Let $A$ be a a two-sided noetherian semiperfect ring and assume that the injective dimension of the left and right regular modules are equal to $n \geq 1$.
Let $\Omega^n(mod A)$ be the category of $n$-th syzygy modules consisting of modules that are projective or a direct summand of a module of the form $\Omega^n(M)$ for some $M$.
For $A$ as above those are the maximal Cohen-Macaulay $A$-modules $CM A$. Let $\underline{CM} A$ denote the stable category.



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> Question: Is there a good reference that the functor $\Omega^1: \underline{CM} A \rightarrow \underline{CM} A$ has the property that $\Omega^1(X) \cong \Omega^1(Y)$ implies $X \cong Y$? $\Omega^1$ should even be an equivalence.
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I only know references for the case when $A$ is an Artin algebra or a local commutative Gorenstein ring.



$\Omega$ for noetherian semiperfect rings
This seems to be answered in Proposition 4.2 of Kameyama, Kimura and Nishida, "On stable equivalences of module subcategories over a semiperfect Noetherian ring".


