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Categories of Modules, Comodules and Contramodules Over Categories of modules, comodules and contramodules over representations Mamta Balodi ∗ Abhishek Banerjee †‡ Samarpita Ray §¶ Abstract We study and relate categories of modules, comodules and contramodules over a representation of a small category taking values in (co)algebras, in a manner similar to modules over a ringed space. As a result, we obtain a categorical framework which incorporates all the adjoint functors between these categories in a natural manner. Various classical properties of coalgebras and their morphisms arise naturally within this theory. We also consider cartesian objects in each of these categories, which may be viewed as counterparts of quasi-coherent sheaves over a scheme. We study their categorical properties using cardinality arguments. Our focus is on generators for these categories and on Grothendieck categories, because the latter may be treated as replacements for noncommutative spaces. MSC(2020) Subject Classification: 16T15, 18E10 Keywords : Modules, comodules, contramodules, cartesian objects, Grothendieck categories Contents 1 Introduction 2 2 Comodules over coalgebra representations 5 2.1 Cis-comodules over coalgebra representations . ..................... 5 2.2 Trans-comodules over coalgebra representations . ....................... 9 3 Coalgebra representations of a poset and projective generatorsforcomodules 11 3.1 Projectivegeneratorsforcis-comodules. ..................... 11 3.2 Projectivegeneratorsfortrans-comodules . ...................... 12 4 Cartesian comodules over coalgebra representations 14 4.1 Cartesian cis-comodules over coalgebra representations . ........................ 14 4.2 Cartesian trans-comodules over coalgebra representations . ......................... 17 5 Contramodules over coalgebra representations 20 arXiv:2106.12237v1 [math.RA] 23 Jun 2021 6 Cartesian trans-contramodules over coalgebra representations 25 7 Rational pairings, torsion classes, functors between comodulesandcontramodules 30 7.1 Modulesoveralgebrarepresentations . ................... 30 7.2 Rational modules, torsion classes, functors between comodulesandcontramodules . 31 ∗Department of Mathematics, Indian Institute of Science, Bangalore 560012, India. Email: [email protected] †Department of Mathematics, Indian Institute of Science, Bangalore 560012, India. Email: [email protected] ‡AB was partially supported by SERB Matrics fellowship MTR/2017/000112 §Department of Mathematics, Indian Institute of Science Education and Research, Pune 411008, India. Email: [email protected] ¶SR was partially supported by SERB National Postdoctoral Fellowship PDF/2020/000670 1 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], To¨en and Vaqui´e[43]), in derived algebraic geometry (see Lurie [23]) and in homotopical algebraic geometry (see To¨en 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).
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