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The Stacks project

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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