Lemma 108.7.4 (0DPE)—The Stacks project

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Table of contents
Part 8: Topics in Moduli Theory
Chapter 108: Moduli Stacks
Section 108.7: Properties of the Hilbert functor
Lemma 108.7.4 (cite)

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Lemma 108.7.4. Let $B$ be an algebraic space. Let $\pi : X \to Y$ be an open immersion of algebraic spaces which are separated and of finite presentation over $B$ . Then $\pi$ induces an open immersion $\mathrm{Hilb}_{X/B} \to \mathrm{Hilb}_{Y/B}$ .

Proof. Omitted. Hint: If $Z \subset X_ T$ is a closed subscheme which is proper over $T$ , then $Z$ is also closed in $Y_ T$ . Thus we obtain the transformation $\mathrm{Hilb}_{X/B} \to \mathrm{Hilb}_{Y/B}$ . If $Z \subset Y_ T$ is an element of $\mathrm{Hilb}_{Y/B}(T)$ and for $t \in T$ we have $|Z_ t| \subset |X_ t|$ , then the same is true for $t' \in T$ in a neighbourhood of $t$ . $\square$

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