We continue the discussion from Criteria for Representability, Section 97.10. In that section, starting with a scheme $S$ and morphisms of algebraic spaces $Z \to B$ and $X \to B$ over $S$ we constructed a functor
We sometimes think of $\mathit{Mor}_ B(Z, X)$ as a functor $(\mathit{Sch}/S)^{opp} \to \textit{Sets}$ endowed with a morphism $\mathit{Mor}_ B(Z, X) \to B$. Namely, if $T$ is a scheme over $S$, then an element of $\mathit{Mor}_ B(Z, X)(T)$ is a pair $(f, h)$ where $h$ is a morphism $h : T \to B$ and $f : Z \times _{B, h} T \to X \times _{B, h} T$ is a morphism of algebraic spaces over $T$. In particular, when we say that $\mathit{Mor}_ B(Z, X)$ is an algebraic space, we mean that the corresponding functor $(\mathit{Sch}/S)^{opp} \to \textit{Sets}$ is an algebraic space.
Lemma 99.12.1. Let $S$ be a scheme. Consider morphisms of algebraic spaces $Z \to B$ and $X \to B$ over $S$. If $X \to B$ is separated and $Z \to B$ is of finite presentation, flat, and proper, then there is a natural injective transformation of functors which maps a morphism $f : Z_ T \to X_ T$ to its graph.
\[ \mathit{Mor}_ B(Z, X) \longrightarrow \mathrm{Hilb}_{Z \times _ B X/B} \]
Proof. Given a scheme $T$ over $B$ and a morphism $f_ T : Z_ T \to X_ T$ over $T$, the graph of $f$ is the morphism $\Gamma _ f = (\text{id}, f) : Z_ T \to Z_ T \times _ T X_ T = (Z \times _ B X)_ T$. Recall that being separated, flat, proper, or finite presentation are properties of morphisms of algebraic spaces which are stable under base change (Morphisms of Spaces, Lemmas 67.4.4, 67.30.4, 67.40.3, and 67.28.3). Hence $\Gamma _ f$ is a closed immersion by Morphisms of Spaces, Lemma 67.4.6. Moreover, $\Gamma _ f(Z_ T)$ is flat, proper, and of finite presentation over $T$. Thus $\Gamma _ f(Z_ T)$ defines an element of $\mathrm{Hilb}_{Z \times _ B X/B}(T)$. To show the transformation is injective it suffices to show that two morphisms with the same graph are the same. This is true because if $Y \subset (Z \times _ B X)_ T$ is the graph of a morphism $f$, then we can recover $f$ by using the inverse of $\text{pr}_1|_ Y : Y \to Z_ T$ composed with $\text{pr}_2|_ Y$. $\square$
Lemma 99.12.2. Assumption and notation as in Lemma 99.12.1. The transformation $\mathit{Mor}_ B(Z, X) \longrightarrow \mathrm{Hilb}_{Z \times _ B X/B}$ is representable by open immersions.
Proof. Let $T$ be a scheme over $B$ and let $Y \subset (Z \times _ B X)_ T$ be an element of $\mathrm{Hilb}_{Z \times _ B X/B}(T)$. Then we see that $Y$ is the graph of a morphism $Z_ T \to X_ T$ over $T$ if and only if $k = \text{pr}_1|_ Y : Y \to Z_ T$ is an isomorphism. By More on Morphisms of Spaces, Lemma 76.49.6 there exists an open subscheme $V \subset T$ such that for any morphism of schemes $T' \to T$ we have $k_{T'} : Y_{T'} \to Z_{T'}$ is an isomorphism if and only if $T' \to T$ factors through $V$. This proves the lemma. $\square$
Proposition 99.12.3. Let $S$ be a scheme. Let $Z \to B$ and $X \to B$ be morphisms of algebraic spaces over $S$. Assume $X \to B$ is of finite presentation and separated and $Z \to B$ is of finite presentation, flat, and proper. Then $\mathit{Mor}_ B(Z, X)$ is an algebraic space locally of finite presentation over $B$.
Proof. Immediate consequence of Lemma 99.12.2 and Proposition 99.9.4. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)