Lemma 76.7.11 (05ZB)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.7: Sheaf of differentials of a morphism
Lemma 76.7.11 (cite)

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Lemma 76.7.11. 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 $B$, and assume $i$ (étale locally) has a left inverse. Then the canonical sequence

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

of Lemma 76.7.10 is (étale locally) split exact.

Proof. Clarification: we claim that if $g : X \to Z$ is a left inverse of $i$ over $B$, then $i^*c_ g$ is a right inverse of the map $i^*\Omega _{X/B} \to \Omega _{Z/B}$. Having said this, the result follows from the corresponding result for morphisms of schemes by étale localization, see Lemmas 76.7.3 and 76.5.2. $\square$

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