Lemma 47.16.9. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring with normalized dualizing complex. Let $I \subset \mathfrak m$ be an ideal of finite length. Set $B = A/I$. Then there is a distinguished triangle

$\omega _ B^\bullet \to \omega _ A^\bullet \to \mathop{\mathrm{Hom}}\nolimits _ A(I, E)[0] \to \omega _ B^\bullet [1]$

in $D(A)$ where $E$ is an injective hull of $\kappa$ and $\omega _ B^\bullet$ is a normalized dualizing complex for $B$.

Proof. Use the short exact sequence $0 \to I \to A \to B \to 0$ and Lemmas 47.16.4 and 47.16.2. $\square$

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