Galois theory for corings and cleft entwining structures
by
Stefaan Caenepeel
Vakgroep Wiskunde, Faculteit Toegepaste Wetenschappen, Vrije Universiteit Brussel, VUB, B-1050 Brussel

Corings were already introduced by Sweedler in 1975. In 1999, Takeuchi observed that the so-called entwined modules, recently introduced by Brzezinski and Majid, are nothing else then comodules over a certain comodules. Some other applications of corings became clear after 1999. For example, one can describe descent theory for categories of modules over rings in terms of corings: the category of descent data corresponding to a ring extension i: B --> A is isomorphic to the ``canonical'' coring C=A\otimes_B A. Another A-coring D together with a fixed grouplike element x is called a Galois coring if it is isomorphic to the canonical coring (x being the element corresponding to 1\otimes_B 1). As special cases, we recover classical Galois extensions, Hopf Galois extensions, and coalgebra Galois extensions. The computations in the general coring case turn out to be more natural and transparent then the corresponding ones in special cases. In the talk, we will survey recent results by Brzezinski, Wisbauer and the author on the general Galois theory of corings.

Date received: June 5, 2002