## 65.31 Locally projective modules

Recall that in Properties, Section 28.21 we defined the notion of a locally projective quasi-coherent module.

Lemma 65.31.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. The following are equivalent

1. for some scheme $U$ and surjective étale morphism $U \to X$ the restriction $\mathcal{F}|_ U$ is locally projective on $U$, and

2. for any scheme $U$ and any étale morphism $U \to X$ the restriction $\mathcal{F}|_ U$ is locally projective on $U$.

Proof. Let $U \to X$ be as in (1) and let $V \to X$ be étale where $V$ is a scheme. Then $\{ U \times _ X V \to V\}$ is an fppf covering of schemes. Hence if $\mathcal{F}|_ U$ is locally projective, then $\mathcal{F}|_{U \times _ X V}$ is locally projective (see Properties, Lemma 28.21.3) and hence $\mathcal{F}|_ V$ is locally projective, see Descent, Lemma 35.7.7. $\square$

Definition 65.31.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. We say $\mathcal{F}$ is locally projective if the equivalent conditions of Lemma 65.31.1 are satisfied.

Lemma 65.31.3. 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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