Colby–Fuller Duality between Coalgebras

w x In 8 , K. Morita introduced a useful notion of duality between categories of modules, usually called ‘‘Morita duality.’’ He proved that every duality is given by contravariant hom functors defined by a bimodule which is an injective cogenerator for both categories of modules. On the other hand, the equivalences between categories of comodules over a coalgebra w x were characterized by M. Takeuchi 12 . In this paper, we study the dualities between categories of comodules. A notion of duality for general Grothendieck categories that seems to extend Morita duality satisfactorily w x was introduced by R. R. Colby and K. R. Fuller in 1 . It has been recently w x investigated by J. L. Gomez Pardo and P. A. Guil Asensio 3, 4, 6 . Section 1 is devoted to obtaining a complete characterization of Colby]Fuller dualities between coalgebras. A coalgebra C over a field k is right semiperfect if the category M C of right C-comodules has enough projectives. If C and D are coalgebras over a field k, then either C and D are left and right semiperfect or there is no Colby]Fuller duality between the category of right C-comodules and the category of left D-comodules Ž . Ž . Theorem 1.11 . This, together with Theorem 1.6 2 , shows that there is a