Igusa--Todorov Algebras and the Auslander--Reiten Conjecture

We prove that every Igusa–Todorov Artin algebra satisfies the Auslander–Reiten conjecture. More precisely, let $V$ be an $n$-Igusa–Todorov witness, and let $t$ be the number of isomorphism classes of nonprojective indecomposable summands of $V$. If a finitely generated module $M$ satisfies ${\text{Ext}}_A^i(M,A)=0$ for every $i>0$ and ${\text{Ext}}_A^q(M,M)=0$ for $1\le q\le 2t+1$, then $M$ is projective. As applications, algebras of representation dimension at most three and algebras satisfying $J^{2m+1}=0$ for which $A\mathrm{/}{J}^m$ has finite representation type satisfy the Auslander–Reiten conjecture.