Preface

Summary

Corings and comodules are fundamental algebraic structures that can be thought of as both dualisations and generalisations of rings and modules. Corings were introduced by Sweedler in 1975 as a generalisation of coalgebras and as a means of presenting a semi-dual version of the Jacobson-Bourbaki Theorem, but their origin can be traced back to 1968 in the work of Jonah on cohomology of coalgebras in monoidal categories. In the late seventies they resurfaced under the name of bimodules over a category with a coalgebra structure, BOCSs for short, in the work of Rojter and Kleiner on algorithms for matrix problems. For a long time, essentially only two types of examples of corings truly generalising coalgebras were known – one associated to a ring extension, the other to a matrix problem. The latter example was also studied in the context of differential graded algebras and categories. This lack of examples hindered the full appreciation of the fundamental role of corings in algebra and obviously hampered their progress in general coring theory.

On the other hand, from the late seventies and throughout the eighties and nineties, various types of Hopf modules were studied. Initially these were typically modules and comodules of a common bialgebra or a Hopf algebra with some compatibility condition, but this evolved to modules of an algebra and comodules of a coalgebra with a compatibility condition controlled by a bialgebra.