Some Studies on QcF-coalgebras

Abstract

Let C be a coalgebra over a field k, {\scr M}\sp C theGrothendieck category of right C comodules, and C\sp * thedual algebra of C. For a left quasi co Frobenius coalgebra(shortly QcF coalgebra) the author gives some characterizationsin terms of left semi perfect coalgebras and right C comodules.Then he proves that for a left and right QcF coalgebra C and afinite dimensional object M\in {\scr M}\sp C, the space ofcomodule morphisms from C to M has dimension at most thedimension of M. This generalizes a result obtained by D. Stefan[Comm. Algebra 23 (1995), no. 5, 1657 1662; MR\Cite{Stefan95:uniqueness:1657--1662}[96f:16047]] for coFrobenius coalgebras. Finally it is proved that properties ofbeing almost connected and QcF are invariant under theequivalence of comodule categories.\par {For the entirecollection see MR\Cite{Birkenmeier01:International:Birkhauser}[2002c:16002].}