Lemma 47.15.12 (0A7L)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 47: Dualizing Complexes
Section 47.15: Dualizing complexes
Lemma 47.15.12 (cite)

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Lemma 47.15.12. Let $A$ be a Noetherian ring. Let $\omega _ A^\bullet$ be a dualizing complex. Let $\mathfrak m \subset A$ be a maximal ideal and set $\kappa = A/\mathfrak m$ . Then $R\mathop{\mathrm{Hom}}\nolimits _ A(\kappa , \omega _ A^\bullet ) \cong \kappa [n]$ for some $n \in \mathbf{Z}$ .

Proof. This is true because $R\mathop{\mathrm{Hom}}\nolimits _ A(\kappa , \omega _ A^\bullet )$ is a dualizing complex over $\kappa$ (Lemma 47.15.9), because dualizing complexes over $\kappa$ are unique up to shifts (Lemma 47.15.5), and because $\kappa$ is a dualizing complex over $\kappa$ . $\square$

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