Lemma 108.5.6. Let $B$ be an algebraic space. Let $j : X \to Y$ be an open immersion of algebraic spaces which are separated and of finite presentation over $B$. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ Y$-module and set $\mathcal{F} = j^*\mathcal{G}$. Then there is an open immersion
of algebraic spaces over $B$.
Proof. If $\mathcal{F}_ T \to \mathcal{Q}$ is an element of $\mathrm{Quot}_{\mathcal{F}/X/B}(T)$ then we can consider $\mathcal{G}_ T \to j_{T, *}\mathcal{F}_ T \to j_{T, *}\mathcal{Q}$. Looking at stalks one finds that this is surjective. By Lemma 108.4.4 we see that $j_{T, *}\mathcal{Q}$ is finitely presented, flat over $B$ with support proper over $B$. Thus we obtain a $T$-valued point of $\mathrm{Quot}_{\mathcal{G}/Y/B}$. This defines the morphism of the lemma. We omit the proof that this is an open immersion. Hint: If $\mathcal{G}_ T \to \mathcal{Q}$ is an element of $\mathrm{Quot}_{\mathcal{G}/Y/B}(T)$ and for $t \in T$ we have $\text{Supp}(\mathcal{Q}_ t) \subset |X_ t|$, then the same is true for $t' \in T$ in a neighbourhood of $t$. $\square$
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