Colby-Fuller duality between coalgebras (Q1125914)

The object of this paper is the study of Morita dualities between categories of comodules over coalgebras. This is done with the help of the concept of Morita duality for Grothendieck categories introduced by \textit{R. R. Colby} and \textit{K. R. Fuller} [in J. Algebra 82, 546-558 (1983; Zbl 0524.16014)]. The authors characterize Colby-Fuller dualities between coalgebras by showing that there exists such a duality between coalgebras \(C\) and \(D\) if and only if \(C\) and \(D\) are right and left semiperfect and their categories of finite-dimensional comodules are (dual) equivalent. Moreover, they show that every Colby-Fuller duality between coalgebras is determined by a bicomodule which is a quasi-finite injective cogenerator for both categories of comodules. In the last section of the paper the authors show that if \(C\) and \(D\) are cocommutative coalgebras such that there exists either an equivalence between the categories of right comodules or a Colby-Fuller duality between the categories of right \(C\)-comodules and left \(D\)-comodules, then \(C\) and \(D\) are isomorphic.