Lemma 76.7.13 (05ZD)—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 76.7.13 (cite)

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Lemma 76.7.13. Let $S$ be a scheme. Let $f : X \to B$ and $g : Y \to B$ be morphisms of algebraic spaces over $S$ with the same target. Let $p : X \times _ B Y \to X$ and $q : X \times _ B Y \to Y$ be the projection morphisms. The maps from Lemma 76.7.6

$p^*\Omega _{X/B} \oplus q^*\Omega _{Y/B} \longrightarrow \Omega _{X \times _ B Y/B}$

give an isomorphism.

Proof. Follows from the schemes version, see Morphisms, Lemma 29.32.11 and étale localization, see Lemma 76.7.3. $\square$

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