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Abstract
Stenström introduced the notion of flat object in a locally finitely presented Grothendieck category 𝒜. 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 flat 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 flat comodules. Left perfect coalgebras are introduced and characterized in analogy with Bass's Theorem P. Coalgebras whose injective left C-comodules are flat are discussed and related to quasi-coFrobenius coalgebras.
ACKNOWLEDGMENTS
The author Juan Cuadra would like to thank Dr. S. Estrada for several enriching conversations about the Flat Cover Theorem.
This research was supported by Spanish grant MTM2005-03227 from MEC and FEDER and by Polish KBN Grant P03A 014 28.
Notes
Communicated by J. L. Gomez Pardo.