Situation 99.9.1. Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Assume that $f$ is of finite presentation. For any scheme $T$ over $B$ we will denote $X_ T$ the base change of $X$ to $T$. Given such a $T$ we set
\[ \mathrm{Hilb}_{X/B}(T) = \left\{ \begin{matrix} \text{closed subspaces }Z \subset X_ T\text{ such that }Z \to T
\\ \text{is of finite presentation, flat, and proper}
\end{matrix} \right\} \]
Since base change preserves the required properties (Spaces, Lemma 65.12.3 and Morphisms of Spaces, Lemmas 67.28.3, 67.30.4, and 67.40.3) we obtain a functor
99.9.1.1
\begin{equation} \label{quot-equation-hilb} \mathrm{Hilb}_{X/B} : (\mathit{Sch}/B)^{opp} \longrightarrow \textit{Sets} \end{equation}
This is the Hilbert functor associated to $X/B$.
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