Dual Gabriel Theorem with applications

Abstract

We introduce the quiver of a bicomodule over a cosemisimple coalgebra. Applying this to the coradical $C_0$ 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 $\operatorname {Ext}$ quiver of $C$ and the link quiver of $C$. The dual Gabriel theorem for a coalgebra with separable coradical is obtained, which generalizes the corresponding result for a pointed coalgebra. We also give a new description of $C_1$ 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.