Flat comodules and perfect coalgebras

Stenstrom introduced the notion of flat object in a locally finitely presented Grothendieck category Alpha. In this article we investigate this notion in the particular case of the category A = C-Comod of left C-comodules, where C is a coalgebra over a field K. Several characterizations of flat left C-comodules are given and coalgebras having enough. at left C-comodules are studied. It is shown how far these coalgebras are from being left semiperfect. As a consequence, we give new characterizations of a left semiperfect coalgebra in terms of. at comodules. Left perfect coalgebras are introduced and characterized in analogy with Bass's Theorem P. Coalgebras whose injective left C-comodules are. at are discussed and related to quasi-coFrobenius coalgebras.