The Stacks project

Lemma 36.36.8. Let $X = \mathop{\mathrm{lim}}\nolimits X_ i$ be a limit of a direct system of quasi-compact and quasi-separated schemes with affine transition morphisms. Then $X$ has the resolution property if and only if $X_ i$ has the resolution properties for some $i$.

Proof. If $X_ i$ has the resolution property, then $X$ does by Lemma 36.36.4. Assume $X$ has the resolution property. Choose a finite affine open covering $X = U_1 \cup \ldots \cup U_ m$. For each $j$ choose a finite type quasi-coherent sheaf of ideals $\mathcal{I}_ j \subset \mathcal{O}_ X$ such that $X \setminus V(\mathcal{I}_ j) = U_ j$, see Properties, Lemma 28.24.1. Choose finite locally free $\mathcal{O}_ X$-modules and surjections $\mathcal{E}_ j \to \mathcal{I}_ j$. By Limits, Lemmas 32.10.3 and 32.10.2 we can find an $i$ and finite locally free $\mathcal{O}_{X_ i}$-modules $\mathcal{E}_{i, j}$ and maps $\mathcal{E}_{i, j} \to \mathcal{O}_{X_ i}$ whose base changes to $X$ recover the maps $\mathcal{E}_ j \to \mathcal{I}_ j$, $j = 1, \ldots , m$. Denote $\mathcal{I}_{i, j} \subset \mathcal{O}_{X_ i}$ the image of these maps. Set $U_{i, j} = X_ i \setminus V(\mathcal{I}_{i, j})$. After increasing $i$ we may assume $U_{i, j}$ is affine, see Limits, Lemma 32.4.13. Then we conclude that $X_ i$ has the resolution property by Lemma 36.36.5. $\square$

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