The Stacks project

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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