Lemma 76.48.13. Let $S$ be a scheme. Let $Z \to Y \to X$ be formally unramified morphisms of algebraic spaces over $S$. Assume that $Z \to Y$ is a local complete intersection morphism. The exact sequence

of Lemma 76.5.6 is short exact.

Lemma 76.48.13. Let $S$ be a scheme. Let $Z \to Y \to X$ be formally unramified morphisms of algebraic spaces over $S$. Assume that $Z \to Y$ is a local complete intersection morphism. The exact sequence

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

of Lemma 76.5.6 is short exact.

**Proof.**
Choose a scheme $U$ and a surjective étale morphism $U \to X$. Choose a scheme $V$ and a surjective étale morphism $V \to U \times _ X Y$. Choose a scheme $W$ and a surjective étale morphism $W \to V \times _ Y Z$. By Lemma 76.15.11 the morphisms $W \to V$ and $V \to U$ are formally unramified. Moreover the sequence $i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0$ restricts to the corresponding sequence $i^*\mathcal{C}_{V/U} \to \mathcal{C}_{W/U} \to \mathcal{C}_{W/V} \to 0$ for $W \to V \to U$. Hence the result follows from the result for schemes (More on Morphisms, Lemma 37.62.23) as by definition the morphism $W \to V$ is a local complete intersection morphism.
$\square$

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