On modules $M$ with $\tau (M) \cong \nu \Omega ^{d+2}(M)$ for isolated singularities of Krull dimension $d$

Abstract:

A classical formula for the Auslander–Reiten translate $\tau$ says that $\tau (M)\cong \nu \Omega ^2(M)$ for every indecomposable module $M$ of a selfinjective Artin algebra. We generalise this by showing that for a $2d$-periodic isolated singularity $A$ of Krull dimension $d$, we have for the Auslander–Reiten translate of an indecomposable nonprojective Cohen-Macaulay $A$-module $M$, $\tau (M)\cong \nu \Omega ^{d+2}(M)$ if and only if $\operatorname {Ext}_A^{d+1}(M,A)=\operatorname {Ext}_A^{d+2}(M,A)=0$. We give several applications for Artin algebras.