Lemma 47.20.3. Let $A$ be a Noetherian ring. If there exists a finite $A$-module $\omega _ A$ such that $\omega _ A[0]$ is a dualizing complex, then $A$ is Cohen-Macaulay.

**Proof.**
We may replace $A$ by the localization at a prime (Lemma 47.15.6 and Algebra, Definition 10.104.6). In this case the result follows immediately from Lemma 47.20.2.
$\square$

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