Lemma 108.7.3 (0DM8)—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 108.7.3 (cite)

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Lemma 108.7.3. Assume $X \to B$ is proper as well as of finite presentation. Then $\mathrm{Hilb}_{X/B} \to B$ satisfies the existence part of the valuative criterion (Morphisms of Spaces, Definition 67.41.1).

Proof. In Quot, Lemma 99.9.2 we have seen that $\mathrm{Hilb}_{X/B} = \mathrm{Quot}_{\mathcal{O}_ X/X/B}$ . Hence this follows from Lemma 108.5.3. $\square$

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