Lemma 47.17.1. Let $A$ be a Noetherian ring. Let $\mathfrak p$ be a minimal prime of $A$. Then $H^ i(\omega _ A^\bullet )_\mathfrak p$ is nonzero for exactly one $i$.

**Proof.**
The complex $\omega _ A^\bullet \otimes _ A A_\mathfrak p$ is a dualizing complex for $A_\mathfrak p$ (Lemma 47.15.6). The dimension of $A_\mathfrak p$ is zero as $\mathfrak p$ is minimal. Hence the result follows from Lemma 47.16.8.
$\square$

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