Lemma 37.11.8 (06B5)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.11: Formally smooth morphisms
Lemma 37.11.8 (cite)

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Lemma 37.11.8. Let $f : X \to Y$ be a formally smooth morphism of schemes. Then $\Omega _{X/Y}$ is locally projective on $X$ .

Proof. Choose $U \subset X$ and $V \subset Y$ affine open such that $f(U) \subset V$ . By Lemma 37.11.5 $f|_ U : U \to V$ is formally smooth. Hence $\Gamma (V, \mathcal{O}_ V) \to \Gamma (U, \mathcal{O}_ U)$ is a formally smooth ring map, see Lemma 37.11.6. Hence by Algebra, Lemma 10.138.7 the $\Gamma (U, \mathcal{O}_ U)$ -module $\Omega _{\Gamma (U, \mathcal{O}_ U)/\Gamma (V, \mathcal{O}_ V)}$ is projective. Hence $\Omega _{U/V}$ is locally projective, see Properties, Section 28.21. $\square$

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