The Stacks project

96.10 Relative morphisms

We continue the discussion started in More on Morphisms, Section 37.66.

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 $a : T \to B$ is a morphism and $b : T \times _{a, B} Z \to T \times _{a, B} X$ is a morphism over $T$. Picture

96.10.0.1
\begin{equation} \label{criteria-equation-hom} \vcenter { \xymatrix{ T \times _{a, B} Z \ar[rd] \ar[rr]_ b & & T \times _{a, B} X \ar[ld] & Z \ar[rd] & & X \ar[ld] \\ & T \ar[rrr]^ a & & & B } } \end{equation}

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

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

commutes. In this situation we can define a functor

96.10.0.2
\begin{equation} \label{criteria-equation-hom-functor} \mathit{Mor}_ B(Z, X) : (\mathit{Sch}/S)^{opp} \longrightarrow \textit{Sets}, \quad T \longmapsto \{ (a, b)\text{ as above}\} \end{equation}

Sometimes we think of this as a functor defined on the category of schemes over $B$, in which case we drop $a$ from the notation.

Lemma 96.10.1. Let $S$ be a scheme. Let $Z \to B$ and $X \to B$ be morphisms of algebraic spaces over $S$. Then

  1. $\mathit{Mor}_ B(Z, X)$ is a sheaf on $(\mathit{Sch}/S)_{fppf}$.

  2. If $T$ is an algebraic space over $S$, then there is a canonical bijection

    \[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf})}(T, \mathit{Mor}_ B(Z, X)) = \{ (a, b)\text{ as in }(05Y1)\} \]

Proof. Let $T$ be an algebraic space over $S$. Let $\{ T_ i \to T\} $ be an fppf covering of $T$ (as in Topologies on Spaces, Section 72.7). Suppose that $(a_ i, b_ i) \in \mathit{Mor}_ B(Z, X)(T_ i)$ such that $(a_ i, b_ i)|_{T_ i \times _ T T_ j} = (a_ j, b_ j)|_{T_ i \times _ T T_ j}$ for all $i, j$. Then by Descent on Spaces, Lemma 73.7.2 there exists a unique morphism $a : T \to B$ such that $a_ i$ is the composition of $T_ i \to T$ and $a$. Then $\{ T_ i \times _{a_ i, B} Z \to T \times _{a, B} Z\} $ is an fppf covering too and the same lemma implies there exists a unique morphism $b : T \times _{a, B} Z \to T \times _{a, B} X$ such that $b_ i$ is the composition of $T_ i \times _{a_ i, B} Z \to T \times _{a, B} Z$ and $b$. Hence $(a, b) \in \mathit{Mor}_ B(Z, X)(T)$ restricts to $(a_ i, b_ i)$ over $T_ i$ for all $i$.

Note that the result of the preceding paragraph in particular implies (1).

Let $T$ be an algebraic space over $S$. In order to prove (2) we will construct mutually inverse maps between the displayed sets. In the following when we say “pair” we mean a pair $(a, b)$ fitting into (96.10.0.1).

Let $v : T \to \mathit{Mor}_ B(Z, X)$ be a natural transformation. Choose a scheme $U$ and a surjective étale morphism $p : U \to T$. Then $v(p) \in \mathit{Mor}_ B(Z, X)(U)$ corresponds to a pair $(a_ U, b_ U)$ over $U$. Let $R = U \times _ T U$ with projections $t, s : R \to U$. As $v$ is a transformation of functors we see that the pullbacks of $(a_ U, b_ U)$ by $s$ and $t$ agree. Hence, since $\{ U \to T\} $ is an fppf covering, we may apply the result of the first paragraph that deduce that there exists a unique pair $(a, b)$ over $T$.

Conversely, let $(a, b)$ be a pair over $T$. Let $U \to T$, $R = U \times _ T U$, and $t, s : R \to U$ be as above. Then the restriction $(a, b)|_ U$ gives rise to a transformation of functors $v : h_ U \to \mathit{Mor}_ B(Z, X)$ by the Yoneda lemma (Categories, Lemma 4.3.5). As the two pullbacks $s^*(a, b)|_ U$ and $t^*(a, b)|_ U$ are equal, we see that $v$ coequalizes the two maps $h_ t, h_ s : h_ R \to h_ U$. Since $T = U/R$ is the fppf quotient sheaf by Spaces, Lemma 64.9.1 and since $\mathit{Mor}_ B(Z, X)$ is an fppf sheaf by (1) we conclude that $v$ factors through a map $T \to \mathit{Mor}_ B(Z, X)$.

We omit the verification that the two constructions above are mutually inverse. $\square$

Lemma 96.10.2. Let $S$ be a scheme. Let $Z \to B$, $X \to B$, and $B' \to B$ be morphisms of algebraic spaces over $S$. Set $Z' = B' \times _ B Z$ and $X' = B' \times _ B X$. Then

\[ \mathit{Mor}_{B'}(Z', X') = B' \times _ B \mathit{Mor}_ B(Z, X) \]

in $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf})$.

Proof. The equality as functors follows immediately from the definitions. The equality as sheaves follows from this because both sides are sheaves according to Lemma 96.10.1 and the fact that a fibre product of sheaves is the same as the corresponding fibre product of pre-sheaves (i.e., functors). $\square$

Lemma 96.10.3. Let $S$ be a scheme. Let $Z \to B$ and $X' \to X \to B$ be morphisms of algebraic spaces over $S$. Assume

  1. $X' \to X$ is étale, and

  2. $Z \to B$ is finite locally free.

Then $\mathit{Mor}_ B(Z, X') \to \mathit{Mor}_ B(Z, X)$ is representable by algebraic spaces and étale. If $X' \to X$ is also surjective, then $\mathit{Mor}_ B(Z, X') \to \mathit{Mor}_ B(Z, X)$ is surjective.

Proof. Let $U$ be a scheme and let $\xi = (a, b)$ be an element of $\mathit{Mor}_ B(Z, X)(U)$. We have to prove that the functor

\[ h_ U \times _{\xi , \mathit{Mor}_ B(Z, X)} \mathit{Mor}_ B(Z, X') \]

is representable by an algebraic space étale over $U$. Set $Z_ U = U \times _{a, B} Z$ and $W = Z_ U \times _{b, X} X'$. Then $W \to Z_ U \to U$ is as in Lemma 96.9.2 and the sheaf $F$ defined there is identified with the fibre product displayed above. Hence the first assertion of the lemma. The second assertion follows from this and Lemma 96.9.1 which guarantees that $F \to U$ is surjective in the situation above. $\square$

Proposition 96.10.4. Let $S$ be a scheme. Let $Z \to B$ and $X \to B$ be morphisms of algebraic spaces over $S$. If $Z \to B$ is finite locally free then $\mathit{Mor}_ B(Z, X)$ is an algebraic space.

Proof. Choose a scheme $B' = \coprod B'_ i$ which is a disjoint union of affine schemes $B'_ i$ and an étale surjective morphism $B' \to B$. We may also assume that $B'_ i \times _ B Z$ is the spectrum of a ring which is finite free as a $\Gamma (B'_ i, \mathcal{O}_{B'_ i})$-module. By Lemma 96.10.2 and Spaces, Lemma 64.5.5 the morphism $\mathit{Mor}_{B'}(Z', X') \to \mathit{Mor}_ B(Z, X)$ is surjective étale. Hence by Bootstrap, Theorem 79.10.1 it suffices to prove the proposition when $B = B'$ is a disjoint union of affine schemes $B'_ i$ so that each $B'_ i \times _ B Z$ is finite free over $B'_ i$. Then it actually suffices to prove the result for the restriction to each $B'_ i$. Thus we may assume that $B$ is affine and that $\Gamma (Z, \mathcal{O}_ Z)$ is a finite free $\Gamma (B, \mathcal{O}_ B)$-module.

Choose a scheme $X'$ which is a disjoint union of affine schemes and a surjective étale morphism $X' \to X$. By Lemma 96.10.3 the morphism $\mathit{Mor}_ B(Z, X') \to \mathit{Mor}_ B(Z, X)$ is representable by algebraic spaces, étale, and surjective. Hence by Bootstrap, Theorem 79.10.1 it suffices to prove the proposition when $X$ is a disjoint union of affine schemes. This reduces us to the case discussed in the next paragraph.

Assume $X = \coprod _{i \in I} X_ i$ is a disjoint union of affine schemes, $B$ is affine, and that $\Gamma (Z, \mathcal{O}_ Z)$ is a finite free $\Gamma (B, \mathcal{O}_ B)$-module. For any finite subset $E \subset I$ set

\[ F_ E = \mathit{Mor}_ B(Z, \coprod \nolimits _{i \in E} X_ i). \]

By More on Morphisms, Lemma 37.66.1 we see that $F_ E$ is an algebraic space. Consider the morphism

\[ \coprod \nolimits _{E \subset I\text{ finite}} F_ E \longrightarrow \mathit{Mor}_ B(Z, X) \]

Each of the morphisms $F_ E \to \mathit{Mor}_ B(Z, X)$ is an open immersion, because it is simply the locus parametrizing pairs $(a, b)$ where $b$ maps into the open subscheme $\coprod \nolimits _{i \in E} X_ i$ of $X$. Moreover, if $T$ is quasi-compact, then for any pair $(a, b)$ the image of $b$ is contained in $\coprod \nolimits _{i \in E} X_ i$ for some $E \subset I$ finite. Hence the displayed arrow is in fact an open covering and we win1 by Spaces, Lemma 64.8.5. $\square$

[1] Modulo some set theoretic arguments. Namely, we have to show that $\coprod F_ E$ 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 More on Morphisms, Lemma 37.66.1. Some details omitted.

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