Category Of Comodules Research Articles

Semicorings and Semicomodules

In this paper, we introduce and investigate semicorings over associative semirings and their categories of semicomodules. Our results generalize old and recent ones on corings over rings and their categories of comodules. The generalization is not straightforward and even subtle at several places due to the nature of the base category of commutative monoids which is neither Abelian (not even additive) nor homological, and has no nonzero injective objects. To overcome these and other difficulties, a combination of methods and techniques from categorical, homological and universal algebra is used including a new notion of exact sequences of semimodules over semirings.

Comodules over weak multiplier bialgebras

This is a sequel paper of [Weak multiplier bialgebras, Trans. Amer. Math. Soc., in press] in which we study the comodules over a regular weak multiplier bialgebra over a field, with a full comultiplication. Replacing the usual notion of coassociative coaction over a (weak) bialgebra, a comodule is defined via a pair of compatible linear maps. Both the total algebra and the base (co)algebra of a regular weak multiplier bialgebra with a full comultiplication are shown to carry comodule structures. Kahng and Van Daele's integrals [The Larson–Sweedler theorem for weak multiplier Hopf algebras, in preparation] are interpreted as comodule maps from the total to the base algebra. Generalizing the counitality of a comodule to the multiplier setting, we consider the particular class of so-called full comodules. They are shown to carry bi(co)module structures over the base (co)algebra and constitute a monoidal category via the (co)module tensor product over the base (co)algebra. If a regular weak multiplier bialgebra with a full comultiplication possesses an antipode, then finite-dimensional full comodules are shown to possess duals in the monoidal category of full comodules. Hopf modules are introduced over regular weak multiplier bialgebras with a full comultiplication. Whenever there is an antipode, the Fundamental Theorem of Hopf Modules is proven. It asserts that the category of Hopf modules is equivalent to the category of firm modules over the base algebra.

Frobenius algebras of corepresentations and group-graded vector spaces

We consider Frobenius algebras in the monoidal category of right comodules over a Hopf algebra H. If H is a group Hopf algebra, we study a more general Frobenius type property, uncover the structure of graded Frobenius algebras, and investigate graded symmetric algebras. The graded Frobenius concept is related to Frobenius functors.

Galois functors and generalised Hopf modules

As shown in a previous paper by the same authors, the theory of Galois functors provides a categorical framework for the characterisation of bimonads on any category as Hopf monads and also for the characterisation of opmonoidal monads on monoidal categories as right Hopf monads in the sense of Bruguieres and Virelizier. Hereby the central part is to describe conditions under which a comparison functor between the base category and the category of Hopf modules becomes an equivalence (Fundamental Theorem). For monoidal categories, Aguiar and Chase extended the setting by replacing the base category by a comodule category for some comonoid and considering a comparison functor to generalised Hopf modules. For duoidal categories, Bohm, Chen and Zhang investigated a comparison functor to the Hopf modules over a bimonoid induced by the two monoidal structures given in such categories. In both approaches fundamental theorems are proved and the purpose of this paper is to show that these can be derived from the theory of Galois functors.

Adjunctions of Hom and Tensor as Endofunctors of (Bi-)Module Categories Over Quasi-Hopf Algebras

For a Hopf algebra H over a commutative ring k and a left H-module V, the tensor functors − ⊗ k V and V ⊗ k − are known to be left adjoint to some kind of Hom-functors as endofunctors of H 𝕄. The units and counits of adjunctions, in this case, are formally trivial as in the classical case. In this paper, we generalize this Hom-tensor adjunction for (bi-)module categories over a quasi-Hopf algebra H and show that these (bi-)module categories are biclosed monoidal. However, the units and counits of adjunctions in these generalized cases are not as trivial as in the Hopf algebra case, and they should be modified in terms of the reassociator and the quasi-antipode. Also, if the H-module V is finitely generated and projective as a k-module, we will obtain a generalized form of adjunction between the tensor functors − ⊗V and − ⊗V* depending on the reassociator and quasi-antipode of H and describe a natural isomorphism between functors − ⊗V* and Hom k (V, −) explicitly. Furthermore, we consider the special case V = A being an H-module algebra. In this case, each tensor functor will be a monad and its corresponding right adjoint is a comonad. We describe isomorphisms between the (Eilenberg–Moore) module categories over these monads and the (Eilenberg–Moore) comodule categories over their corresponding comonads explicitly.

Dualizing Complexes of Noetherian Complete Algebras via Coalgebras

Let A be a noetherian complete basic semiperfect algebra over an algebraically closed field, and C = A°be its dual coalgebra. If A is Artin–Schelter regular, then the local cohomology of A is isomorphic to a shift of twisted bimodule 1 C σ* with σ a coalgebra automorphism. This yields that the balanced dualinzing complex of A is a shift of the twisted bimodule σ* A 1. If σ is an inner automorphism, then A is Calabi–Yau. An appendix is included to prove a duality theorem of the bounded derived category of quasi-finite comodules over an artinian coalgebra.

The homotopy theory of coalgebras over a comonad

Let 𝕂 be a comonad on a model category M. We provide conditions under which the associated category M𝕂 of 𝕂-coalgebras admits a model category structure such that the forgetful functor M𝕂→M creates both cofibrations and weak equivalences. We provide concrete examples that satisfy our conditions and are relevant in descent theory and in the theory of Hopf–Galois extensions. These examples are specific instances of the following categories of comodules over a coring (co-ring). For any semihereditary commutative ring R, let A be a dg R-algebra that is homologically simply connected. Let V be an A-coring that is semifree as a left A-module on a degreewise R-free, homologically simply connected graded module of finite type. We show that there is a model category structure on the category MA of right A-modules satisfying the conditions of our existence theorem with respect to the comonad−⊗AV and conclude that the category MAV of V-comodules in MA admits a model category structure of the desired type. Finally, under extra conditions on R, A and V, we describe fibrant replacements in MAV in terms of a generalized cobar construction.

Valued Gabriel quiver of a wedge product and semiprime coalgebras

We describe the valued Gabriel quiver of a wedge product of coalgebras and study the category of comodules of a semiprime coalgebra. In particular, we prove that any monomial semiprime k-tame fc-tame coalgebra is string. We also prove a version of Eisenbud-Griffith theorem for coalgebras, namely, any hereditary semiprime strictly quasi-finite coalgebra is serial.

Cleft Comodule Categories

We define crossed product categories and we show that they are equivalent with cleft comodule categories. We also prove that a comodule category is cleft if and only if it is Hopf–Galois and has a normal basis. As an application we show that the category of Hopf modules over a cleft linear category and the category of modules over the coinvariant subcategory are equivalent.

Weak Bialgebras of fractions

We construct the algebra of fractions of a Weak Bialgebra relative to a suitable denominator set of group-like elements that is almost central, a condition we introduce in the present article which is sufficient in order to guarantee existence of the algebra of fractions and to render it a Weak Bialgebra. The monoid of all group-like elements of a coquasi-triangular Weak Bialgebra, for example, forms a suitable set of denominators as does any monoid of central group-like elements of an arbitrary Weak Bialgebra. We use this technique in order to construct new Weak Bialgebras whose categories of finite-dimensional comodules relate to SL2-fusion categories in the same way as GL(2) relates to SL(2).

Hochschild homology of Hopf algebras and free Yetter–Drinfeld resolutions of the counit

AbstractWe show that if$A$and$H$are Hopf algebras that have equivalent tensor categories of comodules, then one can transport what we call a free Yetter–Drinfeld resolution of the counit of$A$to the same kind of resolution for the counit of$H$, exhibiting in this way strong links between the Hochschild homologies of$A$and$H$. This enables us to obtain a finite free resolution of the counit of$\mathcal {B}(E)$, the Hopf algebra of the bilinear form associated with an invertible matrix$E$, generalizing an earlier construction of Collins, Härtel and Thom in the orthogonal case$E=I_n$. It follows that$\mathcal {B}(E)$is smooth of dimension 3 and satisfies Poincaré duality. Combining this with results of Vergnioux, it also follows that when$E$is an antisymmetric matrix, the$L^2$-Betti numbers of the associated discrete quantum group all vanish. We also use our resolution to compute the bialgebra cohomology of$\mathcal {B}(E)$in the cosemisimple case.

Invariant rings through categories

We formulate a notion of “geometric reductivity” in an abstract categorical setting which we refer to as adequacy. The main theorem states that the adequacy condition implies that the ring of invariants is finitely generated. This result applies to the category of modules over a bialgebra, the category of comodules over a bialgebra, and the category of quasi-coherent sheaves on an algebraic stack of finite type over an affine base.

De-Equivariantization of Hopf Algebras

We study the de-equivariantization of a Hopf algebra by an affine group scheme and we apply Tannakian techniques in order to realize it as the tensor category of comodules over a coquasi-bialgebra. As an application we construct a family of coquasi-Hopf algebras A(H, G, Φ) attached to a coradically-graded pointed Hopf algebra H and some extra data.

THE CENTER OF THE CATEGORY OF BIMODULES AND DESCENT DATA FOR NONCOMMUTATIVE RINGS

Let A be an algebra over a commutative ring k. We compute the center of the category of A-bimodules. There are six isomorphic descriptions: the center equals the weak center, and can be described as categories of noncommutative descent data, comodules over the Sweedler canonical A-coring, Yetter–Drinfeld type modules or modules with a flat connection from noncommutative differential geometry. All six isomorphic categories are braided monoidal categories: in particular, the category of comodules over the Sweedler canonical A-coring A ⊗ A is braided monoidal. We provide several applications: for instance, if A is finitely generated projective over k then the category of left End k(A)-modules is braided monoidal and we give an explicit description of the braiding in terms of the finite dual basis of A. As another application, new families of solutions for the quantum Yang–Baxter equation are constructed: they are canonical maps Ω associated to any right comodule over the Sweedler canonical coring A ⊗ A and satisfy the condition Ω3 = Ω. Explicit examples are provided.

CATEGORIES OF COMODULES AND CHAIN COMPLEXES OF MODULES

Let [Formula: see text] denote the coendomorphism left R-bialgebroid associated to a left finitely generated and projective extension of rings R → A with identities. We show that the category of left comodules over an epimorphic image of [Formula: see text] is equivalent to the category of chain complexes of left R-modules. This equivalence is monoidal whenever R is commutative and A is an R-algebra. This is a generalization, using entirely new tools, of results by Pareigis and Tambara for chain complexes of vector spaces over fields. Our approach relies heavily on the noncommutative theory of Tannaka reconstruction, and the generalized faithfully flat descent for small additive categories, or rings with enough orthogonal idempotents.

Hom-Tensor Relations for Two-Sided Hopf Modules Over Quasi-Hopf Algebras

For a Hopf algebra H over a commutative ring k, the category of right Hopf modules is equivalent to the category 𝕄 k of k-modules, that is, the comparison functor is an equivalence (Fundamental theorem of Hopf modules). This was proved by Larson and Sweedler via the notion of coinvariants M coH for any . The coinvariants functor is right adjoint to the comparison functor and can be understood as the Hom-functor (without referring to an antipode). For a quasi-Hopf algebra H, the category of quasi-Hopf H-bimodules has been introduced by Hausser and Nill and coinvariants are defined to show that the functor is an equivalence. It is the purpose of this article to show that the related coinvariants functor, right adjoint to the comparison functor, can be seen as the functor More generally, let H be a quasi-bialgebra and 𝒜 an H-comodule algebra (as introduced by Hausser and Nill). Then − ⊗ k H is a comonad on the category 𝒜𝕄 H of (𝒜, H)-bimodules and defines the Eilenberg-Moore comodule category (𝒜𝕄 H )−⊗H , which is just the category of two-sided Hopf modules. Following ideas of Hausser, Nill, Bulacu, Caenepeel, and others, two types of coinvariants are defined to describe right adjoints of the comparison functor and to establish an equivalence between the categories 𝒜𝕄 and provided H has a quasi-antipode. As our main results we show that these coinvariants functors are isomorphic to the functor and give explicit formulas for these isomorphisms.

Skew-monoidal categories and bialgebroids

Skew-monoidal categories arise when the associator and the left and right units of a monoidal category are, in a specific way, not invertible. We prove that the closed skew-monoidal structures on the category of right R-modules are precisely the right bialgebroids over the ring R. These skew-monoidal structures induce quotient skew-monoidal structures on the category of R–R-bimodules and this leads to the following generalization: Opmonoidal monads on a monoidal category correspond to skew-monoidal structures with the same unit object which are compatible with the ordinary monoidal structure by means of a natural distributive law. Pursuing a Theorem of Day and Street we also discuss monoidal lax comonads to describe the comodule categories of bialgebroids beyond the flat case.

A quantum double construction in Rel

We study bialgebras and Hopf algebras in the compact closed categoryRelof sets and binary relations. Various monoidal categories with extra structure arise as the categories of (co)modules of bialgebras and Hopf algebras inRel. In particular, for any groupG, we derive a ribbon category of crossedG-sets as the category of modules of a Hopf algebra inRelthat is obtained by the quantum double construction. This category of crossedG-sets serves as a model of the braided variant of propositional linear logic.

Representable Functors for Corings

We address four problems regarding representable functors and give answers to them for functors connecting the category of comodules over a coring to the category of modules over a ring. A functor's property of being Frobenius is restated as a particular case of its representability by imposing the predefinition of the object of representability. Let R, S be two rings, C an R-coring and the category of left C-comodules. The category of all representable functors is shown to be equivalent to the opposite of the category . For U an (S, R)-bimodule we give necessary and sufficient conditions for the induction functor to be: a representable functor, an equivalence of categories, a separable or a Frobenius functor. The latter results generalize and unify the classical theorems of Morita for categories of modules over rings and the more recent theorems obtained by Brezinski, Caenepeel et al. for categories of comodules over corings.

Weak Bialgebras and Monoidal Categories

We study monoidal structures on the category of (co)modules over a weak bialgebra. Results due to Nill and Szlachányi are unified and extended to infinite algebras. We discuss the coalgebra structure on the source and target space of a weak bialgebra.