Category Of Comodules Research Articles

Triangular matrix coalgebras: Representation theory and recollement

For any triangular matrix coalgebra [Formula: see text], in this paper, we first examine some connections between coalgebra properties of [Formula: see text] and its constituent coalgebras C, D, which contain semiperfectness, computability and row/column-finiteness of their left Cartan matices. Then we devote to considering the coresolution dimensions of recollement of comodule categories by investigating covariantly finite subcategories.

Koszul duality for compactly generated derived categories of second kind

For any dg algebra A we construct a closed model category structure on dg A -modules such that the corresponding homotopy category is compactly generated by dg A -modules that are finitely generated and free over A (disregarding the differential). We prove that this closed model category is Quillen equivalent to the category of comodules over a certain, possibly nonconilpotent dg coalgebra, a so-called extended bar construction of A . This generalises and complements certain aspects of dg Koszul duality for associative algebras.

D4-objects in abelian categories: Transfer via functors

We study the transfer of D4 and dually C4 objects via functors between abelian categories. We also give applications to Grothendieck categories, comodule categories, (graded) module categories.

Transfer of CS-Rickart and dual CS-Rickart properties via functors between Abelian categories

We study the transfer of (dual) relative CS-Rickart properties via functors between abelian categories. We consider fully faithful functors as well as adjoint pairs of functors. We give several applications to Grothendieck categories and, in particular, to (graded) module and comodule categories.

F-Baer objects with respect to a fully invariant short exact sequence in abelian categories

We introduce and study (dual) relative F-Baer objects as specializations of (dual) relative split objects with respect to a fully invariant short exact sequence in (AB3) abelian categories. We analyze their relationship with (dual) relative Baer objects, and we study direct summands and direct sums of (dual) relative F-Baer objects. We give applications to module and comodule categories.

On the E-infinity Algebras

In this paper we study an elementary use of E-infinity modules and E-infinity algebras as together they have a use in terms of describing triangulated categories. Also, we show an interpretation of E-infinity algebras where the modules are fibrant objects within the categories of differential graded co-algebras and co-modules.

English

Tilting theory plays an important role in the representation theory of coalgebras. This paper seeks how to apply the theory of localization and colocalization to tilting torsion theory in the category of comodules. In order to better understand the process, we give the (co)localization for morphisms, (pre)covers and special precovers. For that reason, we investigate the (co)localization in tilting torsion theory for coalgebras.

Gluing Hilbert C⁎-modules over the primitive ideal space

We show that the gluing construction for Hilbert modules introduced by Raeburn in his computation of the Picard group of a continuous-trace C⁎-algebra (1981) [14] can be applied to arbitrary C⁎-algebras, via an algebraic argument with the Haagerup tensor product. We put this result into the context of descent theory by identifying categories of gluing data for Hilbert modules over C⁎-algebras with categories of comodules over C⁎-coalgebras, giving a Hilbert-module version of a standard construction from algebraic geometry. As a consequence we show that if two C⁎-algebras have the same primitive ideal space T, and are Morita equivalent up to a 2-cocycle on T, then their Picard groups relative to T are isomorphic.

An unstable change of rings for Morava E–theory

The Bousfield-Kan (or unstable Adams) spectral sequence can be constructed for various homology theories such as Brown-Peterson homology theory BP, Johnson-Wilson theory $E(n)$, or Morava $E$-theory $E_n$. For nice spaces the $E_2$-term is given by Ext in a category of unstable comodules. We establish an unstable Morava change of rings isomorphism between $\text{Ext}_{\mathcal{U}_{\Gamma_B}}(B,M)$ and $\text{Ext}_{\mathcal{U}_{E_{n*}E_n}/I_{n}}(E_{n*}/I_{n},E_{n*}\otimes_{BP_*} M)$ where $(B,\Gamma_B)$ denotes the Hopf Algebroid $(v_n^{-1}BP_*/I_{n}, v_n^{-1}BP_*BP/I_{n} )$. We show that the latter groups can be interpreted as $\text{Ext}$ in the category of continuous modules over the profinite monoid of endomorphisms of the Honda formal group law. By comparing this with the cohomology of the Morava stabilizer group we obtain an unstable Morava vanishing theorem when $p-1 \nmid n$.

π-Rickart and dual π-Rickart objects in abelian categories

We introduce and study (strongly) [Formula: see text]-Rickart objects and their duals in abelian categories, which generalize (strongly) self-Rickart objects and their duals. We establish general properties of such objects, we analyze their behavior with respect to coproducts, and we study classes all of whose objects are (strongly) [Formula: see text]-Rickart. We derive consequences for module and comodule categories.

Tilting Torsion-Free Classes in the Category of Comodules

A well-known Assem–Smal $$\phi $$ theorem on tilting modules plays an important role in tilting theory. In this paper, we prove that a version of Assem–Smal $$\phi $$ theorem still holds in the category of comodules. Following this result, we characterize the tilting torsion-free classes in the category of comodules using the precover theory.

Geometrically partial actions

We introduce “geometric” partial comodules over coalgebras in monoidal categories as an alternative notion to the notion of partial action and coaction of a Hopf algebra introduced by Caenepeel and Janssen. The name is motivated by the fact that our new notion suits better if one wants to describe phenomena of partial actions in algebraic geometry. Under mild conditions, the category of geometric partial comodules is shown to be complete and cocomplete and the category of partial comodules over a Hopf algebra is lax monoidal. We develop a Hopf-Galois theory for geometric partial coactions to illustrate that our new notion might be a useful additional tool in Hopf algebra theory.

Baer-Kaplansky classes in categories: transfer via functors

We study the transfer of Baer-Kaplansky classes via additive functors between preadditive categories. We show that the Baer-Kaplansky property is well behaved with respect to fully faithful functors, but not with respect to their usual generalizations such as faithful, full, separable, naturally full, Maschke or dual Maschke functors. We also present the transfer of the Baer-Kaplansky property between subclasses of static and adstatic objects induced by adjoint pairs of functors. We mainly give applications to (graded) module categories and comodule categories.

Transfer of splitness with respect to a fully invariant short exact sequence in abelian categories

We study the transfer via functors between abelian categories of the (dual) relative splitness of objects with respect to a fully invariant short exact sequence. We mainly consider fully faithful functors and adjoint pairs of functors. We deduce applications to Grothendieck categories, (graded) module categories and comodule categories.

The Braided Monoidal Structure on the Category of Comodules of Bimonads

We investigate how the category of comodules of bimonads can be made into a monoidal category. It suffices that the monad and comonad in question are bimonads, with some extra compatibility relation. On a monoidal category of comodules of bimonads, we construct a braiding and get the necessary and sufficient conditions making it a braided monoidal category. As an application, we consider the category of comodules of corings and the category of entwined modules.

On the cohomology of comodules over smash coproducts

We consider the category of comodules over a smash coproduct coalgebra [Formula: see text]. We show that there is a Grothendieck spectral sequence connecting the derived functors of the Hom functors coming from [Formula: see text]-colinear, [Formula: see text]-colinear and rational [Formula: see text]-colinear morphisms. We give several applications and connect our results to existing spectral sequences in the literature.

Cohomology of Modules Over -categories and Co--categories

AbstractLet$H$be a Hopf algebra. We consider$H$-equivariant modules over a Hopf module category${\mathcal{C}}$as modules over the smash extension${\mathcal{C}}\#H$. We construct Grothendieck spectral sequences for the cohomologies as well as the$H$-locally finite cohomologies of these objects. We also introduce relative$({\mathcal{D}},H)$-Hopf modules over a Hopf comodule category${\mathcal{D}}$. These generalize relative$(A,H)$-Hopf modules over an$H$-comodule algebra$A$. We construct Grothendieck spectral sequences for their cohomologies by using their rational$\text{Hom}$objects and higher derived functors of coinvariants.

Co-double bosonisation and dual bases of cq[SL2] and cq[SL3

We find a dual version of a previous double-bosonisation theorem whereby each finite-dimensional braided-Hopf algebra B in the category of comodules of a coquasitriangular Hopf algebra A has an associated coquasitriangular Hopf algebra ▪. As an application we find new generators for cq[SL2] reduced at q a primitive odd root of unity with the remarkable property that their monomials are essentially a dual basis to the standard PBW basis of the reduced Drinfeld–Jimbo quantum enveloping algebra uq(sl2). Our methods apply in principle for general cq[G] as we demonstrate for cq[SL3] at certain odd roots of unity.

Strongly Rickart objects in abelian categories: Applications to strongly regular and strongly Baer objects

ABSTRACTWe show how the theory of (dual) strongly relative Rickart objects may be employed in order to study strongly relative regular objects and (dual) strongly relative Baer objects in abelian categories. For each of them, we prove general properties, we analyze the behavior with respect to (co)products, and we study the transfer via functors. We also give applications to Grothendieck categories, (graded) module categories and comodule categories.

Strongly Rickart objects in abelian categories

ABSTRACTWe introduce and study (dual) strongly relative Rickart objects in abelian categories. We prove general properties, we analyze the behaviour with respect to (co)products, and we study the transfer via functors. We also give applications to Grothendieck categories, (graded) module categories and comodule categories. Our theory of (dual) strongly relative Rickart objects may be employed in order to study strongly relative regular objects and (dual) strongly relative Baer objects in abelian categories.