The Stacks project

Lemma 75.19.8. Let $S$ be a scheme. Let $f : X \to Y$ be a formally smooth morphism of algebraic spaces over $S$. Then $\Omega _{X/Y}$ is locally projective on $X$.

Proof. Choose a diagram

\[ \xymatrix{ U \ar[d] \ar[r]_\psi & V \ar[d] \\ X \ar[r]^ f & Y } \]

where $U$ and $V$ are affine(!) schemes and the vertical arrows are étale. By Lemma 75.19.5 we see $\psi : U \to V$ is formally smooth. Hence $\Gamma (V, \mathcal{O}_ V) \to \Gamma (U, \mathcal{O}_ U)$ is a formally smooth ring map, see More on Morphisms, 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. Since $\Omega _{X/Y}|_ U = \Omega _{U/V}$ we see that $\Omega _{X/Y}$ is locally projective too. (Because we can find an étale covering of $X$ by the affine $U$'s fitting into diagrams as above – details omitted.) $\square$


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