This is a first comprehensive monograph on corings and comodules. Corings are generalizations of coalgebras: while a coalgebra is an R-module over a commutative associative unital ring R, equipped with an R-linear coproduct and a comultiplication, a coring is an (A,A)-bimodule over an associative, but not necessarily commutative, unital R-algebra A, equipped with a (A,A)-bilinear coproduct and comultiplication. Corings can also be viewed as coalgebras in a particular monoidal category (the ‘tensor category’). As observed by Takeuchi, important examples of corings are the so-called entwining structures connecting R-algebras with R-coalgebras. Moreover, in the particular case of bialgebra entwinings, entwined modules are exactly the classical Hopf modules. So - and this is the point of the book - the theory of corings and comodules is a natural common generalization of several theories connecting algebra with category theory, noncommutative geometry, and quantum physics. The book consists of six chapters and an Appendix. After developing coalgebra and comodule theory from the module-theoretic point of view in Chapter 1, the authors deal with bialgebras, classical Hopf algebras and their modules in Chapter 2. The module theoretic approach of Chapter 1 pays back in Chapter 3, where basics of the coring and comodule theory are developed. Chapter 4 deals with an important class of corings coming from ring extensions (together with the BOCS's of Rojter and Kleiner, these were the first examples of corings truly generalizing coalgebras). Chapters 5 and 6 deal with the relation to entwinings and weak entwinings. The Appendix recalls basics on the module category σ[M], which is one of the main tools for the theory. The reason is that any right comodule over a coring C is a left module over the left dual ring *C, and, for example, the so called ‘left α-condition’ just says that the category of all right C-comodules coincides with σ[*C C]. The book provides a unified and general treatment of a theory whose pieces were originally developed by people working in rather distinct areas of algebra, category theory, non-commutative geometry, and mathematical physics. The book is a welcome addition to the literature on this young and rapidly developing subject.

Reviewer:

jtrl