Lemma 47.20.2. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring with normalized dualizing complex $\omega _ A^\bullet $ and dualizing module $\omega _ A = H^{-\dim (A)}(\omega _ A^\bullet )$. The following are equivalent

$A$ is Cohen-Macaulay,

$\omega _ A^\bullet $ is concentrated in a single degree, and

$\omega _ A^\bullet = \omega _ A[\dim (A)]$.

In this case $\omega _ A$ is a maximal Cohen-Macaulay module.

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