, we say that a trans-contramodule M over a coflat representation C : X −→ Coalg is α • cartesian if the morphism M : α My −→ Mx is an isomorphism for each α ∈ X (x, y). For each x ∈ Ob(X ), we choose ∗ a regular cardinal λx such that the dual Cx = HomK (Cx,K) is a λx-presentable generator in M[Cx, ]. Since colimits of contramodules do not correspond to the colimits of underlying vector spaces, we rely extensively on presentable objects X X in each M[Cx, ] to set up a transfinite induction argument. When is a poset and C : −→ Coalg is a coflat representation, we show that there is a regular cardinal κ′′ such that any element of a cartesian trans-contramodule over C lies in a cartesian subobject having cardinality ≤ κ′′ (see Theorem 6.11). Finally, we consider modules over an algebra representation A : X −→ Alg and proceed to relate the categories of modules, comodules and contramodules to each other. For any K-algebra A, we denote by MA the category of right A-modules. Corresponding to any morphism α : A −→ B of algebras, we have a system of three adjoint functors

◦ † α : MA −→ MB α◦ : MB −→ MA α : MA −→ MB (1.3)

◦ † Here α◦ is the usual restriction of scalars, α is its left adjoint, while α is its right adjoint. Accordingly, we can define categories Modcs-A and Modtr-A respectively of right cis-modules and right trans-modules over A. We observe in particular that the cis-modules recover the modules of Estrada and Virili [15] (in the case where the representation in [15] takes values in K-algebras). We now relate comodules over a coalgebra representation C : X −→ Coalg to modules over its linear dual representation C∗ : X op −→ Alg as well as modules over an algebra representation A : X op −→ Alg to comodules over its finite dual representation A◦ : X −→ Coalg. More generally, we define a rational pairing (C, A, Φ) of a coalgebra representation C : X −→ Coalg with an algebra representation A : X op −→ Alg. We recall (see, for instance, [11, § 4.18]) that if ϕ : C ⊗ A −→ K is a rational pairing of a coalgebra C with an algebra A, the category of right C-comodules can be embedded as the category of rational left A-modules. Accordingly, we construct pairs of adjoint functors

tr cs IΦ IΦ Comcs-C A-Modtr Comtr-C A-Modcs (1.4) tr cs RΦ RΦ

cs tr cs tr Here, IΦ and IΦ are inclusion functors while their respective right adjoints RΦ and RΦ are “rationalization functors.” Thereafter, using some classical results from [20] on coalgebras and density in module categories, we give conditions for the full subcategory Comcs-C to be a torsion class in A-Modtr. In particular, it follows that if C : X −→ Coalg is a representation taking values in semiperfect coalgebras, then Comcs-C becomes a hereditary torsion class in C∗-Modtr. For any coalgebra representation C : X −→ Coalg, we also construct a canonical functor Conttr-C −→ Modcs-C∗. We conclude with the following result: if C : X −→ Coalg is a quasi-ﬁnite representation taking values in cocommutative tr coalgebras, for each cartesian trans-comodule N ∈ Comc -C, we produce a pair of adjoint functors

⊠C N Conttr-C Comtr-C (1.5) (N , ) determined by N (see Theorem 7.20). Here, the left adjoint is obtained by using the contratensor product of a contramodule and a comodule constructed by Positselski [26]. Acknowledgements: The authors are grateful to L. Positselski for a useful discussion.

4 2 Comodules over coalgebra representations

Throughout this paper, K will denote a ﬁeld and V ect the category of K-vector spaces. Let C be a K-coalgebra. We denote by MC (resp. C M) the category of right C-comodules (resp. left C-comodules). We also denote by ρM : M −→ M ⊗ C (resp. M ρ : M −→ C ⊗ M) the right coaction (resp. the left coaction) for M ∈ MC (resp. M ∈ C M). C C For M ∈ M and N ∈ M, we recall (see, for instance, [11, §10.1]) that their cotensor product MCN is given by the equalizer of the two maps M ρ ⊗idN M ⊗ NM ⊗ C ⊗ N N idM ⊗ ρ (2.1)

C If M ∈ M , we note the isomorphism MCC ≃ M of right C-comodules. A coalgebra morphism α : C −→ D induces a pair of adjoint functors ∗ C D M M α : M −→ M (M,ρ ) 7→ M, (idM ⊗ α) ◦ ρ D C α : M −→ M N 7→ NDC ∗ ∗ The left adjoint α is known as the corestriction functor, while the right adjoint α∗ is known as the coinduction functor. Deﬁnition 2.1. Let X be a small category and let Coalg denote the category of K-coalgebras. By a representation of X on the category of K-coalgebras, we will mean a functor C : X −→ Coalg.

In particular, for each object x ∈ Ob(X ), we have a coalgebra Cx and for any morphism α ∈ X (x, y), we have a morphism Cα : Cx −→ Cy of coalgebras.

∗ By abuse of notation, given a coalgebra representation C : X −→ Coalg, we will simply write α , α∗ and so on for the ∗ X respective functors Cα, Cα∗, etc between corresponding categories of comodules. Corresponding to a representation of on the category of coalgebras, we will now deﬁne two types of comodules, namely, cis-comodules and trans-comodules. We will work with cis-comodules in the ﬁrst subsection and with trans-comodules in the next.

2.1 Cis-comodules over coalgebra representations Deﬁnition 2.2. Let C : X −→ Coalg be a coalgebra representation. A (right) cis-comodule M over C will consist of the following data:

(1) For each object x ∈ Ob(X ), a right Cx-comodule Mx X (2) For each morphism α : x −→ y in , a morphism Mα : Mx −→ My Cy Cx = α∗My of right Cx-comodules α ∗ (equivalently, a morphism M : α Mx −→ My of right Cy-comodules)

α β X We further assume that Midx = idMx and for any pair of composable morphisms x −→ y −→ z in , we have α∗(Mβ) ◦Mα = Mβα : Mx −→ α∗My −→ α∗β∗Mz = (βα)∗Mz. The later condition can be equivalently expressed as Mβα = Mβ ◦ β∗(Mα). A morphism η : M −→ N of cis-comodules over C consists of morphisms ηx : Mx −→ Nx of right Cx-comodules for x ∈ Ob(X ) such that for each morphism x −→ y in X the following diagram commutes

ηx Mx Nx

Mα Nα

α∗ηy α∗My α∗Ny We denote this category of right cis-comodules by Comcs-C. Similarly, we may deﬁne the category C-Comcs of left cis-comodules over C.

5 Proposition 2.3. Let C : X −→ Coalg be a coalgebra representation. Then, Comcs-C is an abelian category. Proof. Clearly, Comcs-C has a zero object. For any morphism η : M −→ N in Comcs-C, we deﬁne the kernel and cokernel of η by setting

Ker(η)x := Ker(ηx : Mx −→ Nx) Coker(η)x := Coker(ηx : Mx −→ Nx) (2.2) for each x ∈ X . For α : x −→ y in X , the exactness of the corestriction functor α∗ : MCx −→ MCy induces the α ∗ α ∗ morphisms Ker(η) : α Ker(η)x −→ Ker(η)y and Coker(η) : α Coker(η)x −→ Coker(η)y. It is also clear from (2.2) .((that Coker(Ker(η) ֒→M)= Ker(N ։ Coker(η We will now study generators in the category Comcs-C. For this, we recall (see, for instance, [13, Corollary 2.2.9]) that any right C-comodule is the sum of its finite dimensional subcomodules. If C is a right semiperfect K-coalgebra, i.e., the category MC has enough projective objects, we know (see, for instance, [13, Corollary 2.4.21]) that for any finite dimensional right N ∈ MC, there exists a finite dimensional projective P ∈ MC and an epimorphism P −→ N in MC . Therefore, it follows that if C is right semiperfect, MC has finitely generated projective generators.

For studying generators in Comcs-C, we will now adapt the steps in [6, § 4], which are motivated by Estrada and Virili [15]. In fact, we will use a similar argument in several contexts throughout this paper. Let C : X −→ Coalg be a coalgebra representation. Let M be a right cis-comodule over C. We consider an object x ∈ Ob(X ) and a morphism

η : V −→ Mx (2.3)

Cx where V is a ﬁnite dimensional projective in M . For each object y ∈ Ob(X ), we set Ny ⊆My to be the image of the family of maps

∗ β ∗ β ∗ β η ∗ M ∗ β η ∗ M Ny = Im( β V −−→ β Mx −−→My) = Im(β V −−→ β Mx −−→My) (2.4) X X β∈M(x,y) β∈ X(x,y)

Cy X ′ β ∗ ∗ in M . Let iy denote the inclusion Ny ֒→My and for each β ∈ (x, y), we denote by ηβ = M ◦ β η : β V −→ Ny the canonical morphism induced from (2.4). Lemma 2.4. Let α ∈ X (y,z) and β ∈ X (x, y). Then, the following composition

η′ ∗ β iy Mα β V −→Ny −→My −−→ α∗Mz (2.5) factors through α∗(iz): α∗Nz −→ α∗Mz. Proof. It is enough to show that the composition

∗ ′ α (η ) ∗ α ∗ ∗ β ∗ α iy ∗ M α β V −−−−→ α Ny −−−→ α My −−→Mz (2.6)

∗ factors through iz : Nz −→ Mz since (α , α∗) is an adjoint pair. By deﬁnition, we know that the composition η′ ∗ β iy ∗ ′ β ∗ ∗ α β V −→ Ny −→My factors through β Mx i.e., we have iy ◦ η β = M ◦ β η. Applying α , composing with M and using Deﬁnition 2.2 we obtain

α ∗ ∗ ′ α ∗ β ∗ ∗ αβ ∗ ∗ M ◦ α (iy) ◦ α (η β)= M ◦ α (M ) ◦ α (β η)= M ◦ α β η (2.7)

αβ ∗ ∗ α By the deﬁnition in (2.4), we know that M ◦ α β η factors through iz : Nz −→ Mz and therefore so does M ◦ ∗ ∗ ′ α (iy) ◦ α (η β).

6 For the rest of this subsection, we suppose that C : X −→ Coalg is a coalgebra representation such that each Cx is right semiperfect, i.e., MCx has a set of projective generators.

Cy cs Proposition 2.5. The objects {Ny ∈ M }y∈Ob(X ) together determine a subobject N ⊆M in Com -C.

Proof. Let α ∈ X (y,z). Since α∗ is a right adjoint, it preserves monomorphisms and it follows that α∗(iz): α∗Nz −→ Cy α∗Mz is a monomorphism in M . We will now show that the morphism Mα : My −→ α∗Mz restricts to a morphism Nα : Ny −→ α∗Nz giving us a commutative diagram

iy Ny My

Nα Mα

α∗(iz ) α∗Nz α∗Mz

Cy Since Cy is right semiperfect, we can ﬁx a set {Gk}k∈K of projective generators for M . Using [6, Lemma 3.2], it ′ suﬃces to show that for any k ∈ K and any morphism ζk : Gk −→ Ny, there exists ζk : Gk −→ α∗Nz such that ′ α∗(iz) ◦ ζk = Mα ◦ iy ◦ ζk. By 2.4, we have an epimorphism

′ ∗ ηβ : β V −→ Ny (2.8) X X β∈M(x,y) β∈M(x,y)

Cy ′′ ∗ in M . Since Gk is projective, the morphism ζk : Gk −→ Ny can be lifted to a morphism ζk : Gk −→ β∈X (x,y) β V such that L ′ ′′ ζk = ηβ ◦ ζk (2.9)  X  β∈M(x,y) ′   X We know from Lemma 2.4, that Mα ◦ iy ◦ ηβ factors through α∗(iz): α∗Nz −→ α∗Mz for each β ∈ (x, y). It now follows from (2.9) that Mα ◦ iy ◦ ζk factors through α∗(iz) as required. ′ X Lemma 2.6. Let η1 : V −→ Nx be the canonical morphism corresponding to the identity map in (x, x). Then, for any y ∈ Ob(X ), we have

∗ ′ β ∗ β η1 ∗ N Ny = Im β V −−−→ β Nx −−→Ny (2.10)  X  β∈M(x,y)   Proof. Let β ∈ X (x, y). We consider the following commutative diagram:

∗ ′ β ∗ β η1 ∗ N β V β Nx Ny

∗ β ix iy (2.11) β ∗ M β Mx My

′ ∗ ∗ ′ ∗ Since ix ◦ η1 = η, we have (β ix) ◦ (β η1)= β η. This gives β ∗ β ∗ ∗ ′ β ∗ ′ β ∗ ′ Im M ◦ (β η) = Im M ◦ (β ix) ◦ (β η1) = Im iy ◦ N ◦ (β η1) = Im N ◦ (β η1) where the last equality follows from the fact that iy is a monomorphism. The result now follows directly from the deﬁnition in (2.4). Lemma 2.7. For any y ∈ Ob(X ), we have

∗ β ∗ β (ix) ∗ M Ny = Im β Nx −−−−→ β Mx −−→My (2.12) X β∈ X(x,y)

7 ∗ β ′ ∗ β (ix) M ′ Proof. We set Ny = β∈X (x,y) Im β Nx −−−−→ β∗Mx −−→My . We want to show that Ny = Ny. It follows

∗ β P ∗ β (ix) ∗ M from the commutative diagram 2.11 that each of the morphisms β Nx −−−−→ β Mx −−→ My factors through the ′ subcomodule Ny ⊆My and hence Ny ⊆ Ny. Also, clearly

∗ ′ ∗ β ∗ β ∗ β η1 ∗ β (ix) M ∗ β (ix) ∗ M Im β V −−−→ β Nx −−−−→ β∗Mx −−→My ⊆ Im β Nx −−−−→ β Mx −−→My (2.13) ′ Using Lemma 2.6, we now have Ny ⊆ Ny . This proves the result. cs For M ∈ Com -C, we denote by elX (M) the union ∪x∈Ob(X )Mx as sets. The cardinality of elX (M) will be denoted by |M|. It is easy to see that for any quotient or subobject N of M ∈ Comcs-C, we have |N | ≤ |M|. Now, we set

κ := sup{ℵ0, |Mor(X )|, |K|} (2.14)

∗ ′ ′ We note in particular that |β V |≤ κ for any ﬁnite dimensional Cx-comodule V . Lemma 2.8. Let N be as constructed in Proposition 2.5. Then, we have |N | ≤ κ. Proof. Let y ∈ Ob(X ). From Lemma 2.6, we have

∗ ′ β ∗ β η1 ∗ N Ny = Im β V −−−→ β Nx −−→Ny (2.15)  X  β∈M(x,y)   ∗ In other words, Ny is an epimorphic image of β∈X (x,y) β V and we have

L ∗ |Ny| ≤ | β V |≤ κ (2.16) X β∈M(x,y)

Thus, |N | = y∈Ob(X ) |Ny|≤ κ. Theorem 2.9.P Let C : X −→ Coalg be a representation taking values in right semiperfect K-coalgebras. Then, the category Comcs-C of right cis-comodules over C is a Grothendieck category. Proof. Since finite limits and filtered colimits in Comcs-C are both computed pointwise, it is clear that they commute in Comcs-C. cs Now, let M be an object in Com -C and m ∈ elX (M). Then, there exists some x ∈ Ob(X ) such that m ∈Mx. By ′ ′ ′ Cx [13, Corollary 2.2.9], there exists a finite dimensional right Cx-comodule V and a morphism η : V −→ Mx in M ′ Cx such that m ∈ Im(η ). Since Cx is semiperfect, we can choose a finite dimensional projective V in M along with an epimorphism V −→ V ′ in MCx (see, for instance, [13, Corollary 2.4.21]). Therefore, we have an induced morphism Cx cs η : V −→ Mx in M such that m ∈ Im(η). We can now define the subobject N ⊆M in Com -C corresponding to η ′ as in (2.4). From the definition of the canonical morphism η1 : V −→ Nx induced by (2.4), it follows that m ∈ Nx. By Lemma 2.8, we also have |N | ≤ κ. We now consider the set of isomorphism classes of objects in Comcs-C having cardinality ≤ κ. From the above, we see that any object in Comcs-C may be expressed as a sum of such objects. This proves the result.

8 2.2 Trans-comodules over coalgebra representations Let C be a K-coalgebra. An object M ∈ MC is said to be quasi-finite (see, for instance, [11, § 12.5]) if the tensor C C functor −⊗ M : V ect −→ M has a left adjoint. In that case, this left adjoint is denoted by HC (M, −): M −→ V ect. We note that the coalgebra C is quasi-finite as a right C-comodule. Suppose that α : C −→ D is a coalgebra morphism such that C is quasi-finite as a right D-comodule. Then, for any D N ∈ M , the space HD(C,N) can be equipped with the structure of a right C-comodule. This determines a functor D C C D HD(C, −): M −→ M which is the left adjoint to the functor −C C : M −→ M (see, for instance, [11, § 12.7]). C D However, −C C : M −→ M is merely the corestriction functor, i.e., we have

C D D ∗ M (HD(C, V ), W ) =∼ M (V, W CC) =∼ M (V, α (W )) for any V ∈ MD and W ∈ MC. In other words, if C is quasi-finite as a right D-comodule, then the corestriction functor ∗ C D ! D C α : M −→ M has a left adjoint α := HD(C, −): M −→ M . Definition 2.10. Let C : X −→ Coalg be a coalgebra representation. Suppose that C is quasi-finite, i.e., for any morphism α : x −→ y in X , the corresponding morphism Cα : Cx −→ Cy of coalgebras makes Cx quasi-finite as a right Cy-comodule. A (right) trans-comodule M over C will consist of the following data:

(1) For each object x ∈ Ob(X ), a right Cx-comodule Mx X α ! (2) For each morphism α : x −→ y in , a morphism M := α My = HCy (Cx, My) −→ Mx of right Cx-comodules ∗ (equivalently, a morphism αM : My −→ α Mx of right Cy-comodules)

α β idx X We further assume that M = idMx and for any pair of composable morphisms x −→ y −→ z in , we have α ! β βα ! ! ! ! M◦ α ( M)= M : (βα) Mz = α β Mz −→ α My −→ Mx. The latter condition can be equivalently expressed as ∗ βαM = β (αM) ◦ βM. A morphism η : M −→ N of trans-comodules over C consists of morphisms ηx : Mx −→ Nx of right Cx-comodules for each x ∈ Ob(X ) such that for each morphism x −→ y in X the following diagram commutes

! ! α ηy ! α My α Ny

αM αN

ηx Mx Nx We denote the category of right trans-comodules by Comtr-C. Similarly, we may define the category C-Comtr of left trans-comodules over C. We remark that if C′ : X −→ Coalg is any coalgebra representation (not necessarily quasi-finite), we can still define the tr ′ ′ category Com -C of trans-comodules over C by considering only the set of maps {αM}α∈Mor(X ) in Definition 2.10 ∗ satisfying βαM = β (αM) ◦ βM for composable morphisms α, β in X . However, the assumption of quasi-finiteness will be necessary to establish most of the properties of trans-comodules that we study in this paper. Let C : X −→ Coalg be a coalgebra representation that is quasi-finite. As in the proof of Proposition 2.3, it follows from the exactness of the corestriction functor α∗ that Comtr-C is an abelian category. We will now study generators tr for the category Com -C. For this, we will suppose that each Cx is a right semiperfect coalgebra. Accordingly, let M ∈ Comtr-C. We consider an object x ∈ Ob(X ) and a morphism

η : V −→ Mx (2.17)

Cx in M , where V is a ﬁnite dimensional projective right comodule over Cx. For each y ∈ Ob(X ), we set Ny ⊆ My to be the image of the family of maps

9 ! β ! β ! β η ! M ! β η ! M Ny = Im( β V −−→ β Mx −−→My) = Im(β V −−→ β Mx −−→My) (2.18) X X β∈M(y,x) β∈ X(y,x) X ′ ! Let iy denote the inclusion Ny ֒→My and for each β ∈ (y, x), we denote by ηβ : β V −→ Ny the canonical morphism induced from (2.18). In a manner similar to the proof of Lemma 2.4, we can show that for any α ∈ X (z,y), the composition η′ ! β iy αM ∗ β V −→Ny −→My −−→ α Mz (2.19) ∗ ∗ ∗ factors through α (iz): α Nz −→ α Mz. Since each Cx is right semiperfect, we can prove the following result, the proof of which is similar to that of Proposition 2.5.

Cy tr Proposition 2.11. The objects {Ny ∈ M }y∈Ob(X ) together determine a subobject N ⊆M in Com -C. We also record here the following two equalities, which can be proved in a manner similar to Lemma 2.6 and Lemma 2.7

! ′ β ! β ! β η1 ! N ! β (ix) ! M Ny = Im β V −−−→ β Nx −−→Ny Im β Nx −−−−→ β Mx −−→My (2.20)  X  X β∈M(y,x) β∈ X(y,x) X  ′  X for any y ∈ Ob( ). Here, η1 : V −→ Nx is the canonical morphism corresponding to the identity map in (x, x). Lemma 2.12. Let α : C −→ D be a morphism of coalgebras such that C is quasi-finite as a right D-comodule. Let V ∈ MD be a finite dimensional comodule. Then, α!V ∈ MC is finite dimensional. Proof. Since every comodule is a colimit of its finite dimensional subcomodules, a C-comodule U is finitely generated as an object of MC (i.e., MC(U, ): MC −→ V ect preserves filtered colimits of systems of monomorphisms) if and only if C U is finite dimensional. Suppose that {Wi}i∈I is a filtered system of comodules in M connected by monomorphisms and let W = lim W . Since α∗ : MC −→ MD preserves all colimits and all finite limits, we note −→ i i∈I lim MC (α!V, W ) = lim MD(V, α∗W )= MD(V, lim α∗W )= MD(V, α∗W )= MC (α!V, W ) (2.21) −→ i −→ i −→ i i∈I i∈I i∈I It follows from (2.21) that α!V ∈ MC is finitely generated as an object in MC , i.e., it is finite dimensional as a K-vector space.

tr For M ∈ Com -C, let elX (M) denote the union ∪x∈Ob(X )Mx. The cardinality of elX (M) will be denoted by |M|. It is easy to see that for any quotient or subobject N of M ∈ Comtr-C, we have |N | ≤ |M|. Now, we set

κ := sup{ℵ0, |K|, |Mor(X )|} (2.22) Lemma 2.13. Let N be as constructed in Proposition 2.11. Then, we have |N | ≤ κ. Proof. We consider some β ∈ X (y, x). Since V is a ﬁnite dimensional projective in MCx , it follows from Lemma 2.12 that β!V ∈ MCy is ﬁnite dimensional. It is now clear from (2.22) that |β!V |≤ κ. From (2.20), we now have

! ′ β ! β η1 ! N Ny = Im β V −−−→ β Nx −−→Ny (2.23)  X  β∈M(y,x)  !  Since Ny is an epimorphic image of β∈X (y,x) β V , we see that

! L |Ny| ≤ | β V |≤ κ (2.24) X β∈M(y,x)

Thus, |N | = y∈Ob(X ) |Ny|≤ κ. P 10 Theorem 2.14. Let C : X −→ Coalg be a representation taking values in right semiperfect K-coalgebras. Suppose that C is quasi-finite, i.e., for each morphism α : z −→ y in X , the induced morphism Cα : Cz −→ Cy of coalgebras makes Cz tr a quasi-finite right Cy-comodule. Then, the category Com -C of right trans-comodules over C is a Grothendieck category. Proof. Since filtered colimits and finite limits in Comtr-C are computed pointwise, it is clear that they commute in tr tr Com -C. Now, let M be an object in Com -C and m ∈ elX (M). Then, there exists some x ∈ Ob(X ) such that ′ ′ ′ m ∈Mx. By [13, Corollary 2.2.9], there exists a finite dimensional right Cx-comodule V and a morphism η : V −→ Mx Cx ′ Cx in M such that m ∈ Im(η ). Since Cx is semiperfect, we can choose a finite dimensional projective V in M along ′ Cx Cx with an epimorphism V −→ V in M . Therefore, we have an induced morphism η : V −→ Mx in M such that m ∈ Im(η). We can now define the subcomodule N ⊆M corresponding to η as in (2.18). From the definition of the ′ canonical morphism η1 : V −→ Nx in (2.20), it follows that m ∈ Nx. By Lemma 2.13, we also have |N | ≤ κ. It follows that isomorphism classes of objects in Comtr-C having cardinality ≤ κ give a set of generators for Comtr-C.

3 Coalgebra representations of a poset and projective generators for comodules

Throughout this section, we assume that X is a partially ordered set. We suppose that C : X −→ Coalg is a coalgebra representation such that Cx is right semiperfect for each x ∈ Ob(X ). Our objective is to show that under these conditions, both Comcs-C and Comtr-C have projective generators.

3.1 Projective generators for cis-comodules Proposition 3.1. Let x ∈ Ob(X ). Then, cs Cx cs X (1) There is a functor exx : M −→ Com -C deﬁned by setting, for any y ∈ Ob( ): ∗ X cs α (M) if α ∈ (x, y) exx (M)y = (0 if X (x, y)= ∅

cs cs Cx (2) The evaluation at x, i.e., evx : Com -C −→ M , M 7→ Mx is an exact functor. cs cs (3) (exx , evx ) is a pair of adjoint functors. cs Cy X ′ cs β Proof. (1) Clearly, exx (M)y ∈ M for each y ∈ Ob( ). We consider β : y −→ y . If x 6≤ y, then exx (M) = 0. ′ cs β ∗ ∗ ∗ Otherwise, if we have α : x −→ y and γ : x −→ y , i.e., βα = γ, we note that id = exx (M) : β α (M) −→ γ (M). It cs cs is now clear that exx (M) ∈ Com -C. (2) This follows from the fact that finite limits and finite colimits in Comcs-C are computed pointwise. cs Cx cs cs ∼ Cx cs (3) Let M ∈ Com -C and N ∈ M . We will show that Com -C(exx (N), M) = M (N,evx (M)). We consider Cx f cs f : N −→ Mx in M . Corresponding to f, we now have η : exx (N) −→ M defined by setting ∗ α f cs ∗ α f ∗ M ηy : exx (N)y = α N −−−−→ α Mx −−−−→ My (3.1) X X f X ′ β ∗ f for y ∈ Ob( ) whenever α ∈ (x, y) and ηy = 0 if x 6≤ y. We now take β ∈ (y,y ) and claim that M ◦ β (ηy )= f cs β X ηy′ ◦exx (N) . If x 6≤ y, both sides of this equality vanish. Otherwise, if α ∈ (x, y), we have the commutative diagram

β∗(ηf ) ∗ cs ∗ ∗ y ∗ β exx (N)y = β α (N) −−−−−−−−−−−→ β My β∗(Mα)◦β∗(α∗f)

cs β β exx (N) id M (3.2)

f  η ′  cs  ∗ ∗ y  exx (N)y′ y= β α (N) −−−−−−−−−→ Myy′ Mβα◦(β∗α∗f)

11 f cs cs which shows that η is a morphism in Com -C. Conversely, given a morphism η : exx(N) −→ M in Com -C, we η cs Cx obtain in particular a morphism f : exx (N)x = N −→ Mx in M . We may verify directly that these two associations are inverse to each other.

cs cs Cx We observe that the functor evx : Com -C −→ M also has a right adjoint which we describe below. Proposition 3.2. Let x ∈ Ob(X ). Then, cs Cx cs X (1) There is a functor coex : M −→ Com -C deﬁned by setting, for any y ∈ Ob( ):

X cs α∗N if α ∈ (y, x) coex (N)y := (0 if X (y, x)= ∅

cs cs (2) (evx ,coex ) is a pair of adjoint functors.

cs Cy X ′ cs Proof. Clearly, coex (N)y ∈ M for each y ∈ Ob( ). We consider β : y −→ y. If y 6≤ x, then exx (N)β = 0. ′ cs Otherwise, if we have α : y −→ x and γ : y −→ x, i.e., αβ = γ, we note that id = coex (N)β : γ∗(N) −→ β∗α∗(N). cs cs It is now clear that coex (N) ∈ Com -C. This proves (1). The adjunction in (2) may be veriﬁed directly in a manner similar to the proof of Proposition 3.1.

Corollary 3.3. Let X be a poset and C : X −→ Coalg be a representation taking values in right semiperfect K- X cs Cx cs coalgebras. Let x ∈ Ob( ). Then, the functor exx : M −→ Com -C preserves projectives. cs cs cs Proof. We know from Proposition 3.1(2) that evx is an exact functor. Since (exx , evx ) is an adjoint pair by Proposition cs 3.1(3), it follows that the left adjoint exx preserves projective objects.

Theorem 3.4. Let X be a poset and C : X −→ Coalg be a representation taking values in right semiperfect K- coalgebras. Then, Comcs-C has a set of projective generators.

f Proof. Let P roj (Cx) denote the set of isomorphism classes of ﬁnite dimensonal projective Cx-comodules. Since Cx is right f Cx f semiperfect, we know (see [13, Corollary 2.4.21]) that P roj (Cx) is a generating set for M . For any V ∈ P roj (Cx), cs cs it follows from Corollary 3.3 that exx (V ) is also projective in Com -C. We will show that the family cs X f G = {exx (V ) | x ∈ Ob( ), V ∈ P roj (Cx)} (3.3) is a set of projective generators for Comcs-C. For this, we consider a non-invertible monomorphism i : N ֒→ M in Comcs-C. Since kernels and cokernels in Comcs-C are constructed pointwise, there exists some x ∈ Ob(X ) such that Cx f Cx ix : Nx ֒→ Mx is a non-invertible monomorphism in M . Since P roj (Cx) is a generating set of M , we can choose f cs cs ( a morphism f : V −→ Mx with V ∈ P roj (Cx) such that f does not factor through ix : Nx ֒→Mx. Since (exx , evx f cs cs is an adjoint pair, this gives us a morphism η : exx (V ) −→ M in Com -C corresponding to f, which does not factor through i : N −→M. It follows from [17, §1.9] that the family G is a set of generators for Comcs-C.

3.2 Projective generators for trans-comodules In this subsection, we will show that the category Comtr-C of trans-comodules over a quasi-ﬁnite representation C : X −→ Coalg has projective generators. Proposition 3.5. Let X be a poset and C : X −→ Coalg be a quasi-ﬁnite representation taking values in right semiperfect K-coalgebras. Let x ∈ Ob(X ). Then,

12 tr Cx tr X (1) There is a functor exx : M −→ Com -C given by setting, for any y ∈ Ob( ):

! X tr α M = HCx (Cy,M) if α ∈ (y, x) exx (M)y := (0 if X (y, x)= ∅

tr tr Cx (2) The evaluation at x, i.e., evx : Com -C −→ M , M 7→ Mx is an exact functor. tr tr (3) (exx , evx ) is a pair of adjoint functors. X tr ! Proof. (1) Since C : −→ Coalg is a quasi-finite representation, it follows that exx (M)y = α M = HCx (Cy,M) is a ′ right Cy-comodule for each y ∈ Ob(X ) and α ∈ X (y, x). We consider a morphism β : y −→ y in X . If y 6≤ x, then β tr ′ β tr ! tr exx (M) = 0. Otherwise, we have α : y −→ x and γ : y −→ x, i.e., αβ = γ, which gives id = exx (M): β exx (M)y = ! ! ! tr tr tr β α (M) −→ γ (M)= exx (M)y′ . It follows that exx (M) ∈ Com -C. tr tr (2) Since finite limits and finite colimits in Com -C are computed pointwise, it follows that the functor evx is exact. tr tr Cx tr Cx (3) We will show that there is an isomorphism Com -C (exx (M), N ) ≃ M (M,evx (N )) for any M ∈ M and tr Cx X X f tr N ∈ Com -C. We start with f : M −→ Nx in M . For each y ∈ Ob( ) and α ∈ (y, x), we set ηy : exx (M)y = ! α M −→ Ny to be the composition ! α ! α f ! N α M α Nx Ny

f ′ f β tr β ! f Clearly, each ηy is a morphism of right Cy-comodules. We now take β : y −→ y and claim that ηy′ ◦ exx (M)= N◦β ηy . If y 6≤ x, then both sides vanish. Otherwise, if α : y −→ x, we have a commutative diagram

β!ηf ! tr ! ! y ! β exx (M)y = β α (M) −−−−−−−−−−→ β Ny β!(αN )◦β!(α!f)

β tr β exx (M) id N (3.4)

f  η ′  tr  ! ! y  exx (M)y′ y= β α (M) −−−−−−−−→ Nyy′ αβ N ◦β!α!(f)

f tr tr tr which shows that η : exx (M) −→ N is a morphism in Com -C. On the other hand, if η : exx (M) −→ N is a tr η Cx morphism in Com -C, we have in particular a morphism f : M −→ Nx in M . It is easily seen that these associations are inverse to each other. This proves the result. Corollary 3.6. Let X be a poset and C : X −→ Coalg be a quasi-ﬁnite representation taking values in right semiperfect X tr Cx tr K-coalgebras. Then, for each x ∈ Ob( ), the functor exx : M −→ Com -C preserves projectives. tr tr Proof. The result follows from the fact that the functor exx is left adjoint to the exact functor evx . tr In a manner similar to Proposition 3.2, we can show that the functor evx also has a right adjoint. Proposition 3.7. Let X be a poset and C : X −→ Coalg be a quasi-ﬁnite representation taking values in right X tr tr Cx tr semiperfect K-coalgebras. For each x ∈ Ob( ), the functor evx : Com -C −→ M has a right adjoint coex : MCx −→ Comtr-C given by setting, for y ∈ Ob(X ):

∗ X tr α M if α ∈ (x, y) coex (M)y := (0 if X (x, y)= ∅

Theorem 3.8. Let X be a poset and C : X −→ Coalg be a quasi-ﬁnite representation taking values in right semiperfect K-coalgebras. Then, the category Comtr-C has a set of projective generators.

13 f Cx Proof. Since Cx is semiperfect for each x ∈ Ob(X ), we know that P roj (Cx) is a generating set for the category M . By tr tr Cx tr tr Corollary 3.6, we also know that exx (V ) is projective in Com -C for any projective object V ∈ M . Since (exx , evx ) is an adjoint pair, we can now show as in the proof of Theorem 3.4 that

tr X f G = {exx (V ) | x ∈ Ob( ), V ∈ P roj (Cx)} is a set of generators for Comtr-C.

4 Cartesian comodules over coalgebra representations 4.1 Cartesian cis-comodules over coalgebra representations

α D C We recall that a coalgebra morphism C −→ D is said to be (left) coflat if the coinduction functor α∗ : M −→ M , M 7→ MDC is exact, i.e., C is coflat as a left D-comodule. We will say that a representation C : X −→ Coalg is left coflat if for each α : x −→ y in X , we have a left coflat morphism Cα : Cx −→ Cy of coalgebras. We will now introduce the category of cartesian cis-comodules over C. Definition 4.1. Let C : X −→ Coalg be a left coflat representation taking values in semiperfect K-coalgebras. Let cs M ∈ Com -C. We will say that M is cartesian if for each α : x −→ y in X , the morphism Mα : Mx −→ α∗My = Cx cs cs My Cy Cx in M is an isomorphism. We let Comc -C denote the full subcategory of Com -C consisting of cartesian comodules.

cs Lemma 4.2. The category Comc -C is a cocomplete abelian category. Explicitly, the colimit M of a family {Mi}i∈I of cs objects in Comc -C is given by setting

Mx := colimi∈I Mix for each x ∈ Ob(X ) (4.1) where the colimit on the right hand side is taken in the category MCx .

cs cs Proof. Let η : M −→ N be a morphism in Comc -C. Then, Ker(η), Coker(η) ∈ Com -C are given by Ker(η)x = Ker(ηx : Mx −→ Nx) and Coker(η)x = Coker(ηx : Mx −→ Nx) for each x ∈ Ob(X ). Since C is a coﬂat representation, for any α ∈ X (x, y), the coinduction functor α∗ is exact. Combining with the fact that Mα and Nα are isomorphisms, we see that Ker(η)α and Coker(η)α are also isomorphisms. From this, it is also clear that Coker(Ker(η) −→ M) = cs Ker(N −→ Coker(η)) in Comc -C. cs Let {Mi}i∈I be a directed family of objects in Comc -C. Since the cotensor product commutes with direct limits (see, for instance, [11, §10.5]), it now follows that

colimi∈I Miα (colimi∈I Mi)x = colimi∈I Mix −−−−−−−−−→ colimi∈I α∗Miy = α∗(colimi∈I Mi)y

cs is an isomorphism. This shows that the expression in (4.1) holds for both directed colimits and cokernels in Comc -C. cs Since any colimit in Comc -C may be expressed in terms of cokernels and directed colimits, this proves the result.

′ Lemma 4.3. Let α : C −→ D be a left coﬂat morphism of semiperfect K-coalgebras. Let κ ≥ max{|K|, ℵ0}. Let D ′ D M ∈ M and A ⊆ α∗M be a set of elements such that |A|≤ κ . Then, there exists a subcomodule N ⊆ M in M with ′ |N|≤ κ such that A ⊆ α∗N.

a Proof. Let a ∈ A ⊆ α∗M. Since C is semiperfect, there exists a finite dimensional projective right C-comodule V a a C a and a morphism η : V −→ α∗M in M such that a ∈ Im(η ). Since D is semiperfect, there exists an epimorphism D ζ : i∈I Vi −→ M in M where each Vi is a finite dimensional projective right D-comodule. Since α is coflat and α∗ a commutes with direct sums, α∗ζ : i∈I α∗Vi −→ α∗M is also an epimorphism. As V is a projective right C-comodule, a L a ′a a a ′a η : V −→ α∗M can be lifted to a morphism η : V −→ α∗Vi. Moreover, since V finitely generated, η factors L i∈I L 14 a a through a finite direct sum of objects from the family {α∗Vi}i∈I , which we denote by {α∗V ,...,α∗Vna }. Thus, we a 1 ′a n a a ′a obtain a morphism ζ : r=1 Vr −→ M such that η factors through α∗ζ . Now, we set

L na ′ ′a a N := Im ζ := ζ : Vr −→ M (4.2) a A a A r=1 ! M∈ M∈ M

Since α∗ is exact and commutes with direct sums, we now obtain

na ′ ′a a α∗(N) := Im α∗ζ = α∗ζ : α∗Vr −→ α∗M (4.3) a A a A r=1 ! M∈ M∈ M a a ′a a Since a ∈ Im(η ) and η factors through α∗ζ , we have A ⊆ α∗(N). Finally, since each Vr is ﬁnite dimensional, we a a ′ n a ′ ′ note that |Vr |≤ κ . Since N is a quotient of a∈A r=1 Vr and |A|≤ κ , it follows that |N|≤ κ . Lemma 4.4. Let α : C −→ D be a left coﬂatL morphismL of semiperfect K-coalgebras and let M ∈ MD. Let κ′ ≥ ′ max{|K|, ℵ0, |C|}. Let A ⊆ M and B ⊆ α∗M be sets of elements such that |A|, |B| ≤ κ . Then, there exists a subcomodule N ⊆ M in MD such that

′ ′ (1) |N|≤ κ , |α∗N|≤ κ

(2) A ⊆ N and B ⊆ α∗N

′ Proof. By Lemma 4.3, we can obtain a right D-comodule N1 such that |N1| ≤ κ and B ⊆ α∗N1. Again, taking ′ α = idD : D −→ D in Lemma 4.3, we can also obtain a right D-comodule N2 such that |N2|≤ κ and A ⊆ N2. We set, ′ N := N1 + N2 ⊆ M. Since N is a quotient of N1 ⊕ N2, we have |N| ≤ κ . Clearly, we also have A ⊆ N2 ⊆ N. Also, since α∗ is a right adjoint, it preserves inclusions and hence B ⊆ α∗N1 ⊆ α∗N. Finally, since α∗N = NDC is a linear ′ ′ subspace of N ⊗ C, it follows from the definition of κ that |α∗N|≤ κ . Let C : X −→ Coalg be a left coflat representation taking values in semiperfect K-coalgebras and let X be a poset. cs We will now show that Comc -C has a generator. We will do this in a manner similar to [15] using transfinite induction (see also [6]). We begin by setting

′ κ := sup{|K|, ℵ0, |Mor(X )|, |Cx|, x ∈ Ob(X )} (4.4)

cs X Let M ∈ Comc -C and let m0 ∈ elX (M). This means that there exists some x ∈ Ob( ) such that m0 ∈ Mx. By ′ ′ ′ Cx [13, Corollary 2.2.9], there exists a finite dimensional right Cx-comodule V and a morphism η : V −→ Mx in M ′ ′ cs such that m0 ∈ im(η ). As in (2.4), corresponding to η , we can define a subcomodule N ⊆M in Com -C such that ′ m0 ∈ Nx and |N | ≤ κ (by Lemma 2.8). We now fix a well ordering of the set Mor(X ) and consider the induced lexicographic ordering on N × Mor(X ). To each (n, α : y −→ z) ∈ N × Mor(X ), we will associate a subcomodule P(n, α) of M in Comcs-C which satisfies the following conditions:

(1) m0 ∈ elX (P(1, α0)), where α0 is the least element of Mor(X ). (2) P(m,β) ⊆P(n, α) for (m,β) ≤ (n, α) in N × Mor(X ).

(3) For each pair (n, α : y −→ z) ∈ N×Mor(X ), the morphism P(n, α)α : P(n, α)y −→ α∗P(n, α)z is an isomorphism in MCy . (4) |P(n, α)|≤ κ′

15 First, we ﬁx (n, α : y −→ z) ∈ N × Mor(X ). To deﬁne P(n, α), we begin by setting

0 Nw if n = 1 and α = α0 A0(w)= (4.5) ( (m,β)<(n,α) P(m,β)w otherwise X 0 0 ′ 0 for each w ∈ Ob( ). Clearly, each A0(Sw) ⊆ Mw and |A0(w)| ≤ κ . For α : y −→ z, we have A0(z) ⊆ Mz and 0 ∼ 0 Cz A0(y) ⊆ My = α∗Mz since M is cartesian. By Lemma 4.4 we can obtain a subcomodule A1(z) ⊆ Mz in M such that 0 ′ 0 ′ 0 0 0 0 |A1(z)|≤ κ |α∗A1(z)|≤ κ A0(z) ⊆ A1(z) A0(y) ⊆ α∗A1(z) (4.6) 0 0 0 0 X X Now, we set A1(y) := α∗A1(z) and A1(w) := A0(w) for any w 6= y,z ∈ Ob( ). We observe that since is a poset, 0 0 X y = z implies α : y −→ z is the identity and hence A1(y)= A1(z). It now follows from (4.6) that for every w ∈ Ob( ), 0 0 0 ′ we have A0(w) ⊆ A1(w) and |A1(w)|≤ κ . cs Lemma 4.5. Let B ⊆ elX (M) with |B|≤ κ′. Then, there is a subcomodule Q⊆M in Com -C such that B ⊆ elX (Q) ′ and |Q| ≤ κ . In particular, if m0 ∈ B is such that m0 ∈Mx for some x ∈ Ob(X ), then m0 ∈Qx.

Proof. As in the proof of Theorem 2.9, for any m0 ∈ B ⊆ elX (M) we can choose a subcomodule Q(m0) ⊆ M in cs ′ X Com -C such that m0 ∈ el (Q(m0)) and |Q(m0)| ≤ κ . Now, we set Q := m0∈B Q(m0). Since Q is a quotient of ′ ′ ⊕m0∈BQ(m0) and |B|≤ κ , we have |Q| ≤ κ . The last statement is clear. P 0 0 ′ X ′ We note that we have A1(w) ⊆ Mw and |A1(w)| ≤ κ for all w ∈ Ob( ). Thus, by the deﬁnition of κ , we have 0 0 ′ w∈Ob(X ) A1(w) ⊆ elX (M) and | w∈Ob(X ) A1(w)| ≤ κ . Therefore, by Lemma 4.5,we can choose a subcomodule cs Q0(n, α) ⊆ M in Com -C such that A0(w) ⊆ elX (Q0(n, α)) and |Q0(n, α)| ≤ κ′. We also have A0(w) ⊆ S S w∈Ob(X ) 1 1 Q0(n, α) ⊆M for each w ∈ Ob(X ) by Lemma 4.5. w w S We will now iterate this construction. Suppose that for each l ≤ r, we have constructed a subcomodule Ql(n, α) ⊆M cs l l l ′ r+1 r in Com -C satisfying w∈Ob(X ) A1(w) ⊆ elX (Q (n, α)) and |Q (n, α)| ≤ κ . Let A0 (w) := Q (n, α)w for each w ∈ Ob(X ). Since Ar+1(y) ⊆ M ∼ α (M ) and Ar+1(z) ⊆ M , using Lemma 4.4, we can obtain a subcomodule 0 S y = ∗ z 0 z r+1 Cz A1 (z) ⊆Mz in M such that

r+1 ′ r+1 ′ r+1 r+1 r+1 r+1 |A1 (z)|≤ κ |α∗A1 (z)|≤ κ A0 (z) ⊆ A1 (z) A0 (y) ⊆ α∗A1 (z) (4.7) r+1 r+1 r+1 r+1 X Now, we set A1 (y) := α∗A1 (z) and A1 (w) := A0 (w) for any w 6= y,z ∈ Ob( ). Combining this with (4.7) it r+1 r+1 X r+1 ′ follows that A0 (w) ⊆ A1 (w) for all w ∈ Ob( ) and each |A1 (w)|≤ κ . Therefore, by Lemma 4.5 we can choose r+1 cs r+1 r+1 r+1 ′ a subcomodule Q (n, α) ⊆ M in Com -C such that w∈Ob(X ) A1 (w) ⊆ elX (Q (n, α)) and |Q (n, α)| ≤ κ . In particular, Ar+1(w) ⊆Qr+1(n, α) for each w ∈ Ob(X ). Finally, we set 1 w S P(n, α) := Qr(n, α) (4.8) r [≥0 Lemma 4.6. The family {P(n, α) | (n, α) ∈ N × Mor(X )} satisfies conditions (1)-(4). Proof. The conditions (1) and (2) follow immediately from the definitions in (4.5) and (4.8). Since each |Qr(n, α)|≤ κ′, the result of (4) follows from (4.8). It remains to show (3). For this, we observe that P(n, α)z can be expressed as the filtered union 0 0 1 1 r+1 r+1 (A1(z) ֒→Q (n, α)z ֒→ A1(z) ֒→Q (n, α)z ֒→ . . . ֒→ A1 (z) ֒→Q (n, α)z ֒→ ... (4.9

Cz of objects in M . Since α∗ preserves monomorphisms as well as direct limits, we can also express α∗P(n, α)z as the ﬁltered union

0 0 1 1 r+1 r+1 (α∗A1(z) ֒→ α∗Q (n, α)z ֒→ α∗A1(z) ֒→ α∗Q (n, α)z ֒→ . . . ֒→ α∗A1 (z) ֒→ α∗Q (n, α)z ֒→ ... (4.10

16 Cy of objects in M . Likewise, P(n, α)y can be expressed as the ﬁltered union

0 0 1 1 r+1 r+1 (A1(y) ֒→Q (n, α)y ֒→ A1(y) ֒→Q (n, α)y ֒→ . . . ֒→ A1 (y) ֒→Q (n, α)y ֒→ ... (4.11

Cy r r Cy of objects in M . Also, by deﬁnition, we have isomorphisms A1(y)= α∗A1(z) in M for each r ≥ 0. Together, these induce an isomorphism P(n, α)α : P(n, α)y −→ α∗P(n, α)z.

cs cs Lemma 4.7. Let M ∈ Comc -C. For any m0 ∈ elX (M) there exists a subcomodule P ⊆M in Comc -C such that ′ m0 ∈ elX (P) and |P| ≤ κ . N X Proof. Clearly, the set ×Mor( ) with the lexicographic ordering is filtered. We set P := (n,α)∈N×Mor(X ) P(n, α) ⊆ cs ′ M in Com -C. By Lemma 4.6, m0 ∈ P(1, α0) which implies m0 ∈ elX (P). Also, since each |P(n, α)| ≤ κ , we have ′ S |P| ≤ κ . Now, we fix any morphism γ : u −→ v in X . We note that the family {(m,γ)}m≥1 is cofinal in N × Mor(X ) and hence we may write P = lim P(m,γ). Since cotensor product commutes with direct limits (see, for instance, −→ m≥1 [11, §10.5]), we have γ∗(Pv) = lim γ∗(P(m,γ)v). Since each P(m,γ)γ : P(m,γ)u −→ γ∗P(m,γ)v is an isomorphism, it −→ m≥1 follows that the filtered colimit Pγ : Pu −→ γ∗(Pv) is an isomorphism. Theorem 4.8. Let X be a poset and let C : X −→ Coalg be a left coflat representation taking values in semiperfect cs K-coalgebras. Then, Comc -C is a Grothendieck category. cs Proof. From the description of direct limits in Comc -C given in Lemma 4.2, it follows that they commute with finite cs limits. Also, given any M ∈ Comc -C, it follows from Lemma 4.7, that M can be expressed as a sum of a family cs ′ {Pm0 }m0∈elX (M) of subcomodules of M such that Pm0 ∈ Comc -C and |Pm0 |≤ κ . Therefore, isomorphism classes of ′ cs cartesian right cis-comodules P with |P| ≤ κ form a set of generators for Comc -C. Theorem 4.9. Let X be a poset and let C : X −→ Coalg be a left coflat representation taking values in semiperfect cs cs .K-coalgebras. Then, the inclusion functor i : Comc -C ֒→ Com -C has a right adjoint cs cs Proof. It follows from Lemma 4.2 that the inclusion functor i : Comc -C ֒→ Com -C preserves all colimits. By Theorem cs 4.8, Comc -C is a Grothendieck category and it follows (see, for instance, [19, Proposition 8.3.27]) that i admits a right adjoint. Remark 4.10. We note that the right adjoint in Theorem 4.9 may be seen as a coalgebraic counterpart of the classical quasi-coherator construction (see [18, Lemme 3.2]), which is right adjoint to the inclusion of quasi-coherent sheaves into the category of modules over a scheme.

4.2 Cartesian trans-comodules over coalgebra representations Let X be a small category and let C : X −→ Coalg be a quasi-ﬁnite representation taking values in right semi-perfect K-coalgebras. In this section, we will introduce the category of cartesian trans-comodules over the representation C. Let α : C −→ D be a coalgebra morphism such that C is quasi-ﬁnite as a right D-comodule. Then, the functor ! D C (Λ) α = HD(C, −): M −→ M is exact if and only if the direct sum C is injective as a right D-comodule for any indexing set Λ (see [3, Corollary 3.10]). In such a situation, we will say that the morphism α : C −→ D of coalgebras is (right) Σ-injective. We note in particular that any identity morphism of coalgebras is Σ-injective.

Deﬁnition 4.11. Let C : X −→ Coalg be a quasi-ﬁnite representation taking values in right semi-perfect coalgebras. Suppose that C is Σ-injective, i.e., for each each α ∈ X (x, y), the morphism Cα : Cx −→ Cy of coalgebras is Σ-injective. tr α ! Let M ∈ Com -C. Then, we say that M is cartesian if for each α ∈ X (x, y), the morphism M : α My −→ Mx is Cx tr an isomorphism in M . We will denote the full subcategory of cartesian trans-comodules by Comc -C.

17 tr Lemma 4.12. The category Comc -C is cocomplete and abelian. Further, if M denotes the colimit of a family {Mi}i∈I tr X of objects in Comc -C, then for each x ∈ Ob( ), we have Mx = colimi∈I Mix, where the colimit on the right is taken in the category MCx .

tr tr Proof. Let η : M −→ N be a morphism in Comc -C. We know that Ker(η) and Coker(η) in Com -C are constructed ! tr pointwise. Since α is exact, it follows as in the proof of Lemma 4.2 that Ker(η) and Coker(η) lie in Comc -C and hence tr ! Comc -C is abelian. Further, since α is left adjoint, it preserves colimits and we may verify as in the proof of Lemma tr 4.2 that Comc -C is cocomplete, with all colimits computed pointwise. tr We will now look at generators in Comc -C. Lemma 4.13. Let α : C −→ D be a quasi-ﬁnite and Σ-injective morphism between right-semiperfect coalgebras. Let ′ D ! ′ κ ≥ max{|K|, ℵ0, |D|}. Let M ∈ M and A ⊆ α M be a set of elements such that |A| ≤ κ . Then, there exists a subcomodule N ⊆ M in MD with |N|≤ κ′ such that A ⊆ α!N. Proof. This follows in a manner similar to Lemma 4.3, using the fact that α! is both a left adjoint and assumed to be exact. Lemma 4.14. Let α : C −→ D be a quasi-ﬁnite and Σ-injective morphism between right semiperfect coalgebras. Let ′ ∗ |D| D ! ′ κ ≥ max{|K|, ℵ0, |C |, 2 }. Let M ∈ M and let A ⊆ M and B ⊆ α M be sets of elements such that |A|, |B| ≤ κ . Then, there exists a subcomodule N ⊆ M in MD such that (1) |N|≤ κ′, |α!N|≤ κ′ (2) A ⊆ N and B ⊆ α!N

′ ! Proof. By Lemma 4.13, we know that there exists a D-subcomodule N1 ⊆ M such that |N1| ≤ κ and B ⊆ α N1. Similarly, taking α = idD : D −→ D in Lemma 4.13, we can also obtain a D-subcomodule N2 ⊆ M such that ′ ′ |N2| ≤ κ and A ⊆ N2. We set N := N1 + N2 ⊆ M. Since N is a quotient of N1 ⊕ N2, we have |N| ≤ κ . Also, ! ! ! A ⊆ N2 ⊆ N. Since α is exact, it preserves monomorphisms and thus B ⊆ α N1 ⊆ α N. Finally, we know that the α! forget functor given by the composition MD −→ MC −−−−→ V ect is exact. Therefore, using [3, Corollary 3.12], we have that α!N = H (C,N) = N H (C,D) as a vector space. Moreover, H (C,D) = lim Hom(D , C)∗, where {D } D D D D −→ i i i∈I i∈I is a directed family of finite dimensional subcoalgebras of D (see, for instance [40, Section 1]). It now follows that |α!N|≤ κ′. Let X be a poset. We continue with C : X −→ Coalg being a quasi-finite and Σ-injective representation taking values in right semiperfect coalgebras. We fix

′ X |Cx| ∗ κ := sup{ℵ0, |K|, |Mor( )|, {2 }x∈Ob(X ), {|Cx|}x∈Ob(X )} (4.12)

We now choose a well ordering of the set Mor(X ) and consider the induced lexicographic ordering of N×Mor(X ). Let tr X M ∈ Comc -C and let m0 ∈ elX (M), i.e. m0 ∈Mx for some x ∈ Ob( ). We will now deﬁne a family of subcomodules {P(n, α)|(n, α : y −→ z) ∈ N × Mor(X )} of M which satisﬁes the following conditions:

(1’) m0 ∈ elX (P(1, α0)), where α0 is the least element of Mor(X ). (2’) P(m,β) ⊆P(n, α) whenever (m,β) ≤ (n, α) in N × Mor(X ).

α ! (3’) For each (n, α : y −→ z) ∈ N × Mor(X ), the morphism P(n, α): α P(n, α)z −→ P(n, α)y is an isomorphism in MCy . (4’) |P(n, α)|≤ κ′.

18 We know that there exists a ﬁnite dimensional Cx-comodule V and a morphism η : V −→ Mx such that m0 ∈ im(η). Then, we can deﬁne the subcomodule N ⊆M corresponding to η as in (2.18) such that m0 ∈ Nx. By Lemma 2.13, we also know that |N | ≤ κ′. For each pair (n, α : y −→ z) ∈ N × Mor(X ), we now start constructing the comodule P(n, α) inductively. We set

Nw if n = 1 and α = α0 A0(w)= (4.13) 0  P(m,β)w otherwise (m,β)<(n,α) S for each w ∈ Ob(X ), where for (n, α) 6= (1, α0), we assume that we have already constructed the subcomodules P(m,β) satisfying all the properties (1’)-(4’) for any (m,β) < (n, α) ∈ N × Mor(X ). 0 0 ′ ! ∼ Clearly, each A0(w) ⊆Mw and |A0(w)|≤ κ . Since M is cartesian, we know that α Mz = My. Using Lemma 4.14, we 0 Cz can obtain a comodule A1(z) ⊆Mz in M such that

0 ′ ! 0 ′ 0 0 0 ! 0 |A1(z)|≤ κ |α A1(z)|≤ κ A0(z) ⊆ A1(z) A0(y) ⊆ α A1(z) (4.14) 0 ! 0 0 0 X 0 0 We now set A1(y) := α A1(z) and A1(w) := A0(w) for any w 6= y,z ∈ Ob( ). It follows from (4.14) that A0(w) ⊆ A1(w) X 0 ′ for every w ∈ Ob( ) and each |A1(w)|≤ κ . tr Lemma 4.15. Let B ⊆ elX (M) with |B|≤ κ′. Then, there is a subcomodule Q⊆M in Com -C such that B ⊆ elX (Q) ′ and |Q| ≤ κ . In particular, if m0 ∈ B is such that m0 ∈Mx for some x ∈ Ob(X ), then m0 ∈Qx. Proof. This is similar to the proof of Lemma 4.5.

0 0 Applying Lemma 4.15 to B = A1(w), we see that there exists a subcomodule Q (n, α) ֒→ M such that w∈Ob(X ) 0 0 X 0 ′ A1(w) ⊆ Q (n, α)w ⊆ Mw for eachS w ∈ Ob( ) and |Q (n, α)| ≤ κ . Suppose now that we have constructed a l l l l ′ subcomodule Q (n, α) ֒→M for every l ≤ r such that A1(w) ⊆ elX (Q (n, α)) and |Q (n, α)|≤ κ . We now set w∈Ob(X ) r+1 r X r+1 S ∼ ! r+1 A0 (w) := Q (n, α)w for each w ∈ Ob( ). Since A0 (y) ⊆My = α Mz and A0 (z) ⊆Mz, using Lemma 4.14, we r+1 Cz can obtain a subcomodule A1 (z) ֒→Mz in M such that

r+1 ′ ! r+1 ′ r+1 r+1 r+1 ! r+1 |A1 (z)|≤ κ |α A1 (z)|≤ κ A0 (z) ⊆ A1 (z) A0 (y) ⊆ α A1 (z) (4.15) r+1 ! r+1 r+1 r+1 X Then, we set A1 (y) := α A1 (z) and A1 (w) := A0 (w) for any w 6= y,z ∈ Ob( ). It now follows from (4.15) r+1 r+1 X r+1 ′ that A0 (w) ⊆ A1 (w) for all w ∈ Ob( ) and each |A1 (w)|≤ κ . Using Lemma 4.15, we can obtain a subcomodule r+1 r+1 r+1 r+1 ′ Q (n, α) ֒→M such that w∈Ob(X ) A1 (w) ⊆ elX (Q (n, α)) and |Q (n, α)|≤ κ . We ﬁnally set S P(n, α) := Qr(n, α) (4.16) r [≥0 Lemma 4.16. The family {P(n, α)|(n, α) ∈ N × Mor(X )} satisﬁes conditions (1’)-(4’). Proof. The idea of the proof is similar to that of Lemma 4.6, using here the fact that α! preserves colimits (being a left adjoint) and the assumption that α! is exact.

tr tr Lemma 4.17. Let M ∈ Comc -C. For any m0 ∈ elX (M) there exists a subcomodule P ⊆M in Comc -C such that ′ m0 ∈ elX (P) and |P| ≤ κ . N X Proof. Since × Mor( ) with the lexicographic ordering is ﬁltered, we set P := (n,α)∈N×Mor(X ) P(n, α) ⊆M. The proof now follows as in Lemma 4.7, using the fact that α! is a left adjoint and that {(m,β)} is coﬁnal in N×Mor(X ) S m≥1 for any morphism β in X .

19 Theorem 4.18. Let X be a poset. Let C : X −→ Coalg be a quasi-finite and Σ-injective representation taking values tr in right semiperfect coalgebras. Then, Comc -C is a Grothendieck category. tr Proof. Since filtered colimits and finite limits in Comc -C are both computed pointwise, it is clear that they commute in tr tr Comc -C. It also follows from Lemma 4.17 that any M ∈ Comc -C can be expressed as a sum of a family {Pm0 }m0∈elX (M) ′ of cartesian subcomodules such that each |Pm0 |≤ κ . Therefore, the isomorphism classes of cartesian comodules P with ′ tr |P| ≤ κ form a set of generators for Comc -C. Theorem 4.19. Let X be a poset. Let C : X −→ Coalg be a quasi-finite and Σ-injective representation taking values tr tr in right semiperfect coalgebras. Then, the inclusion functor i : Comc -C −→ Com -C has a right adjoint. tr tr Proof. Using Lemma 4.12, we know that the inclusion functor i : Comc -C −→ Com -C preserves colimits. Since tr Comc -C is a Grothendieck category, it follows that i has a right adjoint.

5 Contramodules over coalgebra representations

Let C be a K-coalgebra having coproduct ∆C and counit ǫC. A contramodule over C (see, for instance, [26, §0.2.4]) consists of a K-space M along with a K-linear “contraction map” πM : HomK(C,M) −→ M such that the following diagrams

−◦∆C / M HomK(C,HomK (C,M)) =∼ HomK (C ⊗ C,M) / HomK (C,M) (5.1) ▲▲▲ ▲▲ id Hom ǫ ,M ▲▲ π ( C ) ▲▲▲ πM ◦− M ▲▲& &/ / HomK (C,M) M HomK (C,M) / M πM πM commute. If the isomorphism HomK (C,HomK (C,M)) =∼ HomK (C ⊗ C,M) in the diagram above is obtained from the adjointness of −⊗ C and HomK (C, −) (resp. the adjointness of C ⊗− and HomK (C, −)), then M is a right (resp. left) C-contramodule. Equivalently, right C-contramodules may be identiﬁed with objects in the Eilenberg-Moore category of modules over the following monad: let TC denote the endofunctor

TC : V ect −→ Vect M 7→ HomK(C,M) (5.2) on the category of K-vector spaces. Then, there is a natural transformation η :1V ect −→ TC given by

η(M): M −→ Hom(C,M) m 7→ ǫC · m (5.3) for each vector space M, where (ǫC · m)(c) := ǫC (c)m for each c ∈ C. There is also a natural transformation µ : TC ◦ TC −→ TC given by

HomK (C,HomK (C,M)) =∼ HomK (C ⊗ C,M) −→ Hom(C,M) f 7→ f ◦ ∆C (5.4) where the isomorphism HomK (C,HomK (C,M)) =∼ HomK (C ⊗ C,M) in (5.4) comes from the adjointness of −⊗ C and HomK (C, −). It may be veriﬁed that (TC ,µ,η) is a monad on the category of K-vector spaces. Accordingly, a right C-contramodule may be described (see, for instance, [10, § 4.4]) as a pair (M, πM ) satisfying the conditions in (5.1). We will denote the category of right C-contramodules by M[C, ].

The free contradmodules are of the form TC (V ) for some vector space V . Further, the free functor TC : V ect −→ M[C, ] is left adjoint to the forgetful functor, which gives natural isomorphisms (see [26, §0.2.4]) ∼ M[C, ](TC(V ),M) = HomK (V,M) (5.5)

20 for any vector space V and any C-contramodule M. We mention that the forgetful functor M[C, ] −→ V ect is exact (see [26, § 3.1.2]). From (5.5), we also note that direct sums of free contramodules can be expressed as

TC(Vi)= TC Vi (5.6) i I i I ! M∈ M∈ for any family of vector spaces {Vi}i∈I . We know (see [26, §0.2.4]) that the category M[C, ] is abelian, has enough projectives and that the projective objects in M[C, ] correspond to direct summands of free contramodules. We note in ∗ particular that C = HomK (C,K) is a projective (free) C-contramodule. Let α : C −→ D be a morphism of K-coalgebras. Then, we have a functor

C C D ◦α πM α• : M[C, ] −→ M[D, ] (πM : HomK (C,M) −→ M) 7→ (πM : HomK (D,M) −−−→ HomK (C,M) −−→ M) (5.7)

This is called contrarestriction of scalars. We note that α• is exact. The contrarestriction functor α• has a left adjoint α• called the contraextension of scalars [27, § 4.8]. One can ﬁrst deﬁne it on the free D-contramodules by setting

• α (TD(V )) := TC (V ) (5.8)

• • for any vector space V . Then, α may be extended to M[D, ] by using the fact that α is right exact and that every D-contramodule may be expressed as a cokernel of free D-contramodules.

There is an equivalent deﬁnition of the contraextension of scalars which we describe now (see, [31, §5.3]): Let (P,ρP ) be a right C-comodule and (M, πM ) be a right C-contramodule. The K-space of cohomomorphisms CohomC (P,M) is deﬁned to be the coequalizer of the maps

HomK (ρP ,M) HomK (P ⊗ C,M) =∼ HomK (P,HomK (C,M)) HomK (P,M) HomK (P,πM )

• • The contraextension α : M[D, ] −→ M[C, ] may be deﬁned as α M = CohomD(C,M), where C is considered as a right D-comodule. The right C-contramodule structure on CohomD(C,M) is induced by the left C-comodule structure of C. Deﬁnition 5.1. Let C : X −→ Coalg be a coalgebra representation. A right trans-contramodule M over C will consist of the following data:

(1) For each object x ∈ Ob(X ), a right Cx-contramodule Mx

(2) For each morphism α : x −→ y in X , a morphism αM : My −→ α•Mx of right Cy-contramodules (equivalently, α • a morphism M : α My −→ Mx of right Cx-contramodules) α β X We further assume that idx M = idMx and for any pair of composable morphisms x −→ y −→ z in , we have β•(αM) ◦ βM = βαM : Mz −→ β•My −→ β•α•Mx = (βα)•Mx. The latter condition can be equivalently expressed as βαM = αM◦ α•(βM). A morphism η : M −→ N of trans-contramodules over C consists of morphisms ηx : Mx −→ Nx of right Cx- contramodules for each x ∈ Ob(X ) such that for each morphism x −→ y in X the following diagram commutes

ηy My Ny

αM αN

α•ηx α•Mx α•Nx We denote this category of right trans-contramodules by Conttr-C.

21 Remark 5.2. Unlike in the case of comodules, we only deﬁne trans-contramodules over coalgebra representations and not cis-contramodules. This is because, to the knowledge of the authors, there do not appear to be standard conditions in the literature that would make the contrarestriction functor α• a left adjoint. We note that if α = ǫC : C −→ K is the counit, the contrarestriction α• reduces to the forgetful functor M[C, ] −→ V ect which does not preserve colimits in general. Proposition 5.3. Let C : X −→ Coalg be a coalgebra representation. Then, Conttr-C is an abelian category. Proof. For any morphism η : M −→ N in Conttr-C, we deﬁne the kernel and cokernel of η by setting

Ker(η)x := Ker(ηx : Mx −→ Nx) Coker(η)x := Coker(ηx : Mx −→ Nx) (5.9) X X for each x ∈ . For α : x −→ y in , the exactness of the contrarestriction functor α• : M[Cx, ] −→ M[Cy , ] induces the morphisms Ker(η)α : Ker(η)y −→ α•Ker(η)x and Coker(η)α : Coker(η)x −→ α•Coker(η)x. Since each M[Cx, ] is .((abelian, it also follows from (5.9) that Coker(Ker(η) ֒→M)= Ker(N ։ Coker(η

We recall (see [10, § 4.4]) that for any coalgebra C over a ﬁeld K, the category of right C-contramodules is generated ∗ by the free (projective) right C-contramodule C = HomK (C,K). Let C : X −→ Coalg be a coalgebra representation. Let M ∈ Conttr-C. We consider an object x ∈ Ob(X ) and a morphism ∗ η : Cx −→ Mx (5.10) X in M[Cx, ]. For each object y ∈ Ob( ), we set Ny ⊆My to be the image in M[Cy , ] of the family of maps

• β • ∗ ∗ β η • M Ny = Im β (Cx)= Cy −−→ β Mx −−→My (5.11)  X X  β∈M(y,x) β∈M(y,x)   ∋ where the equality in (5.11) follows from (5.8). Let iy denote the inclusion Ny ֒→ My in M[Cy, ] and for each β X ′ • ∗ ∗ (y, x), we denote by ηβ : β (Cx)= Cy −→ Ny the canonical morphism induced from (5.11). For α ∈ X (z,y) and β ∈ X (y, x), we may now verify as in the proof of Lemma 2.4 that the following composition in

M[Cy, ] η′ • ∗ ∗ β iy αM β (Cx)= Cy −→Ny −→My −−→ α•Mz (5.12) factors through α•(iz): α•Nz −→ α•Mz. Proposition 5.4. Let C : X −→ Coalg be a coalgebra representation and let M ∈ Conttr-C. Then, the objects tr {Ny ∈ M[Cy, ]}y∈Ob(X ) together determine a subobject N ⊆M in Cont -C.

Proof. Let α ∈ X (z,y). Since α• is a right adjoint, it preserves monomorphisms and hence α•(iz): α•Nz −→ α•Mz is a monomorphism in M[Cy , ]. We will now show that the morphism αM : My −→ α•Mz restricts to a morphism αN : Ny −→ α•Nz giving us a commutative diagram

iy Ny My

αN αM

α•(iz ) α•Nz α•Mz ∗ We recall that the abelian category M[Cy , ] has a projective generator Cy . It is clear from the proof of [6, Lemma ∗ ′ ∗ 3.2] that it suﬃces to show that for any morphism ζ : Cy −→ Ny in M[Cy , ] there exists ζ : Cy −→ α•Nz such that ′ α•(iz) ◦ ζ = αM◦ iy ◦ ζ. By 5.11, we have an epimorphism

′ ∗ ηβ : Cy −→ Ny (5.13) X X β∈M(y,x) β∈M(y,x)

22 ∗ ∗ ′′ ∗ ∗ in M[Cy, ]. Since Cy is projective, the morphism ζ : Cy −→ Ny can be lifted to a morphism ζ : Cy −→ β∈X (y,x) Cy such that L ′ ′′ ζ = ηβ ◦ ζ (5.14)  X  β∈M(y,x) ′   X We know from (5.12) that αM◦ iy ◦ ηβ factors through α•(iz): α•Nz −→ α•Mz for each β ∈ (y, x). Therefore, it follows from (5.14) that αM◦ iy ◦ ζ factors through α•(iz) as required. This proves the result.

′ ∗ X Proposition 5.5. Let η1 : Cx −→ Nx be the canonical morphism corresponding to the identity map in (x, x). Then, for any y ∈ Ob(X ), we have

• ′ β ∗ • ∗ β η1 • N Ny = Im Cy = β (Cx) −−−→ β Nx −−→Ny (5.15)  X X  β∈M(y,x) β∈M(y,x)   Proof. Let β ∈ X (y, x). We consider the following commutative diagram:

• ′ β η β • ∗ 1 • N β (Cx) β Nx Ny

• β ix iy (5.16) β • M β Mx My

′ • • ′ • Since ix ◦ η1 = η, we have (β ix) ◦ (β η1)= β η. This gives

β • β • • ′ Im β∈X (y,x) M◦ (β η) = Im β∈X (y,x) M◦ (β ix) ◦ (β η1) β • ′ β • ′ L = Im Lβ∈X (y,x) iy ◦ N ◦ (β η1) = Im β∈X (y,x) N ◦ (β η1) L L where the last equality follows from the fact that iy is a monomorphism of contramodules. The result now follows using the deﬁnition in (5.11).

tr For M ∈ Cont -C, let elX (M) denote the union ∪x∈Ob(X )Mx as sets. The cardinality of elX (M) will be denoted by |M|. For any coalgebra C, we recall that the forgetful functor M[C, ] −→ V ect is exact. Hence, epimorphisms and monomorphisms in M[C, ] correspond respectively to epimorphisms and monomorphisms in V ect. It follows that for any quotient or subobject N of M ∈ Conttr-C, we have |N | ≤ |M|. Now, we set

|Mor(X )| |Cy | κ := sup{ℵ0, (|K| ) | y ∈ Ob(X )} (5.17) Lemma 5.6. Let N be as constructed in Proposition 5.4. Then, we have |N | ≤ κ. Proof. Let y ∈ Ob(X ). From Proposition 5.5, we have

• ′ β ∗ β η1 • N Ny = Im Cy −−−→ β Nx −−→Ny (5.18)  X  β∈M(y,x)   From the adjunction in (5.5), we know that the left adjoint TCy : V ect −→ M[Cy , ] preserves direct sums. Since ∗ Cy = HomK (Cy,K) is a free contramodule, we therefore have

∗ Cy = HomK Cy, K (5.19) X  X  β∈M(y,x) β∈M(y,x)   23 ∗ Since Ny is an epimorphic image of β∈X (y,x) Cy , we now have

L ∗ |Ny| ≤ | Cy | = |HomK (Cy, K)|≤ κ (5.20) X X β∈M(y,x) β∈M(y,x)

Thus, |N | = y∈Ob(X ) |Ny|≤ κ. Theorem 5.7.P Let C : X −→ Coalg be a coalgebra representation. Then, Conttr-C is cocomplete and has a set of generators. Proof. We note that colimits in Conttr-C exist and can be computed pointwise. Let M be an object in Conttr-C and X ∗ ∼ m ∈ elX (M). Then, there exists some x ∈ Ob( ) such that m ∈ Mx. From (5.5), we note that M[Cx, ](Cx,M) = ∗ V ect(K, Mx) = Mx and hence there exists a morphism η : Cx −→ Mx in M[Cx, ] such that m ∈ Im(η). Thus, it follows from (5.11) that we can deﬁne N ⊆M in Conttr-C such that

β • ∗ • ∗ M◦β η m ∈ Im (η : Cx −→ Mx) ⊆ Im β (Cx) −−−−−−→Mx = Nx (5.21)  X  β∈M(x,x)   By Lemma 5.6, we also have |N | ≤ κ. For convenience, let us denote this subobject N ⊆M by Nm. We claim that tr ′ ′ ′ ′ m ∈elX (M) Nm −→ M is an epimorphism in Cont -C. For this, it is enough to show m ∈elX (M) Nm x −→ Mx is an epimorphism in M for each x ∈ Ob(X ). If m ∈M for some x ∈ Ob(X ), clearly we have L [Cx, ] x L

(m ∈ Im (Nmx ֒→Mx) ⊆ Im Nm′x −→ Mx (5.22  ′  m ∈MelX (M)   Thus, by choosing one object from each isomorphism class of objects in Conttr-C having cardinality ≤ κ, we obtain a set of generators for Conttr-C. This proves the result.

We continue with C : X −→ Coalg being a coalgebra representation. For the rest of this section, we assume that X is a poset. We will now show that Conttr-C is a locally presentable category with projective generators. Proposition 5.8. Let X be a poset and C : X −→ Coalg be a coalgebra representation. Let x ∈ Ob(X ). Then, tr X (1) There is a functor exx : M[Cx, ] −→ Cont -C deﬁned by setting, for any y ∈ Ob( ),

α•(N) if α ∈ X (y, x) (exx(N))y = (0 if X (y, x)= ∅

tr (2) The evaluation at x, evx : Cont -C −→ M[Cx, ], M 7→ Mx is an exact functor.

(3) (exx, evx) is a pair of adjoint functors. tr (4) The functor evx also has a right adjoint, coex : M[Cx, ] −→ Cont -C given by setting

α•N if α ∈ X (x, y) (coex(N))y := (0 if X (x, y)= ∅ Proof. (1), (2) and (3) follow as in the proof of Proposition 3.5. The result of (4) is proved in a manner similar to Proposition 3.7.

24 Corollary 5.9. Let X be a poset and C : X −→ Coalg be a coalgebra representation. For x ∈ Ob(X ), the functor tr exx : M[Cx, ] −→ Cont -C preserves projectives. tr Proof. We know from Proposition 5.8(2) that evx : Cont -C −→ M[Cx, ] is an exact functor. By Proposition 5.8(3), we know that (exx, evx) is an adjoint pair and hence the left adjoint exx preserves projective objects. We recall that an object X in a category D is said to be λ-presentable for some regular cardinal λ if the representable functor D(X, −) preserves λ-directed colimits (see [2, § 1.13]). We note that for regular cardinals λ ≤ λ′, any λ- presentable object is also λ′-presentable. A category D is called locally λ-presentable if it is cocomplete and it has a set of λ-presentable generators. A category D is said to be locally presentable if D is locally λ-presentable for some regular cardinal λ. For further details on locally presentable categories, we refer the reader to [2]. Theorem 5.10. Let X be a poset and C : X −→ Coalg be a coalgebra representation. Then, Conttr-C is a locally presentable abelian category with projective generators.

Proof. If C is any K-coalgebra, we know (see [28, § 5]) that the category M[C, ] of contramodules is locally presentable. ∗ Indeed, C = HomK (C,K) is itself a λ-presentable generator in M[C, ] for some λ. We suppose therefore that for each X ∗ x ∈ Ob( ), the object Cx is a λx-presentable generator for M[Cx, ]. We set

λ0 := sup{λx | x ∈ X }

∗ ∗ Then, it is clear that each Cx is a λ0-presentable generator in M[Cx, ]. Since Cx is projective in M[Cx, ], it follows by ∗ tr Corollary 5.9 that exx(Cx) is projective in Cont -C. From the deﬁnitions and by (5.8), we know that • ∗ ∗ X ∗ α (Cx)= Cy if α ∈ (y, x) (exx(Cx))y = (0 if X (y, x)= ∅ for y ∈ Ob(X ). We claim that the family ∗ X G = {exx(Cx) | x ∈ Ob( )} (5.23) tr is a set of λ0-presentable generators for Cont -C. For this, we consider a non-invertible monomorphism i : N ֒→ M in Conttr-C. Since kernels and cokernels in Conttr-C are constructed pointwise, there exists some x ∈ Ob(X ) such ∗ that ix : Nx ֒→ Mx is a non-invertible monomorphism in M[Cx, ]. Since Cx is a generator of M[Cx, ], we can choose a ∗ morphism f : Cx −→ Mx = evx(M) such that f does not factor through ix : Nx ֒→Mx. Since (exx, evx) is an adjoint ∗ tr pair, this induces a morphism ηf : exx(Cx) −→ M in Cont -C corresponding to f, which does not factor through i : N −→M. Therefore, it follows from [17, §1.9] that the family G is a set of generators for Conttr-C. tr ∗ Finally, let {Mi}i∈I be a λ0-directed system of objects in Cont -C. Since each Cx is λ0-presentable in M[Cx, ] and colimits in Conttr-C are computed pointwise, we note that Conttr-C(ex (C∗), lim M ) = M (C∗, ev (lim M )) = M (C∗, lim ev (M )) x x −→ i [Cx, ] x x −→ i [Cx, ] x −→ x i i∈I i∈I i∈I = lim M (C∗, ev (M )) = lim Conttr-C(ex (C∗), M ) −→ [Cx, ] x x i −→ x x i i∈I i∈I

∗ tr This shows that each exx(Cx) is λ0-presentable in Cont -C. This proves the result.

6 Cartesian trans-contramodules over coalgebra representations

X tr Let C : −→ Coalg be a coalgebra representation. We will now introduce the category Contc -C of cartesian trans- contramodules over C and study its generators. While the outline is roughly similar to that in Section 4.2, we rely extensively on presentable objects in contramodule categories to set up the transﬁnite induction argument. This is because of the fact that colimits in the category of contramodules do not correspond in general to colimits of underlying vector spaces.

25 Lemma 6.1. Let α : C −→ D be a right coflat morphism of coalgebras, i.e., C is coflat as a right D-comodule. Then, • the contraextension functor α : M[D, ] −→ M[C, ] is exact. Proof. This follows from [30, §3.1, Lemma ]. Definition 6.2. Let C : X −→ Coalg be a right coflat representation. Let M ∈ Conttr-C. We will say that M X α • is cartesian if for each α : x −→ y in , the morphism M : α My = CohomCy (Cx, My) −→ Mx in M[Cx, ] tr tr is an isomorphism. We will denote by Contc -C the full subcategory of Cont -C whose objects are cartesian trans- contramodules over C. X tr Lemma 6.3. Let C : −→ Coalg be a right coflat representation. Then, Contc -C is a cocomplete abelian category. tr tr Proof. Let η : M −→ N be a morphism in Contc -C. Then, its kernel and cokernel in Cont -C are respectively given by Ker(η)x = Ker(ηx : Mx −→ Nx) Coker(η)x = Coker(ηx : Mx −→ Nx) • Since C is a coflat representation, it follows by Lemma 6.1 that the contraextension functor α : M[Cy, ] −→ M[Cx, ] α • α is exact for any α : x −→ y in X . From this, it is clear that Ker(η): α Ker(η)y −→ Ker(η)x and Coker(η): • tr α Coker(η)y −→ Coker(η)x are isomorphisms. It follows that Contc -C is abelian. tr X • Now let {Mi}i∈I be a family of objects in Contc -C and take α ∈ (x, y). Since α is a left adjoint, it preserves direct α • sums. Using the fact that Mi : α Miy −→ Mix is an isomorphism for each i ∈ I, we see that

α L Mi • • i∈I α Mi = α Miy Mix = Mi i∈I y i∈I i∈I i∈I x L L L L tr tr is an isomorphism in M[Cx, ]. Combining with the fact that Contc -C has cokernels, it follows that Contc -C is cocomplete.

′ Lemma 6.4. (a) Let C be a K-coalgebra and let {Mi}i∈I be a system of objects in M[C, ]. Let κ be a cardinal such ′ ′dim(C) that κ ≥ sup{ℵ0, |K|, |I|, |Mi|,i ∈ I} Then, if M = colim Mi, we have |M|≤ κ . i∈I tr ′ (b) Let C : X −→ Coalg be a coalgebra representation and {Mi}i∈I be a system of objects in Cont -C. Let κ ≥ sup{ℵ0, |K|, |I|, |Mor(X )|, |Mi|,i ∈ I} and λ ≥ sup{ℵ0,dim(Cx), x ∈ Ob(X )}. Then, if M = colim Mi, we have i∈I |M| ≤ κ′λ.

Proof. We set N := i∈I Mi. Then, there is an epimorphism N −→ M in M[C, ] and it therefore suﬃces to show that |N|≤ κ′dim(C). For each i ∈ I, any element m ∈ M = V ect(K,M )= M (C∗,M ) corresponds to a morphism L i i [C, ] i ∗ ∗(Mi) C −→ Mi whose image contains m. Together, these induce an epimorphism C −→ Mi in M[C, ]. Since direct ∗(Mi) sums preserve epimorphisms, we see that N becomes a quotient of i∈I C in M[C, ]. Applying (5.6), we see that the direct sum of free contramodules C∗(Mi) = T (V ), where V is a vector space of dimension |M |≤ κ′. It i∈I C L i∈I i follows that |M| ≤ |N| ≤ |T (V )|≤ κ′dim(C). This proves (a). The result of (b) follows directly from (a). C L P ∗ Lemma 6.5. Let α : C −→ D be a right coﬂat morphism of K-coalgebras. Let λC be a regular cardinal such that C is ′ • λC -presentable in M[C, ]. Let κ ≥ max{ℵ0, λC , |K|}. Let M ∈ M[D, ] and A ⊆ α M be a set of elements such that ′ ′dim(D) • |A|≤ κ . Then, there exists a subobject N ⊆ M in M[D, ] with |N|≤ κ such that A ⊆ α N. • ∗ • ∼ • • Proof. Let a ∈ A ⊆ α M. From (5.5), we note that M[C, ](C , α M) = V ect(K, α M) = α M and hence there a ∗ • a ∗ exists a morphism η ∈ M[C, ](C , α M) such that a ∈ Im(η ). Since D is a generator in M[D, ], we can choose an ∗(I) • epimorphism ζ : D −→ M in M[D, ] for some indexing set I. We may suppose that |I| > λC . Since α is a left • • ∗ (I) • ∗ adjoint and therefore preserves colimits, we see that α ζ : (α D ) −→ α M is an epimorphism in M[C, ]. Since C is projective, the morphism ηa can be lifted to a morphism η′a : C∗ −→ (α•D∗)(I) = C∗(I).

26 Because λC is a regular cardinal, we note that the collection of subsets of I with cardinality < λC is a λC -directed ∗ ′a system. Because C is λC -presentable in M[C, ], there exists a subset Ja ⊆ I with |Ja| < λC such that η factors through C∗(Ja) and we obtain the following commutative diagram:

C∗ (α•D∗)(Ja) = C∗(Ja) η′a ηa

α•M (α•D∗)(I) = C∗(I) α•ζ

′a ∗(Ja) a • ′a Thus, we obtain a morphism ζ : D −→ M in M[D, ] such that η factors through α ζ . In M[D, ], we set

N := Im ζ′ = ζ′a : D∗(Ja) −→ M a A a A ! M∈ M∈ Since α• is exact and preserves colimits, we also have

α•N := Im α•ζ′ = α•ζ′a : α•D∗(Ja) −→ α•M a A a A ! M∈ M∈ Since a ∈ Im(ηa) and ηa factors through α•ζ′a, we have a ∈ α•N. Thus, A ⊆ α•N. By (5.6), we know that the direct ∗(Ja) |Ja| sum a∈A D of free contramodules is given by TD(V ), where V is the vector space a∈A K . Since N is a quotient of T (V ) and |V |≤ κ′, it follows that |N|≤ κ′dim(D). L D L

Lemma 6.6. Let α : C −→ D be a right coﬂat morphism of K-coalgebras and let M ∈ M[D, ]. Let λ be a regular ∗ ∗ cardinal such that C and D are λ-presentable in M[C, ] and M[D, ] respectively. We also choose cardinals

′ κ ≥ max{ℵ0, λ, |K|} µ ≥ max{dim(C),dim(D), ℵ0}

• ′µ Let A ⊆ M and B ⊆ α M be such that |A|, |B|≤ κ . Then, there exists a subobject N ⊆ M in M[D, ] such that (1) A ⊆ N and B ⊆ α•N (2) |N|≤ κ′µ, |α•N|≤ κ′µ

′µ dim(D) ′µ Proof. By Lemma 6.5, we know that there exists a D-subcontramodule N1 ⊆ M such that |N1| ≤ (κ ) = κ • and B ⊆ α N1. Taking α = idD : D −→ D in Lemma 6.5, we can also obtain a D-subcontramodule N2 ⊆ M such that ′µ dim(D) ′µ |N2|≤ (κ ) = κ and A ⊆ N2. We set N := N1 + N2 ⊆ M. Since N is a quotient of N1 ⊕ N2, it follows from ′µ dim(D) ′µ • Lemma 6.4(a) that |N| ≤ |N1 ⊕ N2| ≤ (κ ) = κ . We also have A ⊆ N2 ⊆ N. Since α is exact, it preseves • • • monomorphims and hence B ⊆ α N1 ⊆ α N. By deﬁnition, we know that α N = CohomD(C,N) is the cokernel of the diﬀerence of the maps

HomK (C ⊗ C,N) ≃ HomK (C,HomK (C,N)) HomK (C,N)

• dim(C) ′µ dim(C) ′µ Therefore, |α N| = |CohomD(C,N)| ≤ |HomK (C,N)| = |N| ≤ (κ ) = κ . X X ∗ Let be a poset and let C : −→ Coalg be a right coﬂat representation. We choose λ such that each Cx is X λ-presentable in M[Cx, ] for x ∈ Ob( ). We now set

|Mor(X )| |Cx| κ := sup{ℵ0, |K|, λ, |Mor(X )|, (|K| ) , x ∈ Ob(X )}

µ := sup{ℵ0,dim(Cx), x ∈ Ob(X )}

27 We now choose a well ordering of the set Mor(X ) and consider the induced lexicographic ordering of N×Mor(X ). Let tr X M ∈ Contc -C and let m0 ∈ elX (M), i.e. m0 ∈ Mx for some x ∈ Ob( ). We will now deﬁne a family of subobjects {P(n, α)|(n, α : y −→ z) ∈ N × Mor(X )} of M in Conttr-C which satisﬁes the following conditions:

(1”) m0 ∈ elX (P(1, α0)), where α0 is the least element of Mor(X ). (2”) P(n, α) ⊆P(m,β) whenever (n, α) ≤ (m,β) in N × Mor(X ). α • (3”) For each (n, α : y −→ z) ∈ N × Mor(X ), the morphism P(n, α): α P(n, α)z −→ P(n, α)y is an isomorphism in

M[Cy, ]. (4”) |P(n, α)|≤ κµ. For each pair (n, α : y −→ z) ∈ N×Mor(X ), we now start constructing P(n, α). We know that there exists a morphism ∗ tr η : Cx −→ Mx in M[Cx, ] such that m0 ∈ Im(η). Then, we can deﬁne the subobject N ⊆M in Cont -C as in (5.11) µ such that m0 ∈ Nx. We also know by Lemma 5.6 that |N | ≤ κ ≤ κ . For (n, α : y −→ z) ∈ N × Mor(X ), we set

Nw if n = 1 and α = α0 A0(w)= (6.1) 0  P(m,β)w otherwise (m,β)<(n,α) S for each w ∈ Ob(X ), where for (m,β) <(n, α), we assume that P(m,β) satisﬁes all the properties (1”)-(4”). Clearly, 0 0 µ X A0(w) ⊆Mw and |A0(w)|≤ κ for each w ∈ Ob( ). tr • ∼ 0 Since M ∈ Contc -C, we have α Mz = My. Using Lemma 6.6, we can obtain a contramodule A1(z) ⊆Mz in M[Cz, ] such that 0 µ • 0 µ 0 0 0 • 0 |A1(z)|≤ κ |α A1(z)|≤ κ A0(z) ⊆ A1(z) A0(y) ⊆ α A1(z) (6.2) 0 • 0 0 0 X We now set A1(y) := α A1(z) and A1(w) := A0(w) for any w 6= y,z ∈ Ob( ). It now follows from (6.2) that 0 0 X 0 µ A0(w) ⊆ A1(w) for every w ∈ Ob( ) and each |A1(w)|≤ κ . µ tr (Lemma 6.7. Let B ⊆ elX (M) with |B|≤ κ . Then, there is a subobject Q ֒→M in Cont -C such that B ⊆ elX (Q and |Q| ≤ κµ. tr Proof. As in the proof of Theorem 5.7, for any m0 ∈ B ⊆ elX (M) we can choose a subobject Q(m0) ⊆M in Cont -C µ X such that m0 ∈ el (Q(m0)) and |Q(m0)|≤ κ ≤ κ . Now, we set Q := m0∈B Q(m0) ⊆ M. Since Q is a quotient of µ µ µ µ ⊕m0∈BQ(m0) and |B|≤ κ , it follows from Lemma 6.4(b) that |Q| ≤ | ⊕m0∈B Q(m0)|≤ (κ ) = κ . P 0 0 tr Using Lemma 6.7 with B = w∈Ob(X ) A1(w), we know that there exists Q (n, α) ֒→ M in Cont -C such that µ A0(w) ⊆ elX (Q0(n, α)) and |Q0(n, α)| ≤ κ . In particular, we have A0(w) ⊆ Q0(n, α) ⊆ M for each w∈Ob(X ) 1 S 1 w w w ∈ Ob(X ). S We will now iterate this construction. Suppose that we have constructed Ql(n, α) ֒→M in Conttr-C for every l ≤ r such l l l µ r+1 r X that w∈Ob(X ) A1(w) ⊆ elX (Q (n, α)) and |Q (n, α)| ≤ κ . We now set A0 (w) := Q (n, α)w for each w ∈ Ob( ). r+1 • r+1 r+1 Since A (y) ⊆ My =∼ α Mz and A (z) ⊆ Mz, using Lemma 6.6, we can obtain A (z) ֒→ Mz in M such S 0 0 1 [Cz, ] that r+1 µ • r+1 µ r+1 r+1 r+1 • r+1 |A1 (z)|≤ κ |α A1 (z)|≤ κ A0 (z) ⊆ A1 (z) A0 (y) ⊆ α A1 (z) (6.3) r+1 • r+1 r+1 r+1 X Then, we set A1 (y) := α A1 (z) and A1 (w) := A0 (w) for any w 6= y,z ∈ Ob( ). It now follows from (6.3) that r+1 r+1 X r+1 µ A0 (w) ⊆ A1 (w) for all w ∈ Ob( ) and each |A1 (w)|≤ κ . r+1 r+1 tr Again using Lemma 6.7 with B = w∈Ob(X ) A1 (w), we can obtain Q (n, α) ֒→ M in Cont -C such that r+1 r r µ r+1 r A (w) ⊆ elX (Q +1(n, α)) and |Q +1(n, α)| ≤ κ . In particular, A (w) ⊆ Q +1(n, α) for each w∈Ob(X ) 1 S 1 w w ∈ Ob(X ). We now deﬁne S

T (n, α) := lim Qr(n, α) P(n, α) := Im lim Qr(n, α) −→ M = Im (T (n, α) −→ M) (6.4) −→ −→ r≥0 r≥0 !

28 in Conttr-C. α • µ Lemma 6.8. The morphism T (n, α): α T (n, α)z −→ T (n, α)y is an isomorphism in M[Cy , ]. Also, |T (n, α)|≤ κ .

Proof. By definition, T (n, α)z is the following colimit in M[Cz, ]. 0 0 1 1 r+1 r+1 T (n, α)z = colim A1(z) −→ Q (n, α)z −→ A1(z) −→ Q (n, α)z −→ ... −→ A1 (z) −→ Q (n, α)z −→ ... Since α• is a left adjoint, we have therefore • • 0 • 0 • r+1 • r+1 α T (n, α)z = colim α A1(z) −→ α Q (n, α)z −→ ... −→ α A1 (z) −→ α Q (n, α)z −→ ... • r r r in M[Cy , ]. We also know that α A1(z) = A1(y) for each r ≥ 0. Writing T (n, α)y as the colimit of {A1(y)}r≥0 in α r µ M[Cy, ], we see that T (n, α) is an isomorphism. Finally, we note that each |Q (n, α)|≤ κ . Applying Lemma 6.4(b), we now obtain |T (n, α)|≤ (κµ)µ ≤ κµ. Lemma 6.9. The family {P(n, α)|(n, α) ∈ N × Mor(X )} satisfies conditions (1”)–(4”). 0 0 0 Proof. We know that m0 ∈ Nx. For n = 1 and α = α0, we have Nx = A0(x) ⊆ A1(x) ⊆ Q (1, α0)x. Since 0 0 ,Q (1, α0) ֒→M factors through T (1, α0), it follows that m0 ∈Q (1, α0)x ⊆ Im(T (1, α0) −→ M)x = P(1, α0)x. Thus condition (1”) is satisfied. 0 0 0 Now, for any (m,β) < (n, α), it again follows by (6.1) and (6.4) that we have P(m,β)w ⊆ A0(w) ⊆ A1(w) ⊆Q (n, α)w ⊆ Im(T (n, α) −→ M)w = P(n, α)w for every w ∈ Ob(X ). This shows that condition (2”) is satisfied. The condition (4”) follows from Lemma 6.8 and the fact that P(n, α) is a quotient of T (n, α). α • For (n, α : y −→ z) ∈ N×Mor(X ), we know from Lemma 6.8 that T (n, α): α T (n, α)z −→ T (n, α)y is an isomorphism α • • in M[Cy , ]. Since M is cartesian, we also know that M : α Mz −→ My is an isomorphism. Since α is exact and • P(n, α)= Im(T (n, α) −→ M), it follows that α P(n, α)z =∼ P(n, α)y. This proves condition (3”). X X tr Lemma 6.10. Let be a poset and let C : −→ Coalg be a right coflat representation. Let M ∈ Contc -C and tr µ m0 ∈ elX (M). Then, there exists P⊆M in Contc -C with m0 ∈ elX (P) such that |P| ≤ κ . Proof. We note that the set N × Mor(X ) with the lexicographic ordering is filtered. We set T := lim P(n, α) P := Im(T −→M) (6.5) −→ (n,α)∈N×Mor(X )

tr in Cont -C. By Lemma 6.9, we know that m0 ∈ P(1, α0)x ⊆ Im(Tx −→ Mx) = Px. Thus, m0 ∈ elX (P). By Lemma 6.9, we also know that each |P(n, α)| ≤ κµ. Applying Lemma 6.4(b) to the colimit in (6.5), we see that |P| ≤ |T | ≤ (κµ)µ = κµ.

It remains to show that P is cartesian. For this, we consider a morphism β : z −→ w in X . Then, the family {(s,β)}s≥1 is coﬁnal in N × Mor(X ) and therefore we may express T := lim P(s,β) (6.6) −→ s≥1

Since β• is a left adjoint, we have β•T = β• lim P(s,β) = lim β•P(s,β) . By Lemma 6.9, each βP(s,β): w −→ w −→ w s≥1 ! s≥1 • β • β P(s,β)w −→ P(s,β)z is an isomorphism and it follows that T : β Tw −→ Tz is an isomorphism. In other words, tr tr T ∈ Contc -C. It follows that P = Im(T −→M) also lies in Contc -C. X X tr Theorem 6.11. Let be a poset and let C : −→ Coalg be a right coﬂat representation. Then, the category Contc -C of cartesian trans-contramodules over C has a set of generators. tr Proof. It follows from Lemma 6.10 that any M ∈ Contc -C can be expressed as a quotient of a direct sum m0∈elX (M) Pm0 µ of cartesian subcontramodules with each |Pm0 |≤ κ . Therefore, the isomorphism classes of cartesian contramodules P µ tr L with |P| ≤ κ form a set of generators for Contc -C.

29 7 Rational pairings, torsion classes, functors between comodules and contramodules

In this ﬁnal section, we relate modules over algebra representations to comodules and contramodules over coalgebra representations. An algebra representation will be a functor A : X −→ Alg, where Alg is the category of K-algebras. In [15], Estrada and Virili considered modules over a representation of a small category X taking values in small preadditive categories, i.e., in algebras with several objects. Our “cis-modules” over A will be identical to the modules of Estrada and Virili [15] in the case where the representation in [15] takes values in K-algebras. We will observe that cis-modules over A are related to trans-comodules over its ﬁnite dual representation and vice versa.

7.1 Modules over algebra representations

For any algebra A, let MA denote the category of right A-modules. Corresponding to any algebra morphism α : A −→ B, ◦ there is a restriction of scalars α◦ : MB −→ MA and an extension of scalars α : MA −→ MB. The functor α◦ also has a right adjoint

† α : MA −→ MB N 7→ HomA(B,N)

Let A : X −→ Alg be an algebra representation. In particular, for each object x ∈ Ob(X ), we have an algebra Ax and for any α ∈ X (x, y), we have a morphism Aα : Ax −→ Ay of K-algebras. As with coalgebra representations, we will ◦ † ◦ † X abuse notation to write α , α◦ and α respectively for functors Aα, Aα◦, Aα for any morphism α in . Deﬁnition 7.1. Let A : X −→ Alg be an algebra representation. A (right) cis-module M over A will consist of the following data:

(1) For each object x ∈ Ob(X ), a right Ax-module Mx

(2) For each morphism α : x −→ y in X , a morphism Mα : Mx −→ α◦My of right Ax-modules (equivalently, a α ◦ morphism M : α Mx −→ My of right Ay-modules). X βα β ◦ α We further assume that Midx = idMx for any x ∈ Ob( ) and M = M ◦ β (M ) for composable morphisms α, β in X (equivalently, α◦(Mβ) ◦Mα = Mβα). A morphism η : M −→ N of cis-modules over A consists of morphisms ηx : Mx −→ Nx of right Ax-modules for each x ∈ Ob(X ) compatible with the morphisms in (2). We denote this category of right cis-modules by Modcs-A. Similarly, we may define the category A-Modcs of left cis-modules over A. We will say that M ∈ Modcs-A is cartesian if each Mα is an isomorphism. The full subcategory of cartesian cis-modules cs will be denoted by Modc -A. The following results now follow from [15, Theorem 3.18, Theorem 3.23]. Theorem 7.2. Let A : X −→ Alg be an algebra representation. Then, (1) Modcs-A is a Grothendieck category. If X is a poset, then Modcs-A has a projective generator. X ◦ X (2) Suppose that A : −→ Alg is left flat, i.e., each α = ⊗Ax Ay : MAx −→ MAy is exact for α ∈ (x, y). cs Then, Modc -A is a Grothendieck category. We now introduce the notion of trans-module over an algebra representation. Definition 7.3. Let A : X −→ Alg be an algebra representation. A (right) trans-module M over A will consist of the following data:

(1) For each object x ∈ Ob(X ), a right Ax-module Mx

30 X † (2) For each morphism α : x −→ y in , a morphism αM : My −→ α Mx = HomAx (Ay, Mx) of right Ay-modules α (equivalently, a morphism M : α◦My −→ Mx of right Ax-modules) † X We further assume that idx M = idMx and βαM = β (αM) ◦ βM for composable morphisms α, β in (equivalently, βα α M = M◦ α◦(βM)). A morphism η : M −→ N of trans-modules over A consists of morphisms ηx : Mx −→ Nx of right Ax-modules for each x ∈ Ob(X ) compatible with the morphisms in (2). We denote this category of right trans-modules by Modtr-A. Similarly, we may deﬁne the category A-Modtr of left trans-modules over A. tr We will say that M ∈ Mod -A is cartesian if each αM is an isomorphism. The full subcategory of cartesian trans- tr modules will be denoted by Modc -A. We may easily verify that Modtr-A is a cocomplete abelian category. To study generators in Modtr-A, we consider tr M ∈ Mod -A and m0 ∈ Mx for some x ∈ Ob(X ). If we take a morphism η : Ax −→ Mx in Mod-Ax such that m0 ∈ Im(η), we can set for each y ∈ Ob(X )

β β◦η M Ny := Im β◦Ax −−→ β◦Mx −−→My (7.1)  X  β∈M(y,x)   tr As in previous sections, we can show that the family {Ny ∈ MAy }y∈Ob(X ) determines an object N in Mod -A. Moreover, m0 ∈ Nx. Now, we set

κ := sup{ℵ0, |Mor(X )|, |K|, |Ax|, x ∈ Ob(X )}

Then, if |N | denotes the cardinality of the union y∈Ob(X ) Ny, we have |N | ≤ κ. This proves the following result. Theorem 7.4. Let A : X −→ Alg be an algebraS representation. Then, Modtr-A is a Grothendieck category. X tr tr As in Section 3.2 and Section 5, we may show that when is a poset, the evaluation functor evx : Mod -A −→ MAx X tr tr tr tr for each x ∈ Ob( ) has both a left adjoint exx : MAx −→ Mod -A and a right adjoint coex : MAx −→ Mod -A.

Since each Ax is a projective generator in MAx , we may prove the following result in a manner similar to Theorem 3.8. X X tr Theorem 7.5. Let be a poset and A : −→ Alg be an algebra representation. Then, {exx (Ax)}x∈Ob(X ) is a set of projective generators for Modtr-A.

Let X be a poset and A : X −→ Alg be an algebra representation such that for each α ∈ X (x, y), the K-algebra Ay † is ﬁnitely generated and projective as an Ax-module. Then, each α is exact and commutes with colimits. Accordingly, tr we may verify that the category Modc -A of cartesian trans-modules over A is abelian and cocomplete. tr X We take M ∈ Modc -A and consider m0 ∈Mx for some x ∈ Ob( ). Using a transﬁnite induction argument similar to tr Section 4.2, we can obtain a subobject P ֒→M in Modc -A such that m0 ∈Px. Accordingly, we can prove the following result.

Theorem 7.6. Let A : X −→ Alg be an algebra representation such that for each α ∈ X (x, y), the K-algebra Ay is tr ﬁnitely generated and projective as an Ax-module. Then, Modc -A is a Grothendieck category.

7.2 Rational modules, torsion classes, functors between comodules and contramodules Let C be a K-coalgebra and let A be a K-algebra. We recall (see, for instance, [11, § 4.18]) that a rational pairing of C and A consists of a morphism ϕ : C ⊗ A −→ K such that: (i) For any vector space V , the following linear map is injective

δV : V ⊗ C −→ HomK (A, V ) v ⊗ c 7−→ [a 7→ ϕ(c ⊗ a)v]

(ii) The morphism A −→ C∗ induced by ϕ : C ⊗ A −→ K is a morphism of K-algebras.

31 We note in particular that the induced map C −→ A∗ is injective. For more on rational pairings, we refer the reader, for instance, to [1]. C Now let ϕ : C ⊗ A −→ K be a rational pairing, let M denote the category of right C-comodules and AM denote the category of left A-modules. We note that any right C-comodule with structure map ρM : M −→ M ⊗ C may be treated M as a left A-module by setting am := m0ϕ(m1 ⊗a) for each a ∈ A and m ∈ A, where ρ (m)= m0 ⊗m1 ∈ M ⊗C. In C fact, this embeds M as the smallest full Grothendieck subcategory of AM containing C (see [11, § 4.19]). Accordingly, C P C P the inclusion Iϕ : M −→ AM has a right adjoint Rϕ : AM −→ M given by setting (see [11, § 41.1])

C Rϕ(N) := {Im(f) | f ∈ AM(Iϕ(M),N), M ∈ M } (7.2) X A left A-module N is said to be rational if it satisﬁes Rϕ(N) = N. Since the collection of rational modules is closed under quotients and subobjects (see [33, Corollary 2.4]), it follows that for any N ∈ AM, Rϕ(N) is the sum of all rational submodules of N. Deﬁnition 7.7. Let C : X −→ Coalg be a coalgebra representation and let A : X op −→ Alg be an algebra representation. Then, a rational pairing (C, A, Φ= {ϕx}x∈Ob(X )) is a triple such that

(1) for each x ∈ Ob(X ), ϕx : Cx ⊗ Ax −→ K is a rational pairing of a coalgebra with an algebra

(2) for each α ∈ X (x, y), we have ϕy(Cα(c) ⊗ a)= ϕx(c ⊗ Aα(a)) for any c ∈Cx and a ∈ Ay. Definition 7.8. Let (C, A, Φ) be a rational pairing of C : X −→ Coalg with A : X op −→ Alg. A (left) trans-module X M over the algebra representation A is said to be rational if Rϕx (Mx)= Mx for each x ∈ Ob( ). Similarly, we can define rational cis-modules. We will denote by A-Rattr (resp. A-Ratcs) the full subcategory of A-Modtr (resp. A-Modcs) whose objects are rational trans-modules (resp. rational cis-modules) over A. Theorem 7.9. Let (C, A, Φ) be a rational pairing. Then, tr tr tr tr tr tr (1) the inclusion functor IΦ : A-Rat −→ A-Mod admits a right adjoint RΦ : A-Mod −→ A-Rat . tr tr (2) N ∈ A-Mod is rational if and only if RΦ (N )= N . Proof. (1) Let N ∈ A-Modtr. For each y ∈ Ob(X ), we define

tr α X op RΦ (N )y := {ny ∈ Ny | N (ny) ∈ Rϕx (Nx) ∀ α ∈ (x, y)} (7.3)

tr In particular, we note that RΦ (N )y ⊆ Rϕy (Ny). Since the class of rational modules in Ay M is closed under subobjects, tr X op it follows that each RΦ (N )y is a rational Ay-module. From (7.3), it is also clear that for any morphism β ∈ (y,z), β β tr tr tr tr the morphism N : β◦Nz −→ Ny restricts to a morphism RΦ (N ): β◦RΦ (N )z −→ RΦ (N )y. It follows that RΦ (N ) ∈ A-Rattr. tr tr tr We consider now some M ∈ A-Rat and a morphism η : IΦ(M) −→ N in A-Mod . We claim that ηy(My) ⊆ RΦ (N )y X X op for each y ∈ Ob( ). Accordingly, for each α ∈ (x, y), we consider the following commutative diagram in Ax M:

α◦ηy α◦My −−−−→ α◦Ny

αM αN (7.4)

 ηx  Mx −−−−→ Nx y y α α We consider my ∈My. By (7.4), we have ( N )(α◦ηy(my)) = ηx( M(my)). Since Mx is a rational Ax-module, we know α X op tr that Im(ηx) ⊆ Rϕx (Nx). It follows that ( N )(α◦ηy(my)) ∈ Rϕx (Nx) for each α ∈ (x, y), i.e., ηy(my) ∈ RΦ (N )y. The result is now clear. tr X (2) Let N ∈ A-Mod be rational. Then, by deﬁnition, each Nx = Rϕx (Nx) for x ∈ Ob( ). From (7.3), it is immediate tr tr tr that RΦ (N )= N . Conversely, suppose that N = RΦ (N ). Then, each Nx = RΦ (N )x ⊆ Rϕx (Nx) is rational.

32 Theorem 7.10. Let (C, A, Φ) be a rational pairing. Then, the categories Comcs-C and A-Rattr are isomorphic.

cs Proof. Let M ∈ Com -C. Then, for each x ∈ Ob(X ), Mx carries the structure of a rational Ax-module. Further, given op ∗ any α ∈ X (x, y)= X (y, x), the morphism α Mx −→ My of right Cy-comodules induces a morphism α◦Mx −→ My tr of left Ay-modules. As such, M may be treated as an object of A-Rat . tr Conversely, suppose we have N ∈ A-Rat . Then, for each x ∈ Ob(X ), the rational Ax-module Nx can be equipped op α with the structure of a right Cx-comodule. For α ∈ X (x, y)= X (y, x), we also have the morphism N : α◦Nx −→ Ny of left Ay-modules. From condition (2) in Deﬁnition 7.7, it follows that the left Ay-module structure on α◦Nx can be Cy obtained from a right Cy-comodule structure, i.e., α◦Nx is a rational Ay-module. Since M is a full subcategory of α Ay M, we see that N : α◦Nx −→ Ny may now be treated as a morphism of right Cy-comodules. Hence, N may be treated as an object of Comcs-C.

cs cs cs Similarly, given a rational pairing (C, A, Φ), we can show that the inclusion IΦ : A-Rat ֒→ A-Mod admits a right cs tr cs adjoint RΦ and that the categories Com -C and A-Rat are isomorphic. We know in particular that if C is any K-coalgebra, then (C, C∗) is a rational pairing. Similarly, for any K-algebra A, it is well known that we have a rational pairing (A◦, A), where A◦ is the ﬁnite dual coalgebra of A given by setting (see, for instance, [13, Lemma 1.5.2])

◦ ∗ A = {f ∈ A = HomK (A, K) | Ker(f) contains an ideal of ﬁnite codimension} (7.5)

Accordingly, we have the following result. Corollary 7.11. (a) Let C : X −→ Coalg be a coalgebra representation and C∗ : X op −→ Alg be its linear dual representation. Then, Comcs-C≃C∗-Rattr and Comtr-C≃C∗-Ratcs. (b) Let A : X op −→ Alg be an algebra representation and A◦ : X −→ Coalg be its ﬁnite dual representation. Then, Comcs-A◦ ≃ A-Rattr and Comtr-A◦ ≃ A-Ratcs. Proof. Both (a) and (b) follow directly from Theorem 7.10. Our next objective is to give suﬃcient conditions for Comcs-C ≃ A-Rattr to be a torsion class in A-Modtr. For this, we begin by extending some of the classical theory of rational pairings of coalgebras and algebras using torsion theory (see [20], [25]). Let (C, A, ϕ) be a rational pairing. We note (see [33, Proposition 2.2]) that for any left A-module N, the rational submodule Rϕ(N) may also be expressed as

Rϕ(N)= {n ∈ N | Ann(n) is a closed and coﬁnite left ideal in A} (7.6)

∗ Here C = HomK(C,K) carries the finite topology (see, for instance [13, Section 1.2]) and A is equipped with the topology that makes the algebra morphism A −→ C∗ continuous. We also recall here that a subspace V ⊆ A is said to be cofinite if the quotient A/V is finite dimensional as a vector space. The following argument appears in essence in [20, Proposition 22], which we extend here to rational pairings of coalgebras and algebras.

Lemma 7.12. Let ϕ : C ⊗ A −→ K be a rational pairing. Suppose that Rϕ(A) is dense in A. Then, for any N ∈ AM, we have Rϕ(N/Rϕ(N))=0.

Proof. We choose n + Rϕ(N) ∈ Rϕ(N/Rϕ(N)) and a ∈ Rϕ(A). Then, Ann(n + Rϕ(N)) and Ann(a) are coﬁnite and closed left ideals in A. Since Ann(a) ⊆ Ann(an), it follows from [33, § 1.3 (b)] that Ann(an) is also coﬁnite and closed. Hence, an ∈ Rϕ(N) and therefore a ∈ Ann(n + Rϕ(N)). This shows that Rϕ(A) ⊆ Ann(n + Rϕ(N)). Since Rϕ(A) is dense in A and Ann(n + Rϕ(N)) is closed, we get Ann(n + Rϕ(N)) = A, i.e., n + Rϕ(N) = 0. Proposition 7.13. (see [20]) Let ϕ : C ⊗ A −→ K be a rational pairing. Then, the following are equivalent.

(a) For any N ∈ AM, Rϕ(N/Rϕ(N))=0.

(b) The full subcategory of rational modules is a torsion class in AM.

33 Proof. It is immediate that (b) ⇒ (a). To show that (a) ⇒ (b) we consider the two full subcategories T and F of AM given by Ob(T ) := {N ∈ AM | Rϕ(N)= N } Ob(F) := {N ∈ AM | Rϕ(N)=0 } (7.7)

From the adjoint pair (Iϕ, Rϕ), it is clear that AM(N1,N2) =0 for any N1 ∈ T and N2 ∈F. Finally, for any N ∈ AM, we have a short exact sequence 0 −→ Rϕ(N) −→ N −→ N/Rϕ(N) −→ 0 (7.8)

From (7.6), it is clear that Rϕ(N) ∈ T . Using (a), we see that N/Rϕ(N) ∈F and this proves (b). X Theorem 7.14. Let (C, A, Φ) be a rational pairing. Suppose that for each x ∈ Ob( ) and N ∈ Ax M, we have tr tr Rϕx (N/Rϕx (N))=0. Then, the full subcategory A-Rat of rational modules is a torsion class in A-Mod . Proof. Since A-Modtr is an abelian category that is both complete and cocomplete, it suﬃces (see [9, § 1.1]) to show tr that the full subcategory A-Rat is closed under coproducts, quotients and extensions. Since Rϕx (N/Rϕx (N)) = 0 for X every x ∈ Ob( ) and N ∈ Ax M, we know from Proposition 7.13 that rational Ax-modules form a torsion class in Ax M. We consider now a short exact sequence

0 −→ M′ −→ M −→ M′′ −→ 0 (7.9)

tr tr ′′ ′′ in A-Mod . Suppose that M ∈ A-Rat . Then, each Mx −→ Mx is an epimorphism in Ax M. In particular, each Mx ′′ tr is a rational Ax-module and we get M ∈ A-Rat . ′ ′′ tr ′ ′′ Similarly, if M , M ∈ A-Rat , we see that each 0 −→ Mx −→ Mx −→ Mx −→ 0 is exact in Ax M with both ′ ′′ Mx, Mx rational. Again, since rational Ax-modules are closed under extensions in Ax M, it follows that each Mx is rational, i.e., M ∈ A-Rattr. By similar reasoning, we see that A-Rattr is also closed under coproducts. This proves the result. Remark 7.15. Since the submodules of a rational module are always rational (see (7.6)), we note that the condition in Theorem 7.14 makes A-Rattr a hereditary torsion class in A-Modtr. X Corollary 7.16. Let (C, A, Φ) be a rational pairing. Suppose that for each x ∈ Ob( ), Rϕx (Ax) is dense in Ax. Then, the full subcategory A-Rattr of rational modules is a torsion class in A-Modtr. Proof. This follows directly from Lemma 7.12 and Theorem 7.14. Corollary 7.17. Let C : X −→ Coalg be a representation taking values in right semiperfect K-coalgebras. Then, Comcs-C forms a torsion class in C∗-Modtr. Proof. Let C be a coalgebra and ϕ : C ⊗ C∗ −→ K the canonical pairing. If C is right semiperfect, we know from ∗ [20, Theorem 23] that Rϕ(N/Rϕ(N)) = 0 for any left C -module N. The result is now clear from Corollary 7.11 and Theorem 7.14.

C Let C be a K-coalgebra. We note that any C-contramodule (M, πM : HomK (C,M) −→ M) may be treated as a C ∗ ∗ πM C -module by considering C ⊗ M −→ HomK (C,M) −−→ M. We recall from [10, Theorem 3.11] that this determines a functor M[C,−] −→ MC∗ . Proposition 7.18. Let C : X −→ Coalg be a coalgebra representation. Then, we have a functor Conttr-C −→ Modcs-C∗. tr ∗ X Proof. Let M ∈ Cont -C. Then, Mx ∈ M[Cx, ] may be treated as a Cx-module for each x ∈ Ob( ). Additionally, X M M ∗ for each α ∈ (x, y), the morphism αM : My −→ α•Mx in [Cy, ] induces a morphism in Cy . This proves the result.

34 For a K-coalgebra C and a C-bicomodule N, we recall (see [26]) that the functor MC(N, −): MC −→ V ect takes values C C ⊠ C in M[C, ]. Moreover, the functor M (N, −): M −→ M[C, ] has a left adjoint − C N : M[C, ] −→ M known as the contratensor product (see [26]). Explicitly, for any M ∈ M[C, ], we have

M ⊠C N := Coeq HomK (C,M) ⊗ NM ⊗ N (7.10) where the two maps in (7.10) are induced by the structure maps HomK (C,M) −→ M and N −→ C ⊗ N of M and N respectively. Let α : C −→ D be a morphism of cocommutative coalgebras. Let M be a C-contramodule and N be a C-comodule. From the construction of the contratensor product in (7.10), we note that there is a canonical morphism

∗ ∗ α•M ⊠D α N −→ α (M ⊠C N) (7.11) in MD. Lemma 7.19. Let α : C −→ D be a quasi-ﬁnite morphism of cocommutative coalgebras. Then, D C (a) For any M ∈ M[D, ] and N ∈ M , we have a natural isomorphism in M :

! • ! α (M ⊠D N) =∼ (α M) ⊠C (α N) (7.12)

D C (b) For any N ∈ M and P ∈ M , we have a natural isomorphism in M[D, ]

D ∗ C ! M (N, α P ) =∼ α•M (α N, P ) (7.13) ∗ ∗ ⊠ ∼ D • ∗ ∗ Proof. (a) We ﬁrst set M = D in M[D, ]. From [7, § 1], we know that D D N = N in M . Since α D = C in M[C, ], (7.12) reduces to

! ! ∗ ! ∗ ! • ∗ ! • ! α (M ⊠D N)= α (D ⊠D N) =∼ α N =∼ C ⊠C (α N) = (α D ) ⊠C (α N) =∼ (α M) ⊠C (α N) (7.14)

• ! We also know (see [7, § 1]) that ⊠D preserves colimits in both arguments. Since α and α are both left adjoints, it follows from (7.14) that (7.12) holds for any free contramodule M = D∗(I), i.e., any direct sum of copies of D∗. Since any M ∈ M[D, ] can be expressed as a cokernel M = Cok(f : F2 −→ F1) of a morphism of free contramodules and both • ! ! ⊠ ∼ • ⊠ ! α and α preserve colimits, we obtain α (M D N) = (α M) C (α N). By lifting morphisms in M[D, ] to morphisms of their free resolutions, we see that this isomorphism does not depend on the choice of M = Cok(f : F2 −→ F1).

(b) This follows from Yoneda lemma and the isomorphism in part (a), by noting that for any M ∈ M[D, ], we have

D ∗ ∼ D ⊠ ∗ M[D, ](M, M (N, α P )) = M (M D N, α P ) C ! C • ! =∼ M (α (M ⊠D N), P ) =∼ M ((α M) ⊠C (α N), P ) (7.15) ∼ • C ! ∼ C ! = M[C, ](α M, M (α N, P )) = M[D, ](M, α•M (α N, P ))

Theorem 7.20. Let C : X −→ Coalg be a coalgebra representation taking values in cocommutative coalgebras. Let tr N ∈ Comc -C be a cartesian trans-comodule over C. (a) We have a functor F : Conttr-C −→ Comtr-C defined by setting ⊠ X tr F (M)x := Mx Cx Nx ∀ x ∈ Ob( ), M ∈ Cont -C (7.16) (b) Suppose that C : X −→ Coalg is quasi-finite. Then, we have a functor G : Comtr-C −→ Conttr-C defined by setting

Cx tr G(P)x := M (Nx, Px) ∀ x ∈ Ob(X ), P ∈ Com -C (7.17) In that case, (F, G) is a pair of adjoint functors.

35 Proof. (a) We consider α ∈ X (y, x). Then, we have the following composition in MCx

⊠ ⊠ αM Cx αN ⊠ ∗ ∗ ⊠ ∗ αF (M): F (M)x = Mx Cx Nx −−−−−−−−→ (α•My) Cx (α Ny) −−−−→ α (My Cy Ny)= α F (M)y

where the last morphism follows from (7.11). Accordingly, we have a functor F : Conttr-C −→ Comtr-C.

! ∗ ∗ ! Cx (b) Let α ∈ X (y, x). Using the adjoint pair (α , α ), we have a canonical morphism uα(P): Px −→ α α Px in M .

Accordingly, we have the following composition in M[Cx, ]

Cx Cy ! α M (Nx,uα(P)) ∼ α•M (α Nx, P) Cx Cx ∗ ! = Cy ! ! Cy ! M (Nx, Px) −−−−−−−−−−−→ M (Nx, α α Px) −−−−→ α•M (α Nx, α Px) −−−−−−−−−−−−→ α•M (α Nx, Py)

tr where the isomorphism in the middle is obtained from Lemma 7.19(b). Since N ∈ Comc -C is cartesian, we put ! α Nx = Ny. The above therefore gives us a morphism

Cx Cy ! Cy αG(P): G(P)x = M (Nx, Px) −−−−→ α•M (α Nx, Py)= α•M (Ny, Py)= α•G(P)y

corresponding to α ∈ X (y, x). Hence, we have a functor G : Comtr-C −→ Conttr-C. The adjointness of F and G follows Cx ⊠ ∼ Cx X from the fact that M (Mx Cx Nx, Px) = M[Cx, ](Mx, M (Nx, Px)) for each x ∈ Ob( ).

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