The Stacks project

Lemma 70.12.3. Assumptions and notation as in Situation 70.6.1. Let $\mathcal{F}_0$ be a quasi-coherent $\mathcal{O}_{X_0}$-module. Denote $\mathcal{F}$ and $\mathcal{F}_ i$ the pullbacks of $\mathcal{F}_0$ to $X$ and $X_ i$. Assume

  1. $f_0$ is locally of finite type,

  2. $\mathcal{F}_0$ is of finite type,

  3. the scheme theoretic support of $\mathcal{F}$ is proper over $Y$.

Then the scheme theoretic support of $\mathcal{F}_ i$ is proper over $Y_ i$ for some $i$.

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. $\square$

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