Lemma 108.10.1. The diagonal of $\mathit{Mor}_ B(Y, X) \to B$ is a closed immersion of finite presentation.
108.10 Properties of relative morphisms
Let $B$ be an algebraic space. Let $X$ and $Y$ be algebraic spaces over $B$ such that $Y \to B$ is flat, proper, and of finite presentation and $X \to B$ is separated and of finite presentation. Then the functor $\mathit{Mor}_ B(Y, X)$ of relative morphisms is an algebraic space locally of finite presentation over $B$. See Quot, Proposition 99.12.3.
Proof. There is an open immersion $\mathit{Mor}_ B(Y, X) \to \mathrm{Hilb}_{Y \times _ B X/B}$, see Quot, Lemma 99.12.2. Thus the lemma follows from Lemma 108.7.1. $\square$
Lemma 108.10.2. The morphism $\mathit{Mor}_ B(Y, X) \to B$ is separated and locally of finite presentation.
Proof. To check $\mathit{Mor}_ B(Y, X) \to B$ is separated we have to show that its diagonal is a closed immersion. This is true by Lemma 108.10.1. The second statement is part of Quot, Proposition 99.12.3. $\square$
Lemma 108.10.3. With $B, X, Y$ as in the introduction of this section, in addition assume $X \to B$ is proper. Then the subfunctor $\mathit{Isom}_ B(Y, X) \subset \mathit{Mor}_ B(Y, X)$ of isomorphisms is an open subspace.
Proof. Follows immediately from More on Morphisms of Spaces, Lemma 76.49.6. $\square$
Remark 108.10.4 (Numerical invariants). Let $B, X, Y$ be as in the introduction to this section. Let $I$ be a set and for $i \in I$ let $E_ i \in D(\mathcal{O}_{Y \times _ B X})$ be perfect. Let $P : I \to \mathbf{Z}$ be a function. Recall that is an open subspace, see Quot, Lemma 99.12.2. Thus we can define where $\mathrm{Hilb}^ P_{Y \times _ B X/B}$ is as in Remark 108.7.6. 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 $I = \mathbf{Z}$, $E_ i = \mathcal{L}^{\otimes i}$ for some invertible $\mathcal{O}_{Y \times _ B X}$-module $\mathcal{L}$. In the last case we sometimes use the notation $\mathit{Mor}^{P, \mathcal{L}}_ B(Y, X)$.
\[ \mathit{Mor}_ B(Y, X) \subset \mathrm{Hilb}_{Y \times _ B X/B} \]
\[ \mathit{Mor}^ P_ B(Y, X) = \mathit{Mor}_ B(Y, X) \cap \mathrm{Hilb}^ P_{Y \times _ B X/B} \]
\[ \mathit{Mor}^ P_ B(Y, X) \longrightarrow \mathit{Mor}_ B(Y, X) \]
Lemma 108.10.5. With $B, X, Y$ as in the introduction of this section, let $\mathcal{L}$ be ample on $X/B$ and let $\mathcal{N}$ be ample on $Y/B$. See Divisors on Spaces, Definition 71.14.1. Let $P$ be a numerical polynomial. Then is separated and of finite presentation where $\mathcal{M} = \text{pr}_1^*\mathcal{N} \otimes _{\mathcal{O}_{Y \times _ B X}} \text{pr}_2^*\mathcal{L}$.
\[ \mathit{Mor}^{P, \mathcal{M}}_ B(Y, X) \longrightarrow B \]
Proof. By Lemma 108.10.2 the morphism $\mathit{Mor}_ B(Y, X) \to B$ is separated and locally of finite presentation. Thus it suffices to show that the open and closed subspace $\mathit{Mor}^{P, \mathcal{M}}_ B(Y, X)$ of Remark 108.10.4 is quasi-compact over $B$.
The question is étale local on $B$ (Morphisms of Spaces, Lemma 67.8.8). Thus we may assume $B$ is affine.
Assume $B = \mathop{\mathrm{Spec}}(\Lambda )$. Note that $X$ and $Y$ are schemes and that $\mathcal{L}$ and $\mathcal{N}$ are ample invertible sheaves on $X$ and $Y$ (this follows immediately from the definitions). Write $\Lambda = \mathop{\mathrm{colim}}\nolimits \Lambda _ i$ as the colimit of its finite type $\mathbf{Z}$-subalgebras. Then we can find an $i$ and a system $X_ i, Y_ i, \mathcal{L}_ i, \mathcal{N}_ i$ as in the lemma over $B_ i = \mathop{\mathrm{Spec}}(\Lambda _ i)$ whose base change to $B$ gives $X, Y, \mathcal{L}, \mathcal{N}$. This follows from Limits, Lemmas 32.10.1 (to find $X_ i$, $Y_ i$), 32.10.3 (to find $\mathcal{L}_ i$, $\mathcal{N}_ i$), 32.8.6 (to make $X_ i \to B_ i$ separated), 32.13.1 (to make $Y_ i \to B_ i$ proper), and 32.4.15 (to make $\mathcal{L}_ i$, $\mathcal{N}_ i$ ample). Because
\[ \mathit{Mor}_ B(Y, X) = B \times _{B_ i} \mathit{Mor}_{B_ i}(Y_ i, X_ i) \]
and similarly for $\mathit{Mor}^ P_ B(Y, X)$ we reduce to the case discussed in the next paragraph.
Assume $B$ is a Noetherian affine scheme. By Properties, Lemma 28.26.15 we see that $\mathcal{M}$ is ample. By Lemma 108.7.8 we see that $\mathrm{Hilb}^{P, \mathcal{M}}_{Y \times _ B X/B}$ is of finite presentation over $B$ and hence Noetherian. By construction
\[ \mathit{Mor}^{P, \mathcal{M}}_ B(Y, X) = \mathit{Mor}_ B(Y, X) \cap \mathrm{Hilb}^{P, \mathcal{M}}_{Y \times _ B X/B} \]
is an open subspace of $\mathrm{Hilb}^{P, \mathcal{M}}_{Y \times _ B X/B}$ and hence quasi-compact (as an open of a Noetherian algebraic space is quasi-compact). $\square$
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