Move some material on relative morphisms earlier

Also expand a bit on when it is representable

criteria.tex

@@ -1212,6 +1212,10 @@ \section{Relative morphisms}
 \label{section-relative-morphisms}
 
 \noindent
+We continue the discussion started in
+\'Etale Morphisms, Section \ref{etale-section-relative-morphisms}.
+
+\medskip\noindent
 Let $S$ be a scheme. Let $Z \to B$ and $X \to B$ be morphisms of
 algebraic spaces over $S$. Given a scheme $T$ we can consider pairs
 $(a, b)$ where $a : T \to B$
@@ -1371,77 +1375,6 @@ \section{Relative morphisms}
 which guarantees that $F \to U$ is surjective in the situation above.
 \end{proof}
 
-\begin{lemma}
-\label{lemma-hom-from-finite-free-into-affine}
-Let $Z \to B$ and $X \to B$ be morphisms of affine schemes.
-Assume $\Gamma(Z, \mathcal{O}_Z)$ is a finite free
-$\Gamma(B, \mathcal{O}_B)$-module. Then $\mathit{Mor}_B(Z, X)$
-is representable by an affine scheme over $B$.
-\end{lemma}
-
-\begin{proof}
-Write $B = \Spec(R)$. Choose a basis $\{e_1, \ldots, e_m\}$
-for $\Gamma(Z, \mathcal{O}_Z)$. Finally, 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
-$W = \Spec(P/J)$ represents the functor $\mathit{Mor}_B(Z, X)$.
-
-\medskip\noindent
-First, note that by construction $P/J$ is an $R$-algebra, hence
-a morphism $a_{univ} : W \to B$. 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
-$b_{univ} : W \times_{a_{univ}, B} Z \to W \times_{a_{univ}, B} X$.
-By the Yoneda lemma the pair $(a_{univ}, b_{univ})$ determines a
-transformation of functors $W \to \mathit{Mor}_B(Z, X)$ which we
-claim is an isomorphism. To show that it is an isomorphism it suffices
-to show that it induces a bijection of sets
-$W(T) \to \mathit{Mor}_B(Z, X)(T)$ over any affine
-scheme $T$.
-
-\medskip\noindent
-Suppose $T = \Spec(R')$ is an affine scheme
-and $(a, b) \in \mathit{Mor}_B(Z, X)(T)$, then $a$ 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 $(a, b)$ is the pullback of $(a_{univ}, b_{univ})$.
-Some details omitted.
-\end{proof}
-
 \begin{proposition}
 \label{proposition-hom-functor-algebraic-space}
 Let $S$ be a scheme. Let $Z \to B$ and $X \to B$ be morphisms of
@@ -1488,8 +1421,8 @@ \section{Relative morphisms}
 $$
 F_E = \mathit{Mor}_B(Z, \coprod\nolimits_{i \in E} X_i).
 $$
-By
-Lemma \ref{lemma-hom-from-finite-free-into-affine}
+By \'Etale Morphisms,
+Lemma \ref{etale-lemma-hom-from-finite-free-into-affine}
 we see that $F_E$ is an algebraic space. Consider the morphism
 $$
 \coprod\nolimits_{E \subset I\text{ finite}} F_E
@@ -1508,7 +1441,8 @@ \section{Relative morphisms}
 $\coprod F_E$ is an algebraic space. This follows because
 $|I| \leq \text{size}(X)$ and $\text{size}(F_E) \leq \text{size}(X)$
 as follows from the explicit description of $F_E$ in the proof of
-Lemma \ref{lemma-hom-from-finite-free-into-affine}.
+\'Etale Morphisms,
+Lemma \ref{etale-lemma-hom-from-finite-free-into-affine}.
 Some details omitted.} by
 Spaces, Lemma \ref{spaces-lemma-glueing-algebraic-spaces}.
 \end{proof}

etale.tex

@@ -1957,6 +1957,215 @@ \section{Permanence properties}
 formation of the completed local rings -- the details are left to the reader.
 
 
+
+
+
+
+
+\section{Relative morphisms}
+\label{section-relative-morphisms}
+
+\noindent
+We interrupt the discussion of \'etale morphisms to prove a
+representability result which we will use in the next section
+to discuss the category of finite \'etale coverings.
+The material in this section is discussed in the correct
+generality in Criteria for Representability, Section
+\ref{criteria-section-relative-morphisms}.
+
+\medskip\noindent
+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{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{equation-hom-functor}
+\mathit{Mor}_S(Z, X) : (\Sch/S)^{opp} \longrightarrow \textit{Sets},
+\quad
+T \longmapsto \{b\text{ as above}\}
+\end{equation}
+Here is a basic representability result.
+
+\begin{lemma}
+\label{lemma-hom-from-finite-free-into-affine}
+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 $S$.
+\end{lemma}
+
+\begin{proof}
+Write $S = \Spec(R)$. 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
+$W = \Spec(P/J)$ represents the functor $\mathit{Mor}_S(Z, X)$.
+
+\medskip\noindent
+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 $T$.
+
+\medskip\noindent
+Suppose $T = \Spec(R')$ 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.
+\end{proof}
+
+\begin{lemma}
+\label{lemma-hom-from-finite-locally-free-into-affine}
+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 $S$.
+\end{lemma}
+
+\begin{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 \ref{schemes-lemma-glue-functors} which proves the lemma.
+Condition (2)(a) follows from
+Lemma \ref{lemma-hom-from-finite-free-into-affine}
+and the other two follow from straightforward arguments.
+\end{proof}
+
+\noindent
+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 $X$.
+
+\begin{lemma}
+\label{lemma-hom-from-finite-locally-free-representable}
+Let $Z \to S$ and $X \to S$ be morphisms of schemes.
+Assume
+\begin{enumerate}
+\item $Z \to S$ is finite locally free, and
+\item 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 $x_1, \ldots, x_d \in U$.
+\end{enumerate}
+Then $\mathit{Mor}_S(Z, X)$ is representable by a scheme.
+\end{lemma}
+
+\begin{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 \ref{schemes-lemma-glue-functors} which proves the lemma.
+
+\medskip\noindent
+Condition (2)(a) follows from
+Lemma \ref{lemma-hom-from-finite-locally-free-into-affine}.
+
+\medskip\noindent
+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).
+
+\medskip\noindent
+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.
+\end{proof}
+
+
+
+
+
 \input{chapters}
 
 

tags/tags

@@ -7640,7 +7640,7 @@
 05Y3,criteria-lemma-hom-functor-sheaf
 05Y4,criteria-lemma-base-change-hom-functor
 05Y5,criteria-lemma-etale-covering-hom-functor
-05Y6,criteria-lemma-hom-from-finite-free-into-affine
+05Y6,etale-lemma-hom-from-finite-free-into-affine
 05Y7,criteria-proposition-hom-functor-algebraic-space
 05Y8,criteria-section-restriction-of-scalars
 05Y9,criteria-equation-pairs