Corings with decomposition, corings with exact rational functor and Semiperfect corings


Il Prof. Laiachi El Kaoutit dell'Universita' di Granada terra' il giorno venerdi' 26 ottobre 2007 in aula 2BC/30 alle ore 16:00 la conferenza "Corings with decomposition, corings with exact rational functor and Semiperfect corings".
(Joint works with J. Gómez-Torrecillas.)

The notion of corings is in some sense a dual notion of ring. This are bimodules endowed with two compatible external operations, i.e. comultiplication and counit. For instance, if A ! B is any rings extension, then the
B-bimodule B -A B admits a structure of a B-coring. This kind of corings, refereed as Sweedler's canonical corings, were used by M. Sweedler to prove the Jacobson-Bourbaki-Hochschild rst Theorem concerning Galois
theory for division rings. A. V. Roiter introduced the notion of BOCES's in the context of representation theory, in order to reformulate a Matrix Problem. It turns out that, in the small case (i.e. the base category is an additive small category), the BOCES's are nothing but corings over rings with enough orthogonal idempotents. Another framework
where the notion of coring is powerful used, is the context of Brown-Peterson-Theory of homotopy, as was explained in the green book of D. C. Ravenel. In the last years, this
notion was vastly appeared in the literature, since M. Takeuchi observed that the category of entwined modules due to T. Brzezi«ski and S. Majid, is in fact a category of comodules over an adequate coring. Relative modules, like
as Group-graded modules, Hopf modules and Yetter-Drinfeld modules, are entwined modules, and so comodules over corings. In this talk, before recalling some basic denitions, we present an useful characterization of the following classes of corings:
(I) Coring with exact rational functor.
(II) (Right) Semiperfect coring.
(III) Coring with decomposition as direct sum of left comodules which are nitely generated projective as left modules.

Rif. int. S. Bazzoni, R. Colpi, A. Facchini, A. Tonolo