On comatrix corings and bimodules

Comatrix Coring generalized and Equivalences of Categories of Comodules

We extend the comatrix coring to the case of a quasi-finite bicomodule. We also generalize some of its interesting properties. We study equivalences between categories of comodules over rather general corings. We particularize to the case of the adjoint pair of functors associated to a morphism of corings over different base rings. We apply our results to corings coming from entwining structures and graded structures, and we obtain new results in the setting of entwining structures and in the graded ring theory.

On quasi-Frobenius bimodules and corings

It is proved here that any quasi-Frobenius bimodule produces a quasi-Frobenius comatrix coring

Corings with decomposition and semiperfect corings

We give a characterization, in terms of Galois infinite comatrix corings, of the corings that decompose as a direct sum of left comodules which are finitely generated as left modules. Then we show that the associated rational functor is exact. This is the case of a right semiperfect coring which is locally projective and whose Galois comodule is a projective left unital module with superfluous radical.

Compatibility Condition between ring and coring

We introduce the notion of bi-monoid in general monoidal category generalizing by this the notion of bialgebra. In the case of bimodules over a noncommutative algebra, we obtain a compatibility condition between ring and coring whenever both structures admit the same underlying bimodule.

Quasi-Frobenius functors. Applications

We investigate functors between abelian categories having a left adjoint and a right adjoint that are \emph{similar} (these functors are called \emph{quasi-Frobenius functors}). We introduce the notion of a \emph{quasi-Frobenius bimodule} and give a characterization of these bimodules in terms of quasi-Frobenius functors. Some applications to corings and graded rings are presented. In particular, the concept of quasi-Frobenius homomorphism of corings is introduced. Finally, a version of the endomorphism ring Theorem for quasi-Frobenius extensions in terms of corings is obtained.

The bicategories of corings

To a B-coring and a (B,A)-bimodule that is finitely generated and projective as a right A-module an A-coring is associated. This new coring is termed a base ring extension of a coring by a module. We study how the properties of a bimodule such as separability and the Frobenius properties are reflected in the induced base ring extension coring. Any bimodule that is finitely generated and projective on one side, together with a map of corings over the same base ring, lead to the notion of a module-morphism, which extends the notion of a morphism of corings (over different base rings). A module-m…

Bimodule herds

The notion of a bimodule herd is introduced and studied. A bimodule herd consists of a $B$-$A$ bimodule, its formal dual, called a pen, and a map, called a shepherd, which satisfies untiality and coassociativity conditions. It is shown that every bimodule herd gives rise to a pair of corings and coactions. If, in addition, a bimodule herd is tame i.e. it is faithfully flat and a progenerator, then these corings are associated to entwining structures; the bimodule herd is a Galois comodule of these corings. The notion of a bicomodule coherd is introduced as a formal dualisation of the definitio…

Homotopical Morita theory for corings

A coring (A,C) consists of an algebra A and a coalgebra C in the monoidal category of A-bimodules. Corings and their comodules arise naturally in the study of Hopf-Galois extensions and descent theory, as well as in the study of Hopf algebroids. In this paper, we address the question of when two corings in a symmetric monoidal model category V are homotopically Morita equivalent, i.e., when their respective categories of comodules are Quillen equivalent. The category of comodules over the trivial coring (A,A) is isomorphic to the category of A-modules, so the question above englobes that of wh…

Bicategory of entwinings

We define a bicategory in which the 0-cells are the entwinings over variable rings. The 1-cells are triples of a bimodule and two maps of bimodules which satisfy an additional hexagon, two pentagons and two (co)unit triangles; and the 2-cells are the maps of bimodules satisfying two simple compatibilities. The operation of getting the "composed coring" from a given entwining, is promoted here to a canonical morphism of bicategories from a bicategory of entwinings to the Street's bicategory of corings.

The center of the category of bimodules and descent data for non-commutative 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 $A \ot A$ is braided monoidal. We provide several app…

A bicategorical version of Masuoka's theorem. Applications to bimodules over functor categories and to firm bimodules

We give a bicategorical version of the main result of A. Masuoka ({Corings and invertible bimodules,} {\em Tsukuba J. Math.} \textbf{13} (1989), 353--362) which proposes a non-commutative version of the fact that for a faithfully flat extension of commutative rings $R \subseteq S$, the relative Picard group $Pic(S/R)$ is isomorphic to the Amitsur 1--cohomology group $H^1(S/R,U)$ with coefficients in the units functor $U$.

有限生成$(H, {\rm C})$-余模的自同态环上的投射性与平坦性

Projectivity and flatness over the endomorphism ring of a finitely generated $(H, {\mathcal C})$-comodule

设k为交换环，H为k上双代数，C为(H,A)-余环，若C作为左A-模平坦，有限生成的右(H,C)-余模Λ对应的自同态环上的模具有投射性与平坦性的充要条件被给出；当C含固定H-群样元时，可将Λ替换为A。