Lemma 108.7.2 (0DM7)—The Stacks project

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Part 8: Topics in Moduli Theory
Chapter 108: Moduli Stacks
Section 108.7: Properties of the Hilbert functor
Lemma 108.7.2 (cite)

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Lemma 108.7.2. The morphism $\mathrm{Hilb}_{X/B} \to B$ is separated and locally of finite presentation.

Proof. To check $\mathrm{Hilb}_{X/B} \to B$ is separated we have to show that its diagonal is a closed immersion. This is true by Lemma 108.7.1. The second statement is part of Quot, Proposition 99.9.4. $\square$

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