Dual Gabriel theorem with applications

作     者：CHEN Xiaowu, HUANG Hualin & ZHANG Pu Department of Mathematics, University of Science and Technology of China, Hefei 230026, China USTC Shanghai Institute for Advanced Studies, Shanghai 201315, China Mathematical Section, the Abdus Salam ICTP, Strada Costiera 11, Trieste 34014, Italy Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200030, China 

作者机构：Department of Mathematics University of Science and Technology of China Hefei China USTC Shanghai Institute for Advanced Studies Shanghai China Mathematical Section the Abdus Salam ICTP Trieste Italy Department of Mathematics Shanghai Jiao Tong University Shanghai China 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2006年第49卷第1期

页      面：9-26页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：supported by the National Natural Science Foundation of China(Grant Nos.10271113&10301033) the Doctoral Foundation of the Chinese Education Ministry 

主　　题：quivers, cotensor coalgebra, quasi-coFrobenius coalgebra. 

摘      要：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.