Lemma 47.16.10 (0A7T)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 47: Dualizing Complexes
Section 47.16: Dualizing complexes over local rings
Lemma 47.16.10 (cite)

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Lemma 47.16.10. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring with normalized dualizing complex $\omega _ A^\bullet$ . Let $f \in \mathfrak m$ be a nonzerodivisor. Set $B = A/(f)$ . Then there is a distinguished triangle

$\omega _ B^\bullet \to \omega _ A^\bullet \to \omega _ A^\bullet \to \omega _ B^\bullet [1]$

in $D(A)$ where $\omega _ B^\bullet$ is a normalized dualizing complex for $B$ .

Proof. Use the short exact sequence $0 \to A \to A \to B \to 0$ and Lemma 47.16.2. $\square$

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