The Stacks project

Lemma 99.9.3. In Situation 99.9.1. Let $T$ be an algebraic space over $S$. We have

\[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf})}(T, \mathrm{Hilb}_{X/B}) = \left\{ \begin{matrix} (h, Z)\text{ where }h : T \to B,\ Z \subset X_ T \\ \text{finite presentation, flat, proper over }T \end{matrix} \right\} \]

where $X_ T = X \times _{B, h} T$.

Proof. By Lemma 99.9.2 we have $\mathrm{Hilb}_{X/B} = \mathrm{Quot}_{\mathcal{O}_ X/X/B}$. Thus we can apply Lemma 99.8.3 to see that the left hand side is bijective with the set of surjections $\mathcal{O}_{X_ T} \to \mathcal{Q}$ which are finitely presented, flat over $T$, and have support proper over $T$. Arguing exactly as in the proof of Lemma 99.9.2 we see that such quotients correspond exactly to the closed immersions $Z \to X_ T$ such that $Z \to T$ is proper, flat, and of finite presentation. $\square$

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