Lemma 108.6.3. Let $f : X \to B$ be a proper morphism of finite presentation of algebraic spaces. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_ X$-module. 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 $\mathrm{Quot}^ P_{\mathcal{F}/X/B}$ parametrizing quotients of $\mathcal{F}$ having Hilbert polynomial $P$ with respect to $\mathcal{L}$ is proper over $B$.
Proof. The question is étale local over $B$, see Morphisms of Spaces, Lemma 67.40.2. Thus we may assume $B$ is an affine scheme. Then we can find a closed immersion $i : X \to \mathbf{P}^ n_ B$ such that $i^*\mathcal{O}_{\mathbf{P}^ n_ B}(1) \cong \mathcal{L}^{\otimes d}$ for some $d \geq 1$. See Morphisms, Lemma 29.39.3. Changing $\mathcal{L}$ into $\mathcal{L}^{\otimes d}$ and the numerical polynomial $P(t)$ into $P(dt)$ leaves $\mathrm{Quot}^ P_{\mathcal{F}/X/B}$ unaffected; some details omitted. Hence we may assume $\mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^ n_ B}(1)$. Then the isomorphism $\mathrm{Quot}_{\mathcal{F}/X/B} \to \mathrm{Quot}_{i_*\mathcal{F}/\mathbf{P}^ n_ B/B}$ of Lemma 108.5.7 induces an isomorphism $\mathrm{Quot}^ P_{\mathcal{F}/X/B} \cong \mathrm{Quot}^ P_{i_*\mathcal{F}/\mathbf{P}^ n_ B/B}$. Since $\mathrm{Quot}^ P_{i_*\mathcal{F}/\mathbf{P}^ n_ B/B}$ is proper over $B$ by Lemma 108.6.2 we conclude. $\square$
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