## 99.9 The Hilbert functor

In this section we prove the Hilb functor is an algebraic space.

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 $X/B$.

In Situation 99.9.1 we sometimes think of $\mathrm{Hilb}_{X/B}$ as a functor $(\mathit{Sch}/S)^{opp} \to \textit{Sets}$ endowed with a morphism $\mathrm{Hilb}_{X/S} \to B$. Namely, if $T$ is a scheme over $S$, then an element of $\mathrm{Hilb}_{X/B}(T)$ is a pair $(h, Z)$ where $h$ is a morphism $h : T \to B$ and $Z \subset X_ T = X \times _{B, h} T$ is a closed subscheme, flat, proper, and of finite presentation over $T$. In particular, when we say that $\mathrm{Hilb}_{X/B}$ is an algebraic space, we mean that the corresponding functor $(\mathit{Sch}/S)^{opp} \to \textit{Sets}$ is an algebraic space.

Of course the Hilbert functor is just a special case of the Quot functor.

Lemma 99.9.2. In Situation 99.9.1 we have $\mathrm{Hilb}_{X/B} = \mathrm{Quot}_{\mathcal{O}_ X/X/B}$.

**Proof.**
Let $T$ be a scheme over $B$. Given an element $Z \in \mathrm{Hilb}_{X/B}(T)$ we can consider the quotient $\mathcal{O}_{X_ T} \to i_*\mathcal{O}_ Z$ where $i : Z \to X_ T$ is the inclusion morphism. Note that $i_*\mathcal{O}_ Z$ is quasi-coherent. Since $Z \to T$ and $X_ T \to T$ are of finite presentation, we see that $i$ is of finite presentation (Morphisms of Spaces, Lemma 67.28.9), hence $i_*\mathcal{O}_ Z$ is an $\mathcal{O}_{X_ T}$-module of finite presentation (Descent on Spaces, Lemma 74.6.7). Since $Z \to T$ is proper we see that $i_*\mathcal{O}_ Z$ has support proper over $T$ (as defined in Derived Categories of Spaces, Section 75.7). Since $\mathcal{O}_ Z$ is flat over $T$ and $i$ is affine, we see that $i_*\mathcal{O}_ Z$ is flat over $T$ (small argument omitted). Hence $\mathcal{O}_{X_ T} \to i_*\mathcal{O}_ Z$ is an element of $\mathrm{Quot}_{\mathcal{O}_ X/X/B}(T)$.

Conversely, given an element $\mathcal{O}_{X_ T} \to \mathcal{Q}$ of $\mathrm{Quot}_{\mathcal{O}_ X/X/B}(T)$, we can consider the closed immersion $i : Z \to X_ T$ corresponding to the quasi-coherent ideal sheaf $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_{X_ T} \to \mathcal{Q})$ (Morphisms of Spaces, Lemma 67.13.1). By construction of $Z$ we see that $\mathcal{Q} = i_*\mathcal{O}_ Z$. Then we can read the arguments given above backwards to see that $Z$ defines an element of $\mathrm{Hilb}_{X/B}(T)$. For example, $\mathcal{I}$ is quasi-coherent of finite type (Modules on Sites, Lemma 18.24.1) hence $i : Z \to X_ T$ is of finite presentation (Morphisms of Spaces, Lemma 67.28.12) hence $Z \to T$ is of finite presentation (Morphisms of Spaces, Lemma 67.28.2). Properness of $Z \to T$ follows from the discussion in Derived Categories of Spaces, Section 75.7. Flatness of $Z \to T$ follows from flatness of $\mathcal{Q}$ over $T$.

We omit the (immediate) verification that the two constructions given above are mutually inverse.
$\square$

Sanity check: $\mathrm{Hilb}_{X/B}$ sheaf plays the same role among algebraic spaces over $S$.

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.
$\square$

Proposition 99.9.4. Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. If $f$ is of finite presentation and separated, then $\mathrm{Hilb}_{X/B}$ is an algebraic space locally of finite presentation over $B$.

**Proof.**
Immediate consequence of Lemma 99.9.2 and Proposition 99.8.4.
$\square$

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