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
\mathit{Mor}_ B(Y, X) \subset \mathrm{Hilb}_{Y \times _ B X/B}
is an open subspace, see Quot, Lemma 99.12.2. Thus we can define
\mathit{Mor}^ P_ B(Y, X) = \mathit{Mor}_ B(Y, X) \cap \mathrm{Hilb}^ P_{Y \times _ B X/B}
where \mathrm{Hilb}^ P_{Y \times _ B X/B} is as in Remark 108.7.6. The morphism
\mathit{Mor}^ P_ B(Y, X) \longrightarrow \mathit{Mor}_ B(Y, X)
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).
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
\mathit{Mor}^{P, \mathcal{M}}_ B(Y, X) \longrightarrow B
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}.
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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