Lemma 29.32.11 (01V1)—The Stacks project

Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.32: Sheaf of differentials of a morphism
Lemma 29.32.11 (cite)

Lemma 29.32.11. Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes with the same target. Let $p : X \times _ S Y \to X$ and $q : X \times _ S Y \to Y$ be the projection morphisms. The maps from Lemma 29.32.8

\[ p^*\Omega _{X/S} \oplus q^*\Omega _{Y/S} \longrightarrow \Omega _{X \times _ S Y/S} \]

give an isomorphism.

Proof. By Lemma 29.32.10 the composition $p^*\Omega _{X/S} \to \Omega _{X \times _ S Y/S} \to \Omega _{X \times _ S Y/Y}$ is an isomorphism, and similarly for $q$. Moreover, the cokernel of $p^*\Omega _{X/S} \to \Omega _{X \times _ S Y/S}$ is $\Omega _{X \times _ S Y/X}$ by Lemma 29.32.9. The result follows. $\square$

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