Lemma 66.31.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ Y$-module. If $\mathcal{G}$ is locally projective on $Y$, then $f^*\mathcal{G}$ is locally projective on $X$.
Proof. Choose a surjective étale morphism $V \to Y$ with $V$ a scheme. Choose a surjective étale morphism $U \to V \times _ Y X$ with $U$ a scheme. Denote $\psi : U \to V$ the induced morphism. Then
\[ f^*\mathcal{G}|_ U = \psi ^*(\mathcal{G}|_ V) \]
Hence the lemma follows from the definition and the result in the case of schemes, see Properties, Lemma 28.21.3. $\square$
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