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

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