Lemma 48.23.2. Let $X$ be a locally Noetherian scheme. If there exists a coherent sheaf $\omega _ X$ such that $\omega _ X[0]$ is a dualizing complex on $X$, then $X$ is a Cohen-Macaulay scheme.

**Proof.**
This follows immediately from Dualizing Complexes, Lemma 47.20.3 and our definitions.
$\square$

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