The Stacks project

Proof. If $X_ i$ has the resolution property, then $X$ does by Derived Categories of Spaces, Lemma 75.28.3. Assume $X$ has the resolution property. Choose $i \in I$. We may choose an affine scheme $V_ i$ and a surjective étale morphism $V_ i \to X_ i$ (Properties of Spaces, Lemma 66.6.3). We may choose an embedding $j : V_ i \to Y_ i$ with $Y_ i$ finite and finitely presented over $X_ i$ (Lemma 76.34.4). We may choose a finite type quasi-coherent ideal $\mathcal{I}_ i \subset \mathcal{O}_{Y_ i}$ such that $V_ i = Y_ i \setminus V(\mathcal{I}_ i)$ (Limits of Spaces, Lemma 70.14.1). Denote $V \to Y \to X$ the base changes of $V_ i \to Y_ i \to X_ i$ to $X$. Denote $\mathcal{I} \subset \mathcal{O}_ Y$ the pullback of the ideal $\mathcal{I}_ i$. By the easy direction of Lemma 76.56.4 there exists a finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$ and a surjection $\mathcal{E} \to \pi _*\mathcal{I}$. Note that since $\pi _ i : Y_ i \to X_ i$ is finite and of finite presentation we also have that $\pi : Y \to X$ is finite and of finite presentation and that the $\mathcal{O}_{X_ i}$-modules $\pi _{i, *}\mathcal{O}_{Y_ i}$ and $\pi _{i, *}(\mathcal{O}_{Y_ i}/\mathcal{I}_ i)$ are of finite presentation and pullback to $X$ to give $\pi _*\mathcal{O}_ Y$ and $\pi _*(\mathcal{O}_ Y/\mathcal{I})$. Thus by Limits of Spaces, Lemma 70.7.2 after increasing $i$ we can find a finite locally free $\mathcal{O}_{X_ i}$-module $\mathcal{E}_ i$ and a map $\mathcal{E}_ i \to \pi _{i, *}\mathcal{O}_{Y_ i}$ whose base change to $X$ recovers the composition $\mathcal{E} \to \pi _*\mathcal{I} \to \pi _*\mathcal{O}_ Y$. The pullbacks of the finitely presented $\mathcal{O}_{X_ i}$-modules $\mathop{\mathrm{Coker}}(\mathcal{E}_ i \to \pi _{i, *}\mathcal{O}_{Y_ i})$ and $\pi _{i, *}(\mathcal{O}_{Y_ i}/\mathcal{I}_ i)$ to $X$ agree as quotients of $\pi _*\mathcal{O}_ Y$. Hence by Limits of Spaces, Lemma 70.7.2 we may assume that these agree, in other words that the image of $\mathcal{E}_ i \to \pi _{i, *}\mathcal{O}_{X_ i}$ is equal to $\pi _{i, *}\mathcal{I}_ i$. Then we conclude that $X_ i$ has the resolution property by Lemma 76.56.4. $\square$