Lemma 47.16.8. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring with normalized dualizing complex $\omega _ A^\bullet $. If $\dim (A) = 0$, then $\omega _ A^\bullet \cong E[0]$ where $E$ is an injective hull of the residue field.

**Proof.**
Immediate from Lemma 47.16.4.
$\square$

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