Category Of Comodules Research Articles

The Galois group of the category of mixed Hodge–Tate structures

The category of rational mixed Hodge-Tate structures is a mixed Tate category. So thanks to the Tannakian formalism, it is equivalent to the category of finite dimensional graded comodules over a graded commutative Hopf algebra H over Q. Since the category has homological dimension 1, the Hopf algebra H is isomorphic to the commutative graded Hopf algebra provided by the tensor algebra of the graded vector space given by the direct sum of the groups C/Q(n) over n>0. However this isomorphism is not natural in any sense, e.g. does not work in families. We give a natural explicit construction of the Hopf algebra H. Generalizing this, we define a Hopf dg-algebra related to any dg-algebra R over a field k, equipped with an invertible line k(1). When R is the sheaf of algebras given by the holomorphic de Rham complex of a complex manifold X, and the line is Q(1), the related Hopf dg-algebra describes a dg-model of the derived category of variations of Hodge-Tate structures on X. Its cobar complex provides a dg-model for the rational Deligne cohomology of X. We consider a variant of our construction which starts from Fontaine's crystalline / semi-stable period rings and produces graded / dg Hopf algebras, which we relate to the p-adic Hodge theory.

Weak Rickart and dual weak Rickart objects in abelian categories

ABSTRACTWe introduce and investigate weak relative Rickart objects and dual weak relative Rickart objects in abelian categories. Several types of abelian categories are characterized in terms of (dual) weak relative Rickart properties. We relate our theory to the study of relative regular objects and (dual) relative Baer objects. We also give some applications to module and comodule categories.

Rickart and Dual Rickart Objects in Abelian Categories: Transfer via Functors

We study the transfer of (dual) relative Rickart properties via functors between abelian categories, and we deduce the transfer of (dual) relative Baer property. We also give applications to Grothendieck categories, comodule categories and (graded) module categories, with emphasis on endomorphism rings.

Dedualizing Complexes of Bicomodules and MGM Duality Over Coalgebras

This paper is based on the material of Section 4 and Appendix C in arXiv:1503.05523v6, which was excluded from the subsequent versions of arXiv:1503.05523. We present the definition of a dedualizing complex of bicomodules over a pair of cocoherent coassociative coalgebras $\mathcal C$ and $\mathcal D$. Given such a complex $\mathcal B^\bullet$, we construct an equivalence between the (bounded or unbounded) conventional, as well as absolute, derived categories of the abelian categories of left comodules over $\mathcal C$ and left contramodules over $\mathcal D$. Furthermore, we spell out the definition of a dedualizing complex of bisemimodules over a pair of semialgebras, and construct the related equivalence between the conventional or absolute derived categories of the abelian categories of semimodules and semicontramodules. Artinian, co-Noetherian, and cocoherent coalgebras are discussed as a preliminary material.

Recollements of comodule categories over semi-perfect coalgebras

In this paper, we study the recollements of comodule categories over semi-perfect coalgebras. We prove that a recollement whose terms are comodule categories is equivalent to one induced by a co-idempotent subcoalgebra of semi-perfect coalgebras. Furthermore, we present two applications to the Morita-Takeuchi context coalgebras and comodule complex categories in this paper.

Tannakian formalism for fiber functors over tensor categories

In this paper we generalize Tannakian formalism to fiber functors over general tensor categories. We will show that (under some technical conditions) if the fiber functor has a section, then the source category is equivalent to the category of comodules over a Hopf algebra in the target category. We will also give a description of this Hopf algebra using the notion of framed objects.

Calabi–Yau property under monoidal Morita–Takeuchi equivalence

Let $H$ and $L$ be two Hopf algebras such that their comodule categories are monoidal equivalent. We prove that if $H$ is a twisted Calabi-Yau (CY) Hopf algebra, then $L$ is a twisted CY algebra when it is homologically smooth. Especially, if $H$ is a Noetherian twisted CY Hopf algebra and $L$ has finite global dimension, then $L$ is a twisted CY algebra.

On the derived category of quasi-coherent sheaves on an Adams geometric stack

Let X be an Adams geometric stack. We show that D(Aqc(X)), its derived category of quasi-coherent sheaves, satisfies the axioms of a stable homotopy category defined by Hovey, Palmieri and Strickland in [13]. Moreover we show how this structure relates to the derived category of comodules over a Hopf algebroid that determines X.

From motives to comodules over the motivic Hopf algebra

We construct the natural functor from the triangulated category of motives (over a field endowed with a complex embedding) to the triangulated category of homotopy comodules over the motivic homotopy Hopf algebra.

The group of braided autoequivalences of the category of comodules over a coquasi-triangular Hopf algebra

Let H be a coquasi-triangular Hopf algebra. We first show that the group of braided autoequivalences of the category of H -comodules is isomorphic to the group of braided-commutative bi-Galois objects. Next, by investigating the latter, we obtain that the group of braided autoequivalences of the representation category of Lusztig’s quantum group u q ( s l ( 2 ) ) ′ is isomorphic to the projective special linear group P S L ( 2 ) , where q is a root of unity of odd order N > 1 .

INVERTIBLE BIMODULES, MIYASHITA ACTION IN MONOIDAL CATEGORIES AND AZUMAYA MONOIDS

In this paper we introduce and study Miyashita action in the context of monoidal categories aiming by this to provide a common framework of previous studies in the literature. We make a special emphasis of this action on Azumaya monoids. To this end, we develop the theory of invertible bimodules over different monoids (a sort of Morita contexts) in general monoidal categories as well as their corresponding Miyashita action. Roughly speaking, a Miyashita action is a homomorphism of groups from the group of all isomorphic classes of invertible subobjects of a given monoid to its group of automorphisms. In the symmetric case, we show that for certain Azumaya monoids, which are abundant in practice, the corresponding Miyashita action is always an isomorphism of groups. This generalizes Miyashita’s classical result and sheds light on other applications of geometric nature which cannot be treated using the classical theory. In order to illustrate our methods, we give a concrete application to the category of comodules over commutative (flat) Hopf algebroids. This obviously includes the special cases of split Hopf algebroids (action groupoids), which for instance cover the situation of the action of an affine algebraic group on an affine algebraic variety.

Symmetric algebras in categories of corepresentations and smash products

We investigate Frobenius algebras and symmetric algebras in the monoidal category of right comodules over a Hopf algebra H; for the symmetric property H is assumed to be cosovereign. If H is finite dimensional and A is an H-comodule algebra, we uncover the connection between A and the smash product A#H⁎ with respect to the Frobenius and symmetric properties.

The Exchange Property in Grothendieck Categories: Applications

We introduce the concept of an object with the (finite) exchange property in an arbitrary Grothendieck category, and we present the basic properties of such an object. Applications are given for categories of graded modules and for categories of comodules over a coalgebra. Among other results, it is proved that an arbitrary coalgebra 𝒞 over a field has the finite exchange property.

On the Drinfeld Center of the Category of Comodules over a Co-quasitriangular Hopf Algebra

Let $H$ be a co-quasitriangular Hopf algebra with bijective antipode. We prove that the Drinfeld center of the category of $H$-comodules is equivalent to the category of modules over some braided group. In particular, the equivalence holds not only for a finite dimensional $H$, but also for an infinite dimensional one.

Yetter-Drinfeld modules over weak multiplier bialgebras

We continue the study of the representation theory of a regular weak multiplier bialgebra with full comultiplication, started in [4, 2]. Yetter-Drinfeld modules are defined as modules and comodules, with compatibility conditions that are equivalent to a canonical object being (weakly) central in the category of modules, and equivalent also to another canonical object being (weakly) central in the category of comodules. Yetter-Drinfeld modules are shown to constitute a monoidal category via the (co)module tensor product over the base (co)algebra. Finite-dimensional Yetter-Drinfeld modules over a regular weak multiplier Hopf algebra with full comultiplication are shown to possess duals in this monoidal category.

Rickart and Dual Rickart Objects in Abelian Categories

We introduce and study relative Rickart objects and dual relative Rickart objects in abelian categories. We show how our theory may be employed in order to study relative regular objects and (dual) relative Baer objects in abelian categories. We also give applications to module and comodule categories.

Relative Yetter-Drinfeld modules and comodules over braided groups

Let H1 be a quantum group and f : H1⟶H2 a Hopf algebra homomorphism. Assume that B is some braided group obtained by Majid’s transmutation process. We first show that there is a tensor equivalence between the category of comodules over the braided group B and that of relative Yetter-Drinfeld modules. Next, we prove that the Drinfeld centers of the two categories mentioned above are equivalent to the category of modules over some quantum double, namely, the category of ordinary Yetter-Drinfeld modules over some Radford’s biproduct Hopf algebra. Importantly, the above results not only hold for a finite dimensional quantum group but also for an infinite dimensional one.

Braided autoequivalences and quantum commutative bi-Galois objects

Let (H,R) be a quasitriangular weak Hopf algebra over a field k. We show that there is a braided monoidal isomorphism between the Yetter–Drinfeld module category YDHH over H and the category of comodules over some braided Hopf algebra HR in the category MH. Based on this isomorphism, we prove that every braided bi-Galois object A over the braided Hopf algebra HR defines a braided autoequivalence of the category YDHH if and only if A is quantum commutative. In case H is semisimple over an algebraically closed field, i.e. the fusion case, then every braided autoequivalence of YDHH trivializable on MH is determined by such a quantum commutative Galois object. The quantum commutative Galois objects in MH form a group measuring the Brauer group of (H,R) as studied in [21] in the Hopf algebra case.

Multiplier bialgebras in braided monoidal categories

Multiplier bimonoids (or bialgebras) in arbitrary braided monoidal categories are defined. They are shown to possess monoidal categories of comodules and modules. These facts are explained by the structures carried by their induced functors.

Semidualizing and Tilting Adjoint Pairs, Applications to Comodules

The aim of this paper is to introduce the concept of right and left semidualizing adjoint pair of functors and study its main properties. This concept generalizes the concept of semidualizing module and allows one to consider semidualizing comodules, graded modules, etc. We also study tilting adjoint pair of functors as a particular case. We show generalized tilting theorem in this general setting and give some applications to tilting theory in the category of comodules over a coalgebra.