Lemma 76.7.8 (05Z8)—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 76.7.8 (cite)

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Lemma 76.7.8. Let $S$ be a scheme. Let $f : X \to Y$ , $g : Y \to B$ be morphisms of algebraic spaces over $S$ . Then there is a canonical exact sequence

$f^*\Omega _{Y/B} \to \Omega _{X/B} \to \Omega _{X/Y} \to 0$

where the maps come from applications of Lemma 76.7.6.

Proof. Follows from the schemes version, see Morphisms, Lemma 29.32.9, of this result via étale localization, see Lemma 76.7.3. $\square$

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