Lemma 108.7.1. The diagonal of $\mathrm{Hilb}_{X/B} \to B$ is a closed immersion of finite presentation.
108.7 Properties of the Hilbert functor
Let $f : X \to B$ be a morphism of algebraic spaces which is separated and of finite presentation. Then $\mathrm{Hilb}_{X/B}$ is an algebraic space locally of finite presentation over $B$. See Quot, Proposition 99.9.4.
Proof. In Quot, Lemma 99.9.2 we have seen that $\mathrm{Hilb}_{X/B} = \mathrm{Quot}_{\mathcal{O}_ X/X/B}$. Hence this follows from Lemma 108.5.1. $\square$
Lemma 108.7.2. The morphism $\mathrm{Hilb}_{X/B} \to B$ is separated and locally of finite presentation.
Proof. To check $\mathrm{Hilb}_{X/B} \to B$ is separated we have to show that its diagonal is a closed immersion. This is true by Lemma 108.7.1. The second statement is part of Quot, Proposition 99.9.4. $\square$
Lemma 108.7.3. Assume $X \to B$ is proper as well as of finite presentation. Then $\mathrm{Hilb}_{X/B} \to B$ satisfies the existence part of the valuative criterion (Morphisms of Spaces, Definition 67.41.1).
Proof. In Quot, Lemma 99.9.2 we have seen that $\mathrm{Hilb}_{X/B} = \mathrm{Quot}_{\mathcal{O}_ X/X/B}$. Hence this follows from Lemma 108.5.3. $\square$
Lemma 108.7.4. Let $B$ be an algebraic space. Let $\pi : X \to Y$ be an open immersion of algebraic spaces which are separated and of finite presentation over $B$. Then $\pi $ induces an open immersion $\mathrm{Hilb}_{X/B} \to \mathrm{Hilb}_{Y/B}$.
Proof. Omitted. Hint: If $Z \subset X_ T$ is a closed subscheme which is proper over $T$, then $Z$ is also closed in $Y_ T$. Thus we obtain the transformation $\mathrm{Hilb}_{X/B} \to \mathrm{Hilb}_{Y/B}$. If $Z \subset Y_ T$ is an element of $\mathrm{Hilb}_{Y/B}(T)$ and for $t \in T$ we have $|Z_ t| \subset |X_ t|$, then the same is true for $t' \in T$ in a neighbourhood of $t$. $\square$
Lemma 108.7.5. Let $B$ be an algebraic space. Let $\pi : X \to Y$ be a closed immersion of algebraic spaces which are separated and of finite presentation over $B$. Then $\pi $ induces a closed immersion $\mathrm{Hilb}_{X/B} \to \mathrm{Hilb}_{Y/B}$.
Proof. Since $\pi $ is a closed immersion, it is immediate that given a closed subscheme $Z \subset X_ T$, we can view $Z$ as a closed subscheme of $X_ T$. Thus we obtain the transformation $\mathrm{Hilb}_{X/B} \to \mathrm{Hilb}_{Y/B}$. This transformation is immediately seen to be a monomorphism. To prove that it is a closed immersion, you can use Lemma 108.5.8 for the map $\mathcal{O}_ Y \to \mathcal{O}_ X$ and the identifications $\mathrm{Hilb}_{X/B} = \mathrm{Quot}_{\mathcal{O}_ X/X/B}$, $\mathrm{Hilb}_{Y/B} = \mathrm{Quot}_{\mathcal{O}_ Y/Y/B}$ of Quot, Lemma 99.9.2. $\square$
Remark 108.7.6 (Numerical invariants). Let $f : X \to B$ be as in the introduction to this section. Let $I$ be a set and for $i \in I$ let $E_ i \in D(\mathcal{O}_ X)$ be perfect. Let $P : I \to \mathbf{Z}$ be a function. Recall that $\mathrm{Hilb}_{X/B} = \mathrm{Quot}_{\mathcal{O}_ X/X/B}$, see Quot, Lemma 99.9.2. Thus we can define where $\mathrm{Quot}^ P_{\mathcal{O}_ X/X/B}$ is as in Remark 108.5.9. The morphism is a flat closed immersion which is an open and closed immersion for example if $I$ is finite, or $B$ is locally Noetherian, or $I = \mathbf{Z}$ and $E_ i = \mathcal{L}^{\otimes i}$ for some invertible $\mathcal{O}_ X$-module $\mathcal{L}$. In the last case we sometimes use the notation $\mathrm{Hilb}^{P, \mathcal{L}}_{X/B}$.
\[ \mathrm{Hilb}^ P_{X/B} = \mathrm{Quot}^ P_{\mathcal{O}_ X/X/B} \]
\[ \mathrm{Hilb}^ P_{X/B} \longrightarrow \mathrm{Hilb}_{X/B} \]
Lemma 108.7.7. Let $f : X \to B$ be a proper morphism of finite presentation of algebraic spaces. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module ample on $X/B$, see Divisors on Spaces, Definition 71.14.1. The algebraic space $\mathrm{Hilb}^ P_{X/B}$ parametrizing closed subschemes having Hilbert polynomial $P$ with respect to $\mathcal{L}$ is proper over $B$.
Proof. Recall that $\mathrm{Hilb}_{X/B} = \mathrm{Quot}_{\mathcal{O}_ X/X/B}$, see Quot, Lemma 99.9.2. Thus this lemma is an immediate consequence of Lemma 108.6.3. $\square$
Lemma 108.7.8. Let $f : X \to B$ be a separated morphism of finite presentation of algebraic spaces. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module ample on $X/B$, see Divisors on Spaces, Definition 71.14.1. The algebraic space $\mathrm{Hilb}^ P_{X/B}$ parametrizing closed subschemes having Hilbert polynomial $P$ with respect to $\mathcal{L}$ is separated of finite presentation over $B$.
Proof. Recall that $\mathrm{Hilb}_{X/B} = \mathrm{Quot}_{\mathcal{O}_ X/X/B}$, see Quot, Lemma 99.9.2. Thus this lemma is an immediate consequence of Lemma 108.6.4. $\square$
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