Lemma 48.20.7. Let $(S, \omega _ S^\bullet )$ be as in Situation 48.20.1. Let $j : X \to Y$ be an open immersion of schemes of finite type over $S$. Let $\omega _ X^\bullet $ and $\omega _ Y^\bullet $ be dualizing complexes normalized relative to $\omega _ S^\bullet $. Then there is a canonical isomorphism $\omega _ X^\bullet = \omega _ Y^\bullet |_ X$.

**Proof.**
Immediate from the construction of normalized dualizing complexes given just above Definition 48.20.5.
$\square$

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