The Stacks project

Proof. Let $U$ be a scheme and let $U \to X$ be a surjective étale morphism. By Lemma 67.28.4 we see that $i' : Z \times _ X U \to U$ is of finite presentation if and only if $i$ is. By Properties of Spaces, Section 66.30 we see that $\mathcal{I}$ is of finite type if and only if $\mathcal{I}|_ U = \mathop{\mathrm{Ker}}(\mathcal{O}_ U \to i'_*\mathcal{O}_{Z \times _ X U})$ is. Hence the result follows from the case of schemes, see Morphisms, Lemma 29.21.7. $\square$