Lemma 108.6.4. Let $f : X \to B$ be a separated 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 separated of finite presentation over $B$.
Proof. We have already seen that $\mathrm{Quot}_{\mathcal{F}/X/B} \to B$ is separated and locally of finite presentation, see Lemma 108.5.2. Thus it suffices to show that the open subspace $\mathrm{Quot}^ P_{\mathcal{F}/X/B}$ of Remark 108.5.9 is quasi-compact over $B$.
The question is étale local on $B$ (Morphisms of Spaces, Lemma 67.8.8). Thus we may assume $B$ is affine.
Assume $B = \mathop{\mathrm{Spec}}(\Lambda )$. Write $\Lambda = \mathop{\mathrm{colim}}\nolimits \Lambda _ i$ as the colimit of its finite type $\mathbf{Z}$-subalgebras. Then we can find an $i$ and a system $X_ i, \mathcal{F}_ i, \mathcal{L}_ i$ as in the lemma over $B_ i = \mathop{\mathrm{Spec}}(\Lambda _ i)$ whose base change to $B$ gives $X, \mathcal{F}, \mathcal{L}$. This follows from Limits of Spaces, Lemmas 70.7.1 (to find $X_ i$), 70.7.2 (to find $\mathcal{F}_ i$), 70.7.3 (to find $\mathcal{L}_ i$), and 70.5.9 (to make $X_ i$ separated). Because
\[ \mathrm{Quot}_{\mathcal{F}/X/B} = B \times _{B_ i} \mathrm{Quot}_{\mathcal{F}_ i/X_ i/B_ i} \]
and similarly for $\mathrm{Quot}^ P_{\mathcal{F}/X/B}$ we reduce to the case discussed in the next paragraph.
Assume $B$ is affine and Noetherian. We may replace $\mathcal{L}$ by a positive power, see Lemma 108.5.11. Thus we may assume there exists an immersion $i : X \to \mathbf{P}^ n_ B$ such that $i^*\mathcal{O}_{\mathbf{P}^ n}(1) = \mathcal{L}$. By Morphisms, Lemma 29.7.7 there exists a closed subscheme $X' \subset \mathbf{P}^ n_ B$ such that $i$ factors through an open immersion $j : X \to X'$. By Properties, Lemma 28.22.5 there exists a finitely presented $\mathcal{O}_{X'}$-module $\mathcal{G}$ such that $j^*\mathcal{G} = \mathcal{F}$. Thus we obtain an open immersion
\[ \mathrm{Quot}_{\mathcal{F}/X/B} \longrightarrow \mathrm{Quot}_{\mathcal{G}/X'/B} \]
by Lemma 108.5.6. Clearly this open immersion sends $\mathrm{Quot}^ P_{\mathcal{F}/X/B}$ into $\mathrm{Quot}^ P_{\mathcal{G}/X'/B}$. Now $\mathrm{Quot}^ P_{\mathcal{G}/X'/B}$ is proper over $B$ by Lemma 108.6.3. Therefore it is Noetherian and since any open of a Noetherian algebraic space is quasi-compact we win. $\square$
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