Dual Gabriel theorem with applications

We introduce the quiver of a bicomodule over a cosemisimple coalgebra. Applying this to the coradical C0 of an arbitrary coalgebra C, we give an alternative definition of the Gabriel quiver of C, and then show that it coincides with the known Ext quiver of C and the link quiver of C. The dual Gabriel theorem for a coalgebra with a separable coradical is obtained, which generalizes the corresponding result for a pointed coalgebra. We also give a new description of C1 = C0 ∧C C0 of any coalgebra C, which can be regarded as a generalization of the first part of the well-known Taft-Wilson Theorem for pointed coal-gebras. As applications, we give a characterization of locally finite coalgebras via their Gabriel quivers, and a property of the Gabriel quiver of a quasi-coFrobenius coalgebra.

We introduce the quiver of a bicomodule over a cosemisimple coalgebra. Applying this to the coradical C0 of an arbitrary coalgebra C, we give an alternative definition of the Gabriel quiver of C, and then show that it coincides with the known Ext quiver of C and the link quiver of C. The dual Gabriel theorem for a coalgebra with a separable coradical is obtained, which generalizes the corresponding result for a pointed coalgebra. We also give a new description of C1 = C0 ∧X C0 of any coalgebra C, which can be regarded as a generalization of the first part of the well-known Taft-Wilson Theorem for pointed coalgebras. As applications, we give a characterization of locally finite coalgebras via their Gabriel quivers, and a property of the Gabriel quiver of a quasi-coFrobenius coalgebra.