Lemma 42.59.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $i : X \to Y$ and $j : Y \to Z$ be regular immersions of schemes locally of finite type over $S$. Then $j \circ i$ is a regular immersion and $(j \circ i)^! = i^! \circ j^!$.

**Proof.**
The first statement is Divisors, Lemma 31.21.7. By Divisors, Lemma 31.21.6 there is a short exact sequence

\[ 0 \to i^*(\mathcal{C}_{Y/Z}) \to \mathcal{C}_{X/Z} \to \mathcal{C}_{X/Y} \to 0 \]

Thus the result by the more general Lemma 42.54.10. $\square$

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