The Stacks project

Lemma 76.7.10. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $i : Z \to X$ be an immersion of algebraic spaces over $B$. There is a canonical exact sequence

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

where the first arrow is induced by $\text{d}_{X/B}$ and the second arrow comes from Lemma 76.7.6.

Proof. This is the algebraic spaces version of Morphisms, Lemma 29.32.15 and will be a consequence of that lemma by étale localization, see Lemmas 76.7.3 and 76.5.2. However, we should make sure we can define the first arrow globally. Hence we explain the meaning of “induced by $\text{d}_{X/B}$” here. Namely, we may assume that $i$ is a closed immersion after replacing $X$ by an open subspace. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the quasi-coherent 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 Morphisms of Spaces, Lemma 67.14.1 this corresponds to a map $\mathcal{C}_{Z/X} \to i^*\Omega _{X/S}$ as desired. $\square$

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