共查询到20条相似文献,搜索用时 31 毫秒
1.
Hiroshi Kihara 《Archiv der Mathematik》2016,106(6):507-513
The fundamental theorem on coalgebras asserts that coalgebras are locally finite in the case where the ground ring is a field. We prove the local finiteness theorem of corings under the semihereditarity condition on the base algebra and the projectivity condition on a coring. This result generalizes not only the fundamental theorem on coalgebras but also Hazewinkel’s result on the local finiteness of coalgebras over a principal ideal domain and Bergman’s unpublished result on the local finiteness of corings over a semisimple Artinian ring. 相似文献
2.
3.
We investigate functors between abelian categories having a left adjoint and a right adjoint that are similar (these functors are called quasi-Frobenius functors). We introduce the notion of a 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. 相似文献
4.
Mohssin Zarouali-Darkaoui 《代数通讯》2013,41(2):689-724
We investigate adjoint and Frobenius pairs 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, which leads to a reasonable notion of Frobenius coring extension. When applied to corings stemming from entwining structures, we obtain new results in this setting and in graded ring theory. 相似文献
5.
We introduce group corings, and study functors between categories of comodules over group corings, and the relationship to
graded modules over graded rings. Galois group corings are defined, and a Structure Theorem for the G-comodules over a Galois group coring is given. We study (graded) Morita contexts associated to a group coring. Our theory
is applied to group corings associated to a comodule algebra over a Hopf group coalgebra.
This research was supported by the research project G.0622.06 “Deformation quantization methods for algebras and categories
with applications to quantum mechanics” from Fonds Wetenschappelijk Onderzoek-Vlaanderen. The third author was partially supported
by the SRF (20060286006) and the FNS (10571026). 相似文献
6.
We propose a class of infinite comatrix corings, and describe them as colimits of systems of usual comatrix corings. The infinite
comatrix corings of El Kaoutit and Gómez Torrecillas are special cases of our construction, which in turn can be considered
as a special case of the comatrix corings introduced recently by Gómez Torrecillas and the third author.
相似文献
7.
Mohssin Zarouali-Darkaoui 《代数通讯》2013,41(10):3912-3943
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. 相似文献
8.
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-morphism of corings induces functors between the categories of comodules. These functors are termed pull-back and push-out functors, respectively, and thus relate categories of comodules of different corings. We study when the pull-back functor is fully faithful and when it is an equivalence. A generalised descent associated to a morphism of corings is introduced. We define a category of module-morphisms, and show that push-out functors are naturally isomorphic to each other if and only if the corresponding module-morphisms are mutually isomorphic. All these topics are studied within a unifying language of bicategories and the extensive use is made of interpretation of corings as comonads in the bicategory Bim of bimodules and module-morphisms as 1-cells in the associated bicategories of comonads in Bim. 相似文献
9.
Mohssin Zarouali-Darkaoui 《代数通讯》2013,41(12):4018-4031
We introduce and study the Picard group of a coring. We give an exact sequence relating the Picard group of a coring and its automorphisms generalizing the known exact sequences associated to an algebra and a coalgebra over a field. We also extend to corings the Aut–Pic property and we give some new corings satisfying this property. 相似文献
10.
L. El Kaoutit 《Mathematische Nachrichten》2009,282(5):726-747
We show that the bicategory whose 0‐cells are corings over rings with local units is bi‐equivalent to the bicategory of comonads in (right) unital modules whose underlying functors are right exact and preserve direct sums. A base ring extension of a coring by an adjunction is introduced as well (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
11.
S. Caenepeel E. De Groot J. Vercruysse 《Transactions of the American Mathematical Society》2007,359(1):185-226
El Kaoutit and Gómez-Torrecillas introduced comatrix corings, generalizing Sweedler's canonical coring, and proved a new version of the Faithfully Flat Descent Theorem. They also introduced Galois corings as corings isomorphic to a comatrix coring. In this paper, we further investigate this theory. We prove a new version of the Joyal-Tierney Descent Theorem, and generalize the Galois Coring Structure Theorem. We associate a Morita context to a coring with a fixed comodule, and relate it to Galois-type properties of the coring. An affineness criterion is proved in the situation where the coring is coseparable. Further properties of the Morita context are studied in the situation where the coring is (co)Frobenius.
12.
We investigate the Morita context and graded cases for weak group corings and derive some equivalent conditions for μ to be
surjective. Furthermore, we develop Galois theory for weak group corings. As an application, we give Galois theory for comodulelike
algebras over a weak Hopf group coalgebra. 相似文献
13.
We study co-Frobenius and more generally quasi-co-Frobenius corings over arbitrary base rings and over PF base rings in particular. We generalize some results about co-Frobenius and quasi-co-Frobenius coalgebras to the case of non-commutative base rings and give several new characterizations for co-Frobenius and more generally quasi-co-Frobenius corings, some of them are new even in the coalgebra situation. We construct Morita contexts to study Frobenius properties of corings and a second kind of Morita contexts to study adjoint pairs. Comparing both Morita contexts, we obtain our main result that characterizes quasi-co-Frobenius corings in terms of a pair of adjoint functors (F,G) such that (G,F) is locally quasi-adjoint in a sense defined in this note. 相似文献
14.
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. 相似文献
15.
16.
J. Gómez-Torrecillas 《Applied Categorical Structures》2006,14(5-6):579-598
We investigate which aspects of recent developments on Galois corings and comodules admit a formulation in terms of comonads.
The general theory is applied to the study of Galois comodules over corings over firm rings.
Supported by the research project “Algebraic Methods in Non Commutative Geometry,” with financial support of the grant MTM2004-01406
from the DGICYT and FEDER. 相似文献
17.
We construct comatrix corings on bimodules without finiteness conditions by using firm rings. This leads to the formulion
of a notion of Galois coring which plays a key role in the statement of a Noncommutative Faithfully Flat Descent for comodules
which generalizes previous versions. In particular, infinite comatrix corings fit in our general theory.
Presented by A. Verschoren. 相似文献
18.
On Comatrix Corings and Bimodules 总被引:5,自引:0,他引:5
19.
Hans-E. Porst 《Applied Categorical Structures》2008,16(1-2):223-238
We study the various categories of corings, coalgebras, and comodules from a categorical perspective. Emphasis is given to
the question which properties of these categories can be seen as instances of general categorical resp. algebraic results.
However, we also obtain new results concerning the existence of limits and of factorizations of morphisms.
相似文献
20.
本文研究了环上模范畴与余环上余模范畴.运用可裂叉与余可分余环的性质,得到了以上两个范畴等价的一些充分条件,从而推广了文献[6]中的一些结果.Abstract: In this article,we consider the categories of modules over rings and categories of comodules over corings.By properties of split forks and coseparable corings,we get some sufficient conditions for the equivalence between above two categories.As a consequence,we generalize some results in[6]. 相似文献