The Stacks project

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

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

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. $\square$