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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.

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.

I only know references for the case when $A$ is an Artin algebra or a local commutative Gorenstein ring.

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    $\begingroup$ There's a paper by Kameyama, Kimura and Nishida, "On stable equivalences of module subcategories over a semiperfect Noetherian ring" that seems relevant. I haven't looked closely at it, so I don't know whether it answers your question. $\endgroup$
    – Dave Benson
    Mar 8, 2023 at 21:09
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    $\begingroup$ @DaveBenson Thank you. It seems proposition 4.2 in that article gives the desired result. There is a reference to the book by Auslander Brider that I will check now. $\endgroup$
    – Mare
    Mar 9, 2023 at 9:18
  • $\begingroup$ In that case, I'll add it as an "answer". $\endgroup$
    – Dave Benson
    Mar 9, 2023 at 11:41

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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".

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