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

A classical formula for the Auslander-Reiten translate $\tau$ says that $\tau (M)\cong \nu {\mathrm{\Omega }}^2(M)$ 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 non-projective Cohen-Macaulay $A$-module $M$, $\tau (M)\cong \nu {\mathrm{\Omega }}^{d+2}(M)$ if and only if $Ex{t}_A^{d+1}(M,A)=Ex{t}_A^{d+2}(M,A)=0$. We give several applications for Artin algebras.