Takeuchis observation has lead to a revived interest in the theory of corings, which goes back to sweedler 16. Acoring c, a an associative ring, is a subcategory of the category of left modules cm. Journal of algebra and its applications vol 09, no 01. Fundamental constructions for coalgebras, corings, and comodules. Vercruysse faculty of engineering vrije universiteit brussel, vub b1015 brussels, belgium email. For a long period the theory of modules over rings on the one hand and comodules and hopf modules for coalgebras and bialgebras on the other side developed quite independently. When you submit your registration, you can quickly join the corings. Fill up this form to submit for a founder application or be able to contribute to this startup. This canonical map allows to consider not necessarily. Emphasis is given to the question which properties of these categories can be seen as instances of general. Osofsky received june 25, 1987 introduction in this paper we introduce cointegrations from one bicomodule into another, a notion. It is well known that the category mc of right comodules over an. As coalgebra is an example of a coring, a summary of properties of comodules of a coring provides also the reader with the knowledge of properties of comodules of coalgebra.
Galois corings, descent theory, and a structure theorem for cosemisimple corings l. In this paper we introduce and investigate what turns out to be a suitable generalization of simple corings simple comodules, namely fully coprime corings fully coprime comodules. Our theory is applied to group corings associated to a comodule algebra over. The notion of a coring appeared first in the algebra literature in 1975 in sweedlers paper on a predual version of the jacobsonbourbaki correspondence. This book is intended to provide a reasonably selfcontained account of a major portion of the general theory of rings and modules suitable as a text for introductory and more advanced graduate courses. It is shown that the categories of rcoalgebras for a commutative unital ring r and the category of acorings for some ralgebra a as well as their respective.
The braided monoidal structure on the category of comodules. Algebra, pdf, 39 on the category of comodules over corings, in proceedings of the 3rd palestinian conference 2000, pdf. This page intentionally left blank london mathematical society lecture note series managing editor. We include the proof for completeness and reference in the proofs of parts 2 and 3. Apr 17, 2007 read fundamental constructions for coalgebras, corings, and comodules, applied categorical structures on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Nov 15, 2006 prime objects were defined as generalization of simple objects in the categories of rings modules. Moreover, we consider several primeness notions in the category of comodules of a given coring and investigate their relations with the fully coprimeness and the simplicity of these comodules. Corings and comodules tomasz brzezinski, robert wisbauer. Group corings introduction group coalgebras and hopf group. Pdf on the category of comodules over corings robert. Corings and comodules tomasz brzezinski university of wales swansea robert wisbauer heinrich heine universitat dusseldorf. A coring c, a an associative ring, is a subcategory of the category of left modules cm.
Comatrix corings were introduced in 16 to make out the structure of cosemisimple corings over an arbitrary. We introduce group corings, and study functors between categories of comodules over group corings, and the relationship to graded modules over graded rings. Prime objects were defined as generalization of simple objects in the categories of rings modules. Coalgebras and comodules are dualisations of algebras and modules. Miodrag c iovanov location mathematical sciences the university of iowa 14 maclean hall. We discuss when the rat functor associated to a coring satisfying the left. This leads to the formulion of a notion of galois coring which plays a key role in the statement of a noncommutative. As an application, we consider the category of comodules of corings and the category of entwined modules. S p c is an isomorphism of corings where s end c p. Pdf modules and comodules for corings robert wisbauer. Robert wisbauer department of mathematics, heinrich heine. We study the category of comodules over a semiperfect coring. We characterize semiperfect corings over artinian rings and over qfrings.
The classical notion of a prime ring was generalized, in di erent ways, to introduce prime objects in the category of modules of a given ring see wis1996, section. Fundamental constructions for coalgebras, corings, and comodules fundamental constructions for coalgebras, corings, and comodules porst, hanse. The phrase \provided imf is a pure submodule of c0 should be added. Introduction corings and their comodules were introduced by sweedler in 11. Fundamental constructions for coalgebras, corings, and. Request pdf comodules and corings a coring is one of the most basic algebraic structures dual to that of a ring but also a ring itself can be viewed as a coring. Our general approach is categorical rather than arithmetical. Comodules of this coring provide one with an equiv. Moreover, we consider several primeness notions in the category of comodules of a given coring and investigate. Osofsky received june 25, 1987 introduction in this paper we introduce cointegrations from one bicomodule into another, a notion extending that. Introduction the aim of this note is to generalize properties of semiperfect coalgebras over. If p 2 mc is a small projective generator, then, by the adjunction u a.
The 23 articles in this volume encompass the proceedings of the international conference on modules and comodules held in porto portugal in 2006 and dedicated to robert wisbauer on the occasion of his 65th birthday. A note on coinduction functors between categories of. Apr 14, 2007 read comatrix corings and galois comodules over firm rings, algebras and representation theory on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. On the category of comodules over corings heinrichheine. In section 2 we have collected basic properties of galois comodules over. Hopf algebras and their actions on rings download ebook.
We characterize the corings whose category of comodules has a generating set of small projective comodules in terms of the non commutative descent theory. Click download or read online button to get hopf algebras and their actions on rings book now. On galois comodules 2685 of comodules of this type. Journal of algebra 126, 211224 1989 cointegrations, relative cohomology for comodules, and coseparable corings fernando guzman department of mathematical sciences, state university of new york, binghamton, new york 901 communicated by barbara l. Buy corings and comodules london mathematical society lecture note series on free shipping on qualified orders. Takeuchis observation has caused a revival of the theory of corings. This chapter discusses the basic definitions and the basic properties of corings and comodules, focussing on recent developments.
In light of this observation, it is no surprise that attempts have been made to interpret the above. This formula looks like the corresponding formula for usual coalgebras. Request pdf on jan 1, 2003, tomasz brzezinski and others published corings and comodules find, read and cite all the research you need. This point of view allows one to apply results from module theory to the investigation of coalgebras and comodules. Introduction usc dana and david dornsife college of. One application is for the coproduct of comodules indexed by a set ifunctor. Comodules over coalgebras or corings are familiar to many algebraists. Galois corings from the descent theory point of view 3 where the summation is implicitely understood.
Comatrix corings and galois comodules over firm rings. Published by the press syndicate of the university of cambridge the pitt building, trumpington street, cambridge, united kingdom. Coalgebras, corings, comodules, limits, factorizations introduction the categories coalg r of rcoalgebras, for a commutative ring r, and comod a of acomodules for a given rcoalgebra a have attracted attention and quite a number of their categorical properties have been obtained, as well as some others labelled. Galois corings from the descent theory point of view. A main goal of this project is to introduce coprime comodules coprime corings, which generalize simple comodules simple cor. Hitchin, mathematical institute, university of oxford, 2429 st giles, oxford ox1 3lb, united kingdom.
The main purpose of this note is to show that mcis a full subcatgeory in cmif and only if c. 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. Miodrag c iovanov associate professor, mathematics phd. These articles reflect professor wisbauers wide interests and give an overview. The 2003 mono graph 21, which was supposed to contain state of the art on corings and comodules. A note on coinduction functors between categories of comodules for corings. Corings and their comodules were introduced by sweedler in 11. Many types of comodules of coalgebras considered in hopf algebra theory turn out to be the comodules of specific corings.
Cointegrations, relative cohomology for comodules, and. Caenepeel proceedings of the international conference on mathematics and its applications. We assume the famil iarity with rings usually acquired in standard undergraduate algebra courses. Takeuchi 31 remarked that en twined modules introduced in 11 can be viewed as examples of comodules over a coring. On the category of comodules over corings robert wisbauer abstract it is well known that the category mcof right comodules over an acoring c, aan associative ring, is a subcategory of the category of left modules cmover the dual ring c. Corings and comodules london mathematical society lecture. On coseparable and biseparable corings 3 then m is called a right ccomodule.
In fact, under weak restrictions on the amodule properties of c, the category of right ccomodules can be identified with the subcategory. Generalising the notion of galois corings, galois comodules were introduced as comodules p over an acoring c for which pa is finitely generated and projective and the evaluation map c. For a quick look at my cv, you can access this file cv. Corings and comodules tomasz brzezinski university of wales swansea robert wisbauer heinrich heine. It is well known that the categorymc of right comodules over an acoringc, a an associative ring, is a subcategory of the category of left modules cm over the dual ring c. In partic ular we specialise to corings which are built on a tensor product of. Wisbauer, corings and comodules, london mathematical society. It became clear that corings provide a unifying and simplifying framework to various topics, such as galois theory, descent theory, frobenius. For each situation of frobenius extension of corings we consider we give a. These notions are applied then to study primeness and coprimeness properties of a given coring, considered as an.
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