The Stacks project

37.68 Relative morphisms

In this section we prove a representability result which we will use in Fundamental Groups, Section 58.5 to prove a result on the category of finite ├ętale coverings of a scheme. The material in this section is discussed in the correct generality in Criteria for Representability, Section 97.10.

Let $S$ be a scheme. Let $Z$ and $X$ be schemes over $S$. Given a scheme $T$ over $S$ we can consider morphisms $b : T \times _ S Z \to T \times _ S X$ over $S$. Picture
\begin{equation} \label{more-morphisms-equation-hom} \vcenter { \xymatrix{ T \times _ S Z \ar[rd] \ar[rr]_ b & & T \times _ S X \ar[ld] & Z \ar[rd] & & X \ar[ld] \\ & T \ar[rrr] & & & S } } \end{equation}

Of course, we can also think of $b$ as a morphism $b : T \times _ S Z \to X$ such that

\[ \xymatrix{ T \times _ S Z \ar[r] \ar[d] \ar@/^1pc/[rrr]_-b & Z \ar[rd] & & X \ar[ld] \\ T \ar[rr] & & S } \]

commutes. In this situation we can define a functor
\begin{equation} \label{more-morphisms-equation-hom-functor} \mathit{Mor}_ S(Z, X) : (\mathit{Sch}/S)^{opp} \longrightarrow \textit{Sets}, \quad T \longmapsto \{ b\text{ as above}\} \end{equation}

Here is a basic representability result.

Lemma 37.68.1. Let $Z \to S$ and $X \to S$ be morphisms of affine schemes. Assume $\Gamma (Z, \mathcal{O}_ Z)$ is a finite free $\Gamma (S, \mathcal{O}_ S)$-module. Then $\mathit{Mor}_ S(Z, X)$ is representable by an affine scheme over $S$.

Proof. Write $S = \mathop{\mathrm{Spec}}(R)$. Choose a basis $\{ e_1, \ldots , e_ m\} $ for $\Gamma (Z, \mathcal{O}_ Z)$ over $R$. Choose a presentation

\[ \Gamma (X, \mathcal{O}_ X) = R[\{ x_ i\} _{i \in I}]/(\{ f_ k\} _{k \in K}). \]

We will denote $\overline{x}_ i$ the image of $x_ i$ in this quotient. Write

\[ P = R[\{ a_{ij}\} _{i \in I, 1 \leq j \leq m}]. \]

Consider the $R$-algebra map

\[ \Psi : R[\{ x_ i\} _{i \in I}] \longrightarrow P \otimes _ R \Gamma (Z, \mathcal{O}_ Z), \quad x_ i \longmapsto \sum \nolimits _ j a_{ij} \otimes e_ j. \]

Write $\Psi (f_ k) = \sum c_{kj} \otimes e_ j$ with $c_{kj} \in P$. Finally, denote $J \subset P$ the ideal generated by the elements $c_{kj}$, $k \in K$, $1 \leq j \leq m$. We claim that $W = \mathop{\mathrm{Spec}}(P/J)$ represents the functor $\mathit{Mor}_ S(Z, X)$.

First, note that by construction $P/J$ is an $R$-algebra, hence a morphism $W \to S$. Second, by construction the map $\Psi $ factors through $\Gamma (X, \mathcal{O}_ X)$, hence we obtain an $P/J$-algebra homomorphism

\[ P/J \otimes _ R \Gamma (X, \mathcal{O}_ X) \longrightarrow P/J \otimes _ R \Gamma (Z, \mathcal{O}_ Z) \]

which determines a morphism $b_{univ} : W \times _ S Z \to W \times _ S X$. By the Yoneda lemma $b_{univ}$ determines a transformation of functors $W \to \mathit{Mor}_ S(Z, X)$ which we claim is an isomorphism. To show that it is an isomorphism it suffices to show that it induces a bijection of sets $W(T) \to \mathit{Mor}_ S(Z, X)(T)$ over any affine scheme $T$.

Suppose $T = \mathop{\mathrm{Spec}}(R')$ is an affine scheme over $S$ and $b \in \mathit{Mor}_ S(Z, X)(T)$. The structure morphism $T \to S$ defines an $R$-algebra structure on $R'$ and $b$ defines an $R'$-algebra map

\[ b^\sharp : R' \otimes _ R \Gamma (X, \mathcal{O}_ X) \longrightarrow R' \otimes _ R \Gamma (Z, \mathcal{O}_ Z). \]

In particular we can write $b^\sharp (1 \otimes \overline{x}_ i) = \sum \alpha _{ij} \otimes e_ j$ for some $\alpha _{ij} \in R'$. This corresponds to an $R$-algebra map $P \to R'$ determined by the rule $a_{ij} \mapsto \alpha _{ij}$. This map factors through the quotient $P/J$ by the construction of the ideal $J$ to give a map $P/J \to R'$. This in turn corresponds to a morphism $T \to W$ such that $b$ is the pullback of $b_{univ}$. Some details omitted. $\square$

Lemma 37.68.2. Let $Z \to S$ and $X \to S$ be morphisms of schemes. If $Z \to S$ is finite locally free and $X \to S$ is affine, then $\mathit{Mor}_ S(Z, X)$ is representable by a scheme affine over $S$.

Proof. Choose an affine open covering $S = \bigcup U_ i$ such that $\Gamma (Z \times _ S U_ i, \mathcal{O}_{Z \times _ S U_ i})$ is finite free over $\mathcal{O}_ S(U_ i)$. Let $F_ i \subset \mathit{Mor}_ S(Z, X)$ be the subfunctor which assigns to $T/S$ the empty set if $T \to S$ does not factor through $U_ i$ and $\mathit{Mor}_ S(Z, X)(T)$ otherwise. Then the collection of these subfunctors satisfy the conditions (2)(a), (2)(b), (2)(c) of Schemes, Lemma 26.15.4 which proves the lemma. Condition (2)(a) follows from Lemma 37.68.1 and the other two follow from straightforward arguments. $\square$

The condition on the morphism $f : X \to S$ in the lemma below is very useful to prove statements like it. It holds if one of the following is true: $X$ is quasi-affine, $f$ is quasi-affine, $f$ is quasi-projective, $f$ is locally projective, there exists an ample invertible sheaf on $X$, there exists an $f$-ample invertible sheaf on $X$, or there exists an $f$-very ample invertible sheaf on $X$.

Lemma 37.68.3. Let $Z \to S$ and $X \to S$ be morphisms of schemes. Assume

  1. $Z \to S$ is finite locally free, and

  2. for all $(s, x_1, \ldots , x_ d)$ where $s \in S$ and $x_1, \ldots , x_ d \in X_ s$ there exists an affine open $U \subset X$ with $x_1, \ldots , x_ d \in U$.

Then $\mathit{Mor}_ S(Z, X)$ is representable by a scheme.

Proof. Consider the set $I$ of pairs $(U, V)$ where $U \subset X$ and $V \subset S$ are affine open and $U \to S$ factors through $V$. For $i \in I$ denote $(U_ i, V_ i)$ the corresponding pair. Set $F_ i = \mathit{Mor}_{V_ i}(Z_{V_ i}, U_ i)$. It is immediate that $F_ i$ is a subfunctor of $\mathit{Mor}_ S(Z, X)$. Then we claim that conditions (2)(a), (2)(b), (2)(c) of Schemes, Lemma 26.15.4 which proves the lemma.

Condition (2)(a) follows from Lemma 37.68.2.

To check condition (2)(b) consider $T/S$ and $b \in \mathit{Mor}_ S(Z, X)$. Thinking of $b$ as a morphism $T \times _ S Z \to X$ we find an open $b^{-1}(U_ i) \subset T \times _ S Z$. Clearly, $b \in F_ i(T)$ if and only if $b^{-1}(U_ i) = T \times _ S Z$. Since the projection $p : T \times _ S Z \to T$ is finite hence closed, the set $U_{i, b} \subset T$ of points $t \in T$ with $p^{-1}(\{ t\} ) \subset b^{-1}(U_ i)$ is open. Then $f : T' \to T$ factors through $U_{i, b}$ if and only if $b \circ f \in F_ i(T')$ and we are done checking (2)(b).

Finally, we check condition (2)(c) and this is where our condition on $X \to S$ is used. Namely, consider $T/S$ and $b \in \mathit{Mor}_ S(Z, X)$. It suffices to prove that every $t \in T$ is contained in one of the opens $U_{i, b}$ defined in the previous paragraph. This is equivalent to the condition that $b(p^{-1}(\{ t\} )) \subset U_ i$ for some $i$ where $p : T \times _ S Z \to T$ is the projection and $b : T \times _ S Z \to X$ is the given morphism. Since $p$ is finite, the set $b(p^{-1}(\{ t\} )) \subset X$ is finite and contained in the fibre of $X \to S$ over the image $s$ of $t$ in $S$. Thus our condition on $X \to S$ exactly shows a suitable pair exists. $\square$

Lemma 37.68.4. Let $Z \to S$ and $X \to S$ be morphisms of schemes. Assume $Z \to S$ is finite locally free and $X \to S$ is separated and locally quasi-finite. Then $\mathit{Mor}_ S(Z, X)$ is representable by a scheme.

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