The Stacks project

Lemma 107.6.2. Let $B$ be an algebraic space. Let $X = B \times \mathbf{P}^ n_\mathbf {Z}$. Let $\mathcal{L}$ be the pullback of $\mathcal{O}_{\mathbf{P}^ n}(1)$ to $X$. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module of finite presentation. The algebraic space $\mathrm{Quot}^ P_{\mathcal{F}/X/B}$ parametrizing quotients of $\mathcal{F}$ having Hilbert polynomial $P$ with respect to $\mathcal{L}$ is proper over $B$.

Proof. The question is ├ętale local over $B$, see Morphisms of Spaces, Lemma 66.40.2. Thus we may assume $B$ is an affine scheme. In this case $\mathcal{L}$ is an ample invertible module on $X$ (by Constructions, Lemma 27.10.6 and the definition of ample invertible modules in Properties, Definition 28.26.1). Thus we can find $r' \geq 0$ and $r \geq 0$ and a surjection

\[ \mathcal{O}_ X^{\oplus r} \longrightarrow \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes r'} \]

by Properties, Proposition 28.26.13. By Lemma 107.5.10 we may replace $\mathcal{F}$ by $\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes r'}$ and $P(t)$ by $P(t + r')$. By Lemma 107.5.8 we obtain a closed immersion

\[ \mathrm{Quot}^ P_{\mathcal{F}/X/B} \longrightarrow \mathrm{Quot}^ P_{\mathcal{O}_ X^{\oplus r}/X/B} \]

Since we've shown that $\mathrm{Quot}^ P_{\mathcal{O}_ X^{\oplus r}/X/B} \to B$ is proper in Lemma 107.6.1 we conclude. $\square$

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