Lemma 108.7.8 (0DPI)—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 108.7.8 (cite)

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Lemma 108.7.8. Let $f : X \to B$ be a separated morphism of finite presentation of algebraic spaces. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$ -module ample on $X/B$ , see Divisors on Spaces, Definition 71.14.1. The algebraic space $\mathrm{Hilb}^ P_{X/B}$ parametrizing closed subschemes having Hilbert polynomial $P$ with respect to $\mathcal{L}$ is separated of finite presentation over $B$ .

Proof. Recall that $\mathrm{Hilb}_{X/B} = \mathrm{Quot}_{\mathcal{O}_ X/X/B}$ , see Quot, Lemma 99.9.2. Thus this lemma is an immediate consequence of Lemma 108.6.4. $\square$

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