Lemma 76.44.7 (09RZ)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.44: Regular immersions
Lemma 76.44.7 (cite)

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Lemma 76.44.7. Let $S$ be a scheme. Let $Z \to Y \to X$ be immersions of algebraic spaces over $S$ . Assume that $Z \to Y$ is $H_1$ -regular. Then the canonical sequence of Lemma 76.5.6

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

is exact and (étale) locally split.

Proof. Since $\mathcal{C}_{Z/Y}$ is finite locally free (see Lemma 76.44.6 and Lemma 76.44.3) it suffices to prove that the sequence is exact. It suffices to show that the first map is injective as the sequence is already right exact in general. After étale localization on $X$ this reduces to the case of schemes, see Divisors, Lemma 31.21.6. $\square$

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