The Stacks project

Proof. We may replace $X_0$ by the scheme theoretic support of $\mathcal{F}_0$. By Morphisms of Spaces, Lemma 67.15.2 this guarantees that $X_ i$ is the support of $\mathcal{F}_ i$ and $X$ is the support of $\mathcal{F}$. Then, if $Z \subset X$ denotes the scheme theoretic support of $\mathcal{F}$, we see that $Z \to X$ is a universal homeomorphism. We conclude that $X \to Y$ is proper as this is true for $Z \to Y$ by assumption, see Morphisms, Lemma 29.41.9. By Lemma 70.6.13 we see that $X_ i \to Y$ is proper for some $i$. Then it follows that the scheme theoretic support $Z_ i$ of $\mathcal{F}_ i$ is proper over $Y$ by Morphisms of Spaces, Lemmas 67.40.5 and 67.40.4. $\square$