The Stacks project

Lemma 76.7.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Then $\Omega _{X/Y}$ is a quasi-coherent $\mathcal{O}_ X$-module.

Proof. Choose a diagram as in Lemma 76.7.3 with $a$ and $b$ surjective and $U$ and $V$ schemes. Then we see that $\Omega _{X/Y}|_ U = \Omega _{U/V}$ which is quasi-coherent (for example by Morphisms, Lemma 29.32.7). Hence we conclude that $\Omega _{X/Y}$ is quasi-coherent by Properties of Spaces, Lemma 66.29.6. $\square$

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