The Stacks project

Proof. If $X_ i$ has the resolution property, then $X$ does by Lemma 36.36.4. Assume $X$ has the resolution property. Choose $i \in I$. Choose a finite affine open covering $X_ i = U_{i, 1} \cup \ldots \cup U_{i, m}$. For each $j$ choose a finite type quasi-coherent sheaf of ideals $\mathcal{I}_{i, j} \subset \mathcal{O}_{X_ i}$ such that $X_ i \setminus V(\mathcal{I}_{i, j}) = U_{i, j}$, see Properties, Lemma 28.24.1. Denote $U_ j \subset X$ the inverse image of $U_{i, j}$ and denote $\mathcal{I}_ j \subset \mathcal{O}_ X$ the pullback of $\mathcal{I}_{i, j}$. Since $X$ has the resolution property, we may choose finite locally free $\mathcal{O}_ X$-modules $\mathcal{E}_ j$ and surjections $\mathcal{E}_ j \to \mathcal{I}_ j$. By Limits, Lemmas 32.10.3 and 32.10.2 after increasing $i$ we can find 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 compositions $\mathcal{E}_ j \to \mathcal{I}_ j \to \mathcal{O}_ X$, $j = 1, \ldots , m$. The pullbacks of the finitely presented $\mathcal{O}_{X_ i}$-modules $\mathop{\mathrm{Coker}}(\mathcal{E}_{i, j} \to \mathcal{O}_{X_ i})$ and $\mathcal{O}_{X_ i}/\mathcal{I}_{i, j}$ to $X$ agree as quotients of $\mathcal{O}_ X$. Hence by Limits, Lemma 32.10.2 we may assume that these agree, in other words that the image of $\mathcal{E}_{i, j} \to \mathcal{O}_{X_ i}$ is equal to $\mathcal{I}_{i, j}$. Then we conclude that $X_ i$ has the resolution property by Lemma 36.36.6. $\square$