The Stacks project

Lemma 76.5.6. Let $S$ be a scheme. Let $Z \to Y \to X$ be immersions of algebraic spaces. Then there is a canonical exact sequence

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

where the maps come from Lemma 76.5.3 and $i : Z \to Y$ is the first morphism.

Proof. Let $U$ be a scheme and let $U \to X$ be a surjective ├ętale morphism. Via Lemmas 76.5.2 and 76.5.4 the exactness of the sequence translates immediately into the exactness of the corresponding sequence for the immersions of schemes $Z \times _ X U \to Y \times _ X U \to U$. Hence the lemma follows from Morphisms, Lemma 29.31.5. $\square$


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