Lemma 47.17.1 (0A7X)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 47: Dualizing Complexes
Section 47.17: Dualizing complexes and dimension functions
Lemma 47.17.1 (cite)

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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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