Corings and Comodules

by Tomasz Brzezinski , Robert Wisbauer

• ISBN13: 9780521539319
• ISBN10: 0521539315
• Format: Paperback
• Copyright: 2003-09-15
• Publisher: Cambridge University Press

Summary

Corings and comodules are fundamental algebraic structures, which can be thought of as both dualisations and generalisations of rings and modules. Introduced by Sweedler in 1975, only recently they have been shown to have far reaching applications ranging from the category theory including differential graded categories through classical and Hopf-type module theory to non-commutative geometry and mathematical physics. This is the first extensive treatment of the theory of corings and their comodules. In the first part, the module-theoretic aspects of coalgebras over commutative rings are described. Corings are then defined as coalgebras over non-commutative rings. Topics covered include module-theoretic aspects of corings, such as the relation of comodules to special subcategories of the category of modules (sigma-type categories), connections between corings and extensions of rings, properties of new examples of corings associated to entwining structures, generalisations of bialgebras such as bialgebroids and weak bialgebras, and the appearance of corings in non-commutative geometry.

Table of Contents

Preface	vii
Notations	xi
1 Coalgebras and comodules	1	(128)
1 Coalgebras	1	(7)
2 Coalgebra morphisms	8	(12)
3 Comodules	20	(21)
4 C-comodules and C*-modules	41	(14)
5 The finite dual of an algebra	55	(7)
6 Annihilators and bilinear forms	62	(4)
7 The rational functor	66	(9)
8 Structure of comodules	75	(5)
9 Coalgebras over QF rings	80	(13)
10 Cotensor product of comodules	93	(8)
11 Bicomodules	101	(9)
12 Functors between comodule categories	110	(19)
2 Bialgebras and Hopf algebras	129	(40)
13 Bialgebras	129	(5)
14 Hopf modules	134	(10)
15 Hopf algebras	144	(14)
16 Trace ideal and integrals for Hopf algebras	158	(11)
3 Corings and comodules	169	(82)
17 Corings and their morphisms	170	(10)
18 Comodules over corings	180	(15)
19 C-comodules and C*modules	195	(22)
20 The rational functor for corings 212
21 Cotensor producover corings	217	(6)
22 Bicomodules over corings	223	(7)
23 Functors between comodule categories	230	(21)
24 The category of corings 240
4 Corings and extensions of rings	251	(72)
25 Canonical corings and ring extensions	251	(5)
26 Coseparable and cosplit corings	256	(8)
27 Frobenius extensions and corings	264	(12)
28 Corings with a grouplike element	276	(12)
29 Amitsur complex and connections	288	(11)
30 Cartier and Hochschild cohomology	299	(9)
31 Bialgebroids	308	(15)
5 Corings and entwining structures	323	(34)
32 Entwining structures and corings	323	(12)
33 Entwinings and Hopf-type modules	335	(7)
34 Entwinings and Galois-type extensions	342	(15)
6 Weak corings and entwinings	357	(38)
35 Weak corings	357	(9)
36 Weak bialgebras	366	(16)
37 Weak entwining structures	382	(13)
Appendix	395	(62)
38 Categories and functors	395	(14)
39 Modules and Abchan categories	409	(7)
40 Algebras over commutative rings	416	(9)
41 The category σ [M]	425	(10)
42 Torsion-theoretic aspects	435	(10)
43 Cogenerating and generating conditions	445	(6)
44 Decompositions of σ [M]	451	(6)
Bibliography	457	(14)
Index	471