Flat Comodules and Perfect Coalgebras |
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Authors: | Juan Cuadra Daniel Simson |
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Affiliation: | 1. Department of Algebra and Mathematical Analysis , University of Almería , Almería, Spain jcdiaz@ual.es;3. Faculty of Mathematics and Computer Science , Nicolaus Copernicus University , Toruń, Poland |
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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. |
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Keywords: | Flat comodule Flat cover Flat object Perfect coalgebra Quasi-coFrobenius coalgebra Semiperfect coalgebra |
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