Lemma 36.36.3. Let $f : X \to Y$ be a morphism of schemes. Assume

1. $Y$ is quasi-compact and quasi-separated and has the resolution property,

2. there exists an $f$-ample invertible module on $X$.

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 open covering $Y = V_1 \cup \ldots \cup V_ m$. Set $U_ j = f^{-1}(V_ j)$. By Properties, Proposition 28.26.13 for each $j$ we know there exists finitely many maps $s_{j, i} : \mathcal{L}^{\otimes n_{j, i}}|_{U_ j} \to \mathcal{F}|_{U_ j}$ which are jointly surjective. Consider the quasi-coherent $\mathcal{O}_ Y$-modules

$\mathcal{H}_ n = f_*(\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n})$

We may think of $s_{j, i}$ as a section over $V_ j$ of the sheaf $\mathcal{H}_{-n_{j, i}}$. Suppose we can find finite locally free $\mathcal{O}_ Y$-modules $\mathcal{E}_{i, j}$ and maps $\mathcal{E}_{i, j} \to \mathcal{H}_{-n_{j, i}}$ such that $s_{j, i}$ is in the image. Then the corresponding maps

$f^*\mathcal{E}_{i, j} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n_{i, j}} \longrightarrow \mathcal{F}$

are going to be jointly surjective and the lemma is proved. By Properties, Lemma 28.22.3 for each $i, j$ we can find a finite type quasi-coherent submodule $\mathcal{H}'_{i, j} \subset \mathcal{H}_{-n_{j, i}}$ which contains the section $s_{i, j}$ over $V_ j$. Thus the resolution property of $Y$ produces surjections $\mathcal{E}_{i, j} \to \mathcal{H}'_{j, i}$ and we conclude. $\square$

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