Lemma 76.19.8. 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 76.19.5 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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