Lemma 99.8.3. In Situation 99.8.1. Let $T$ be an algebraic space over $S$. We have
\[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf})}(T, \mathrm{Quot}_{\mathcal{F}/X/B}) = \left\{ \begin{matrix} (h, \mathcal{F}_ T \to \mathcal{Q}) \text{ where } h : T \to B \text{ and}
\\ \mathcal{Q}\text{ is of finite presentation and}
\\ \text{flat over }T\text{ with support proper over }T
\end{matrix} \right\} \]
where $\mathcal{F}_ T$ denotes the pullback of $\mathcal{F}$ to the algebraic space $X \times _{B, h} T$.
Proof.
Observe that the left and right hand side of the equality are subsets of the left and right hand side of the second equality in Lemma 99.7.5. To see that these subsets correspond under the identification given in the proof of that lemma it suffices to show: given $h : T \to B$, a surjective étale morphism $U \to T$, a finite type quasi-coherent $\mathcal{O}_{X_ T}$-module $\mathcal{Q}$ the following are equivalent
the scheme theoretic support of $\mathcal{Q}$ is proper over $T$, and
the scheme theoretic support of $(X_ U \to X_ T)^*\mathcal{Q}$ is proper over $U$.
This follows from Descent on Spaces, Lemma 74.11.19 combined with Morphisms of Spaces, Lemma 67.30.10 which shows that taking scheme theoretic support commutes with flat base change.
$\square$
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