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Dual Gabriel theorem with applications

2006; Springer Nature; Volume: 49; Issue: 1 Linguagem: Inglês

10.1007/s11425-004-5235-4

2095-0535

Xiaowu Chen, Hua-Lin Huang, Pu Zhang,

Homotopy and Cohomology in Algebraic Topology

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 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 C 1 = C 0 ∧C C 0 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.