对偶Gabriel定理及其应用
Dual Gabriel Theorem with applications
摘要：
引入余半单余代数上双余代数的可积性。将其应用于任意余代数$C$的余根$C_0$，给出了$C$的Gabriel算子的另一种定义，并证明了它与已知的$C$的$\operatorname {Ext}$算子和$C$的链算子是一致的.得到了具有可分余根的余代数的对偶Gabriel定理，推广了点余代数的相应结果。我们还给出了余代数$C$的$C_1$的一个新的描述，它可以看作是著名的点余代数的Taft-Wilson定理第一部分的推广.作为应用，我们给出了局部有限余代数的Gabriel箭图的一个刻画，以及拟余Frobenius余代数的Gabriel箭图的一个性质.
英文摘要：
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.
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