Approach to Artinian algebras via natural quivers

Li, Fang; Lin, Zongzhu

doi:10.1090/S0002-9947-2011-05410-3

Approach to Artinian algebras via natural quivers
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by Fang Li and Zongzhu Lin PDF

Trans. Amer. Math. Soc. 364 (2012), 1395-1411 Request permission

Abstract:

Given an Artinian algebra $A$ over a field $k$, there are several combinatorial objects associated to $A$. They are the diagram $D_A$ as defined by Drozd and Kirichenko, the natural quiver $\Delta _A$ defined by Li (cf. Section 2), and a generalized version of $k$-species $(A/r, r/r^2)$ with $r$ being the Jacobson radical of $A$. When $A$ is splitting over the field $k$, the diagram $D_A$ and the well-known Ext-quiver $\Gamma _A$ are the same. The main objective of this paper is to investigate the relations among these combinatorial objects and in turn to use these relations to give a characterization of the algebra $A$.

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