The Stacks project

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

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

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. $\square$