Lemma 47.20.2 (0AWS)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 47: Dualizing Complexes
Section 47.20: Cohen-Macaulay rings
Lemma 47.20.2 (cite)

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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.

Proof. Follows immediately from Lemma 47.16.7. $\square$

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