发表机构
Jiangsu Normal University; Changsha University of Science and Technology（江苏师范大学; 长沙理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明根立方为零的有限维代数满足Auslander-Reiten猜想，即每个自正交生成元为投射模，通过构造非投射模在1到3s+1度间的非零自扩张实现。
AI 中文摘要
设$A$是域上的可裂有限维代数，其根$J$满足$J^3=0$，并设$s$为简单$A$-模的同构类数。我们证明，一个非投射模$M$若对所有$i>0$都有${\ m Ext}_A^i(M,A)=0$，则在$1$到$3s+1$之间的某个次数上具有非零自扩张。特别地，$A$满足Auslander-Reiten猜想，该猜想断言每个自正交生成元都是投射的。作为推论，代数闭域上每个根立方为零的有限维代数都满足Auslander-Reiten猜想。
英文摘要
Let $A$ be a split finite-dimensional algebra over a field whose radical $J$ satisfies $J^3=0$, and let $s$ be the number of isomorphism classes of simple $A$-modules. We prove that a non-projective module $M$ with ${\rm Ext}_A^i(M,A)=0$ for all $i>0$ has a non-zero self-extension in some degree between $1$ and $3s+1$. In particular, $A$ satisfies the Auslander--Reiten conjecture, which asserts that every self-orthogonal generator is projective. As a consequence, every finite-dimensional algebra over an algebraically closed field with radical cube zero satisfies the Auslander-Reiten conjecture.
Comments9 pages
