The Auslander-Reiten conjecture for algebras with radical cube zero

Abstract

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.