Lemma 76.15.13 (060F)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.15: Universal first order thickenings
Lemma 76.15.13 (cite)

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Lemma 76.15.13. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$ . Let $h : Z \to X$ be a formally unramified morphism of algebraic spaces over $B$ . There is a canonical exact sequence

$\mathcal{C}_{Z/X} \to h^*\Omega _{X/B} \to \Omega _{Z/B} \to 0.$

The first arrow is induced by $\text{d}_{Z'/B}$ where $Z'$ is the universal first order neighbourhood of $Z$ over $X$ .

Proof. We know that there is a canonical exact sequence

$\mathcal{C}_{Z/Z'} \to \Omega _{Z'/S} \otimes \mathcal{O}_ Z \to \Omega _{Z/S} \to 0.$

see Lemma 76.7.10. Hence the result follows on applying Lemma 76.15.12. $\square$

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