Lemma 29.32.15. Let $i : Z \to X$ be an immersion of schemes over $S$. There is a canonical exact sequence

where the first arrow is induced by $\text{d}_{X/S}$ and the second arrow comes from Lemma 29.32.8.

Lemma 29.32.15. Let $i : Z \to X$ be an immersion of schemes over $S$. There is a canonical exact sequence

\[ \mathcal{C}_{Z/X} \to i^*\Omega _{X/S} \to \Omega _{Z/S} \to 0 \]

where the first arrow is induced by $\text{d}_{X/S}$ and the second arrow comes from Lemma 29.32.8.

**Proof.**
This is the sheafified version of Algebra, Lemma 10.131.9. However we should make sure we can define the first arrow globally. Hence we explain the meaning of “induced by $\text{d}_{X/S}$” here. Namely, we may assume that $i$ is a closed immersion by shrinking $X$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the sheaf of ideals corresponding to $Z \subset X$. Then $\text{d}_{X/S} : \mathcal{I} \to \Omega _{X/S}$ maps the subsheaf $\mathcal{I}^2 \subset \mathcal{I}$ to $\mathcal{I}\Omega _{X/S}$. Hence it induces a map $\mathcal{I}/\mathcal{I}^2 \to \Omega _{X/S}/\mathcal{I}\Omega _{X/S}$ which is $\mathcal{O}_ X/\mathcal{I}$-linear. By Lemma 29.4.1 this corresponds to a map $\mathcal{C}_{Z/X} \to i^*\Omega _{X/S}$ as desired.
$\square$

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