Lemma 75.28.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume
$Y$ is quasi-compact and quasi-separated and has the resolution property,
there exists an $f$-ample invertible module on $X$ (Divisors on Spaces, Definition 71.14.1).
Then $X$ has the resolution property.
Proof. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Let $\mathcal{L}$ be an $f$-ample invertible module. Choose an affine scheme $V$ and a surjective étale morphism $V \to Y$. Set $U = V \times _ Y X$. Then $\mathcal{L}|_ U$ is ample on $U$. By Properties, Proposition 28.26.13 we know there exists finitely many maps $s_ i : \mathcal{L}^{\otimes n_ i}|_ U \to \mathcal{F}|_ U$ which are jointly surjective. Consider the quasi-coherent $\mathcal{O}_ Y$-modules
We may think of $s_ i$ as a section over $V$ of the sheaf $\mathcal{H}_{-n_ i}$. Suppose we can find finite locally free $\mathcal{O}_ Y$-modules $\mathcal{E}_ i$ and maps $\mathcal{E}_ i \to \mathcal{H}_{-n_ i}$ such that $s_ i$ is in the image. Then the corresponding maps
are going to be jointly surjective and the lemma is proved. By Limits of Spaces, Lemma 70.9.2 for each $i$ we can find a finite type quasi-coherent submodule $\mathcal{H}'_ i \subset \mathcal{H}_{-n_ i}$ which contains the section $s_ i$ over $V$. Thus the resolution property of $Y$ produces surjections $\mathcal{E}_ i \to \mathcal{H}'_ i$ and we conclude. $\square$
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