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criteria.tex
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\input{preamble}
% OK, start here.
%
\begin{document}
\title{Criteria for Representability}
\maketitle
\phantomsection
\label{section-phantom}
\tableofcontents
\section{Introduction}
\label{section-introduction}
\noindent
The purpose of this chapter is to find criteria guaranteeing that a
stack in groupoids over the category of schemes with the fppf topology
is an algebraic stack. Historically, this often involved proving that
certain functors were representable, see Grothendieck's lectures
\cite{Gr-I},
\cite{Gr-II},
\cite{Gr-III},
\cite{Gr-IV},
\cite{Gr-V}, and
\cite{Gr-VI}.
This explains the title of this chapter. Another important source
of this material comes from the work of Artin, see
\cite{ArtinI},
\cite{ArtinII},
\cite{Artin-Theorem-Representability},
\cite{Artin-Construction-Techniques},
\cite{Artin-Algebraic-Spaces},
\cite{Artin-Algebraic-Approximation},
\cite{Artin-Implicit-Function}, and
\cite{ArtinVersal}.
\medskip\noindent
Some of the notation, conventions and terminology in this chapter is awkward
and may seem backwards to the more experienced reader. This is intentional.
Please see Quot, Section \ref{quot-section-introduction} for an
explanation.
\section{Conventions}
\label{section-conventions}
\noindent
The conventions we use in this chapter are the same as those in the
chapter on algebraic stacks, see
Algebraic Stacks, Section \ref{algebraic-section-conventions}.
\section{What we already know}
\label{section-done-so-far}
\noindent
The analogue of this chapter for algebraic spaces is the chapter entitled
``Bootstrap'', see
Bootstrap, Section \ref{bootstrap-section-introduction}.
That chapter already contains some representability results.
Moreover, some of the preliminary material treated there we already
have worked out in the chapter on algebraic stacks.
Here is a list:
\begin{enumerate}
\item We discuss morphisms of presheaves representable by algebraic spaces in
Bootstrap, Section
\ref{bootstrap-section-morphism-representable-by-spaces}.
In
Algebraic Stacks, Section
\ref{algebraic-section-morphisms-representable-by-algebraic-spaces}
we discuss the notion of a $1$-morphism of categories fibred in groupoids
being representable by algebraic spaces.
\item We discuss properties of morphisms of presheaves representable by
algebraic spaces in
Bootstrap, Section
\ref{bootstrap-section-representable-by-spaces-properties}.
In
Algebraic Stacks, Section
\ref{algebraic-section-representable-properties}
we discuss the notion of a $1$-morphism of categories fibred in groupoids
being representable by algebraic spaces.
\item We proved that if $F$ is a sheaf whose diagonal is representable
by algebraic spaces and which has an \'etale covering by an algebraic
space, then $F$ is an algebraic space, see
Bootstrap, Theorem \ref{bootstrap-theorem-bootstrap}.
(This is a weak version of the result in the next item on the list.)
\item
\label{item-bootstrap-final}
We proved that if $F$ is a sheaf and if there exists an algebraic
space $U$ and a morphism $U \to F$ which is representable by algebraic
spaces, surjective, flat, and locally of finite presentation, then
$F$ is an algebraic space, see
Bootstrap, Theorem \ref{bootstrap-theorem-final-bootstrap}.
\item We have also proved the ``smooth'' analogue of
(\ref{item-bootstrap-final}) for algebraic
stacks: If $\mathcal{X}$ is a stack in groupoids over
$(\Sch/S)_{fppf}$ and if there exists a stack in groupoids
$\mathcal{U}$ over $(\Sch/S)_{fppf}$ which is representable
by an algebraic space and a $1$-morphism $u : \mathcal{U} \to \mathcal{X}$
which is representable by algebraic spaces, surjective, and smooth
then $\mathcal{X}$ is an algebraic stack, see
Algebraic Stacks, Lemma
\ref{algebraic-lemma-smooth-surjective-morphism-implies-algebraic}.
\end{enumerate}
Our first task now is to prove the analogue of
(\ref{item-bootstrap-final}) for algebraic
stacks in general; it is
Theorem \ref{theorem-bootstrap}.
\section{Morphisms of stacks in groupoids}
\label{section-1-morphisms}
\noindent
This section is preliminary and should be skipped on a first reading.
\begin{lemma}
\label{lemma-etale-permanence}
Let $\mathcal{X} \to \mathcal{Y} \to \mathcal{Z}$
be $1$-morphisms of categories fibred in groupoids over
$(\Sch/S)_{fppf}$.
If $\mathcal{X} \to \mathcal{Z}$ and $\mathcal{Y} \to \mathcal{Z}$ are
representable by algebraic spaces and \'etale so is
$\mathcal{X} \to \mathcal{Y}$.
\end{lemma}
\begin{proof}
Let $\mathcal{U}$ be a representable category fibred in groupoids over $S$.
Let $f : \mathcal{U} \to \mathcal{Y}$ be a $1$-morphism. We have to show that
$\mathcal{X} \times_\mathcal{Y} \mathcal{U}$ is representable by an
algebraic space and \'etale over $\mathcal{U}$.
Consider the composition $h : \mathcal{U} \to \mathcal{Z}$. Then
$$
\mathcal{X} \times_\mathcal{Z} \mathcal{U}
\longrightarrow
\mathcal{Y} \times_\mathcal{Z} \mathcal{U}
$$
is a $1$-morphism between categories fibres in groupoids which are both
representable by algebraic spaces and both \'etale over $\mathcal{U}$.
Hence by
Properties of Spaces, Lemma \ref{spaces-properties-lemma-etale-permanence}
this is represented by an \'etale morphism of algebraic spaces.
Finally, we obtain the result we want as the morphism $f$ induces
a morphism $\mathcal{U} \to \mathcal{Y} \times_\mathcal{Z} \mathcal{U}$
and we have
$$
\mathcal{X} \times_\mathcal{Y} \mathcal{U} =
(\mathcal{X} \times_\mathcal{Z} \mathcal{U})
\times_{(\mathcal{Y} \times_\mathcal{Z} \mathcal{U})}
\mathcal{U}.
$$
\end{proof}
\begin{lemma}
\label{lemma-stack-in-setoids-descent}
Let $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ be stacks in groupoids
over $(\Sch/S)_{fppf}$. Suppose that $\mathcal{X} \to \mathcal{Y}$
and $\mathcal{Z} \to \mathcal{Y}$ are $1$-morphisms.
If
\begin{enumerate}
\item $\mathcal{Y}$, $\mathcal{Z}$ are representable by algebraic spaces
$Y$, $Z$ over $S$,
\item the associated morphism of algebraic spaces $Y \to Z$ is surjective,
flat and locally of finite presentation, and
\item $\mathcal{Y} \times_\mathcal{Z} \mathcal{X}$ is a stack in
setoids,
\end{enumerate}
then $\mathcal{X}$ is a stack in setoids.
\end{lemma}
\begin{proof}
This is a special case of
Stacks, Lemma \ref{stacks-lemma-stack-in-setoids-descent}.
\end{proof}
\noindent
The following lemma is the analogue of
Algebraic Stacks, Lemma
\ref{algebraic-lemma-smooth-surjective-morphism-implies-algebraic}
and will be superseded by the stronger
Theorem \ref{theorem-bootstrap}.
\begin{lemma}
\label{lemma-flat-finite-presentation-surjective-diagonal}
Let $S$ be a scheme.
Let $u : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of
stacks in groupoids over $(\Sch/S)_{fppf}$. If
\begin{enumerate}
\item $\mathcal{U}$ is representable by an algebraic space, and
\item $u$ is representable by algebraic spaces, surjective, flat and
locally of finite presentation,
\end{enumerate}
then
$\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$
representable by algebraic spaces.
\end{lemma}
\begin{proof}
Given two schemes $T_1$, $T_2$ over $S$ denote
$\mathcal{T}_i = (\Sch/T_i)_{fppf}$ the associated representable
fibre categories. Suppose given $1$-morphisms
$f_i : \mathcal{T}_i \to \mathcal{X}$.
According to
Algebraic Stacks, Lemma \ref{algebraic-lemma-representable-diagonal}
it suffices to prove that the $2$-fibered
product $\mathcal{T}_1 \times_\mathcal{X} \mathcal{T}_2$
is representable by an algebraic space. By
Stacks, Lemma
\ref{stacks-lemma-2-fibre-product-stacks-in-setoids-over-stack-in-groupoids}
this is in any case a stack in setoids. Thus
$\mathcal{T}_1 \times_\mathcal{X} \mathcal{T}_2$ corresponds
to some sheaf $F$ on $(\Sch/S)_{fppf}$, see
Stacks, Lemma \ref{stacks-lemma-stack-in-setoids-characterize}.
Let $U$ be the algebraic space which represents $\mathcal{U}$.
By assumption
$$
\mathcal{T}_i' = \mathcal{U} \times_{u, \mathcal{X}, f_i} \mathcal{T}_i
$$
is representable by an algebraic space $T'_i$ over $S$. Hence
$\mathcal{T}_1' \times_\mathcal{U} \mathcal{T}_2'$ is representable
by the algebraic space $T'_1 \times_U T'_2$.
Consider the commutative diagram
$$
\xymatrix{
&
\mathcal{T}_1 \times_{\mathcal X} \mathcal{T}_2 \ar[rr]\ar'[d][dd] & &
\mathcal{T}_1 \ar[dd] \\
\mathcal{T}_1' \times_\mathcal{U} \mathcal{T}_2' \ar[ur]\ar[rr]\ar[dd] & &
\mathcal{T}_1' \ar[ur]\ar[dd] \\
&
\mathcal{T}_2 \ar'[r][rr] & &
\mathcal X \\
\mathcal{T}_2' \ar[rr]\ar[ur] & &
\mathcal{U} \ar[ur] }
$$
In this diagram the bottom square, the right square, the back square, and
the front square are $2$-fibre products. A formal argument then shows
that $\mathcal{T}_1' \times_\mathcal{U} \mathcal{T}_2' \to
\mathcal{T}_1 \times_{\mathcal X} \mathcal{T}_2$
is the ``base change'' of $\mathcal{U} \to \mathcal{X}$, more precisely
the diagram
$$
\xymatrix{
\mathcal{T}_1' \times_\mathcal{U} \mathcal{T}_2' \ar[d] \ar[r] &
\mathcal{U} \ar[d] \\
\mathcal{T}_1 \times_{\mathcal X} \mathcal{T}_2 \ar[r] &
\mathcal{X}
}
$$
is a $2$-fibre square.
Hence $T'_1 \times_U T'_2 \to F$ is representable by algebraic spaces,
flat, locally of finite presentation and surjective, see
Algebraic Stacks, Lemmas
\ref{algebraic-lemma-map-fibred-setoids-representable-algebraic-spaces},
\ref{algebraic-lemma-base-change-representable-by-spaces},
\ref{algebraic-lemma-map-fibred-setoids-property}, and
\ref{algebraic-lemma-base-change-representable-transformations-property}.
Therefore $F$ is an algebraic space by
Bootstrap, Theorem \ref{bootstrap-theorem-final-bootstrap}
and we win.
\end{proof}
\begin{lemma}
\label{lemma-second-diagonal}
Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$.
The following are equivalent
\begin{enumerate}
\item $\Delta_\Delta : \mathcal{X} \to
\mathcal{X} \times_{\mathcal{X} \times \mathcal{X}} \mathcal{X}$
is representable by algebraic spaces,
\item for every $1$-morphism $\mathcal{V} \to \mathcal{X} \times \mathcal{X}$
with $\mathcal{V}$ representable (by a scheme) and the fibre product
$\mathcal{Y} =
\mathcal{X} \times_{\Delta, \mathcal{X} \times \mathcal{X}} \mathcal{V}$
has diagonal representable by algebraic spaces.
\end{enumerate}
\end{lemma}
\begin{proof}
Although this is a bit of a brain twister, it is completely formal.
Namely, recall that
$\mathcal{X} \times_{\mathcal{X} \times \mathcal{X}} \mathcal{X} =
\mathcal{I}_\mathcal{X}$ is the inertia of $\mathcal{X}$ and that
$\Delta_\Delta$ is the identity section of $\mathcal{I}_\mathcal{X}$, see
Categories, Section \ref{categories-section-inertia}.
Thus condition (1) says the following: Given a scheme $V$, an object $x$ of
$\mathcal{X}$ over $V$, and a morphism $\alpha : x \to x$ of $\mathcal{X}_V$
the condition ``$\alpha = \text{id}_x$'' defines an algebraic space over $V$.
(In other words, there exists a monomorphism of algebraic spaces $W \to V$
such that a morphism of schemes $f : T \to V$ factors through $W$
if and only if $f^*\alpha = \text{id}_{f^*x}$.)
\medskip\noindent
On the other hand, let $V$ be a scheme and let $x, y$ be objects of
$\mathcal{X}$ over $V$. Then $(x, y)$ define a morphism
$\mathcal{V} = (\Sch/V)_{fppf} \to \mathcal{X} \times \mathcal{X}$.
Next, let $h : V' \to V$ be a morphism of schemes and let
$\alpha : h^*x \to h^*y$ and $\beta : h^*x \to h^*y$ be morphisms
of $\mathcal{X}_{V'}$. Then $(\alpha, \beta)$ define a morphism
$\mathcal{V}' = (\Sch/V)_{fppf} \to \mathcal{Y} \times \mathcal{Y}$.
Condition (2) now says that (with any choices as above) the
condition ``$\alpha = \beta$'' defines an algebraic space over $V$.
\medskip\noindent
To see the equivalence, given $(\alpha, \beta)$ as in (2) we see that
(1) implies that ``$\alpha^{-1} \circ \beta = \text{id}_{h^*x}$''
defines an algebraic space. The implication (2) $\Rightarrow$ (1)
follows by taking $h = \text{id}_V$ and $\beta = \text{id}_x$.
\end{proof}
\section{Limit preserving on objects}
\label{section-limit-preserving}
\noindent
Let $S$ be a scheme. Let $p : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism
of categories fibred in groupoids over $(\Sch/S)_{fppf}$. We will say that
$p$ is {\it limit preserving on objects} if the following condition holds:
Given any data consisting of
\begin{enumerate}
\item an affine scheme $U = \lim_{i \in I} U_i$ which is written as the
directed limit of affine schemes $U_i$ over $S$,
\item an object $y_i$ of $\mathcal{Y}$ over $U_i$ for some $i$,
\item an object $x$ of $\mathcal{X}$ over $U$, and
\item an isomorphism $\gamma : p(x) \to y_i|_U$,
\end{enumerate}
then there exists an $i' \geq i$, an object $x_{i'}$ of
$\mathcal{X}$ over $U_{i'}$, an isomorphism
$\beta : x_{i'}|_U \to x$, and an isomorphism
$\gamma_{i'} : p(x_{i'}) \to y_i|_{U_{i'}}$
such that
\begin{equation}
\label{equation-limit-preserving}
\vcenter{
\xymatrix{
p(x_{i'}|_U) \ar[d]_{p(\beta)} \ar[rr]_{\gamma_{i'}|_U} & &
(y_i|_{U_{i'}})|_U \ar@{=}[d] \\
p(x) \ar[rr]^\gamma & & y_i|_U
}
}
\end{equation}
commutes. In this situation we say that ``$(i', x_{i'}, \beta, \gamma_{i'})$
is a {\it solution} to the problem posed by our data (1), (2), (3), (4)''.
The motivation for this definition comes from
Limits of Spaces,
Lemma \ref{spaces-limits-lemma-characterize-relative-limit-preserving}.
\begin{lemma}
\label{lemma-base-change-limit-preserving}
Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$
be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$.
If $p : \mathcal{X} \to \mathcal{Y}$ is limit preserving on objects, then so
is the base change
$p' : \mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to \mathcal{Z}$
of $p$ by $q$.
\end{lemma}
\begin{proof}
This is formal. Let $U = \lim_{i \in I} U_i$ be the directed limit
of affine schemes $U_i$ over $S$, let $z_i$ be an object of $\mathcal{Z}$
over $U_i$ for some $i$, let $w$ be an object of
$\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$ over $U$, and let
$\delta : p'(w) \to z_i|_U$ be an isomorphism.
We may write
$w = (U, x, z, \alpha)$ for some object $x$ of $\mathcal{X}$ over $U$
and object $z$ of $\mathcal{Z}$ over $U$ and isomorphism
$\alpha : p(x) \to q(z)$. Note that $p'(w) = z$ hence
$\delta : z \to z_i|_U$. Set $y_i = q(z_i)$ and
$\gamma = q(\delta) \circ \alpha : p(x) \to y_i|_U$.
As $p$ is limit preserving on objects there exists an $i' \geq i$
and an object $x_{i'}$ of $\mathcal{X}$ over $U_{i'}$ as well as
isomorphisms $\beta : x_{i'}|_U \to x$ and
$\gamma_{i'} : p(x_{i'}) \to y_i|_{U_{i'}}$ such that
(\ref{equation-limit-preserving}) commutes. Then we consider the object
$w_{i'} = (U_{i'}, x_{i'}, z_i|_{U_{i'}}, \gamma_{i'})$ of
$\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$ over $U_{i'}$
and define isomorphisms
$$
w_{i'}|_U = (U, x_{i'}|_U, z_i|_U, \gamma_{i'}|_U)
\xrightarrow{(\beta, \delta^{-1})}
(U, x, z, \alpha) = w
$$
and
$$
p'(w_{i'}) = z_i|_{U_{i'}} \xrightarrow{\text{id}} z_i|_{U_{i'}}.
$$
These combine to give a solution to the problem.
\end{proof}
\begin{lemma}
\label{lemma-composition-limit-preserving}
Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Y} \to \mathcal{Z}$
be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$.
If $p$ and $q$ are limit preserving on objects, then so is the composition
$q \circ p$.
\end{lemma}
\begin{proof}
This is formal. Let $U = \lim_{i \in I} U_i$ be the directed limit
of affine schemes $U_i$ over $S$, let $z_i$ be an object of $\mathcal{Z}$
over $U_i$ for some $i$, let $x$ be an object of $\mathcal{X}$ over $U$,
and let $\gamma : q(p(x)) \to z_i|_U$ be an isomorphism. As $q$ is
limit preserving on objects there exist an $i' \geq i$, an object
$y_{i'}$ of $\mathcal{Y}$ over $U_{i'}$, an isomorphism
$\beta : y_{i'}|_U \to p(x)$, and an isomorphism
$\gamma_{i'} : q(y_{i'}) \to z_i|_{U_{i'}}$
such that (\ref{equation-limit-preserving}) is commutative. As $p$ is
limit preserving on objects there exist an $i'' \geq i'$, an object
$x_{i''}$ of $\mathcal{X}$ over $U_{i''}$, an isomorphism
$\beta' : x_{i''}|_U \to x$, and an isomorphism
$\gamma'_{i''} : p(x_{i''}) \to y_{i'}|_{U_{i''}}$
such that (\ref{equation-limit-preserving}) is commutative.
The solution is to take $x_{i''}$ over $U_{i''}$ with isomorphism
$$
q(p(x_{i''})) \xrightarrow{q(\gamma'_{i''})}
q(y_{i'})|_{U_{i''}} \xrightarrow{\gamma_{i'}|_{U_{i''}}}
z_i|_{U_{i''}}
$$
and isomorphism $\beta' : x_{i''}|_U \to x$. We omit the verification
that (\ref{equation-limit-preserving}) is commutative.
\end{proof}
\begin{lemma}
\label{lemma-representable-by-spaces-limit-preserving}
Let $p : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories
fibred in groupoids over $(\Sch/S)_{fppf}$. If $p$ is
representable by algebraic spaces, then the following are equivalent:
\begin{enumerate}
\item $p$ is limit preserving on objects, and
\item $p$ is locally of finite presentation (see
Algebraic Stacks,
Definition \ref{algebraic-definition-relative-representable-property}).
\end{enumerate}
\end{lemma}
\begin{proof}
Assume (2). Let $U = \lim_{i \in I} U_i$ be the directed limit
of affine schemes $U_i$ over $S$, let $y_i$ be an object of $\mathcal{Y}$
over $U_i$ for some $i$, let $x$ be an object of $\mathcal{X}$ over $U$,
and let $\gamma : p(x) \to y_i|_U$ be an isomorphism. Let
$X_{y_i}$ denote an algebraic space over $U_i$ representing the $2$-fibre
product
$$
(\Sch/U_i)_{fppf} \times_{y_i, \mathcal{Y}, p} \mathcal{X}.
$$
Note that $\xi = (U, U \to U_i, x, \gamma^{-1})$ defines an object of
this $2$-fibre product over $U$. Via the $2$-Yoneda lemma $\xi$ corresponds
to a morphism $f_\xi : U \to X_{y_i}$ over $U_i$. By
Limits of Spaces, Proposition
\ref{spaces-limits-proposition-characterize-locally-finite-presentation}
there exists an $i' \geq i$ and a morphism $f_{i'} : U_{i'} \to X_{y_i}$
such that $f_\xi$ is the composition of $f_{i'}$ and the projection
morphism $U \to U_{i'}$. Also, the $2$-Yoneda lemma tells us that
$f_{i'}$ corresponds to an object
$\xi_{i'} = (U_{i'}, U_{i'} \to U_i, x_{i'}, \alpha)$ of
the displayed $2$-fibre product over $U_{i'}$ whose restriction to
$U$ recovers $\xi$. In particular we obtain an isomorphism
$\gamma : x_{i'}|U \to x$. Note that $\alpha : y_i|_{U_{i'}} \to p(x_{i'})$.
Hence we see that taking $x_{i'}$, the isomorphism
$\gamma : x_{i'}|U \to x$, and the isomorphism
$\beta = \alpha^{-1} : p(x_{i'}) \to y_i|_{U_{i'}}$
is a solution to the problem.
\medskip\noindent
Assume (1). Choose a scheme $T$ and a $1$-morphism
$y : (\Sch/T)_{fppf} \to \mathcal{Y}$. Let
$X_y$ be an algebraic space over $T$ representing the $2$-fibre product
$(\Sch/T)_{fppf} \times_{y, \mathcal{Y}, p} \mathcal{X}$.
We have to show that $X_y \to T$ is locally of finite presentation.
To do this we may use
Limits of Spaces, Proposition
\ref{spaces-limits-proposition-characterize-locally-finite-presentation}
in the form described in
Limits of Spaces,
Remark \ref{spaces-limits-remark-limit-preserving}.
Hence it suffices to show that given an affine scheme
$U = \lim_{i \in I} U_i$ written as the directed limit of affine schemes
over $T$, then $X_y(U) = \colim_i X_y(U_i)$.
Pick any $i \in I$ and set $y_i = y|_{U_i}$. Also denote $i'$ an element
of $I$ which is bigger than or equal to $i$. By the $2$-Yoneda lemma
morphisms $U \to X_y$ over $T$ correspond bijectively
to isomorphism classes of pairs $(x, \alpha)$ where $x$ is an object
of $\mathcal{X}$ over $U$ and $\alpha : y|_U \to p(x)$ is an isomorphism.
Of course giving $\alpha$ is, up to an inverse, the same thing as giving
an isomorphism $\gamma : p(x) \to y_i|_U$.
Similarly for morphisms $U_{i'} \to X_y$ over $T$. Hence (1) guarantees
that
$$
X_y(U) = \colim_{i' \geq i} X_y(U_{i'})
$$
in this situation and we win.
\end{proof}
\begin{lemma}
\label{lemma-open-immersion-limit-preserving}
Let $p : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories
fibred in groupoids over $(\Sch/S)_{fppf}$. Assume $p$ is representable
by algebraic spaces and an open immersion. Then $p$ is limit preserving
on objects.
\end{lemma}
\begin{proof}
This follows from
Lemma \ref{lemma-representable-by-spaces-limit-preserving}
and (via the general principle
Algebraic Stacks, Lemma
\ref{algebraic-lemma-representable-transformations-property-implication})
from the fact that an open immersion of algebraic spaces is
locally of finite presentation, see
Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-open-immersion-locally-finite-presentation}.
\end{proof}
\noindent
Let $S$ be a scheme. In the following lemma we need the notion of the
{\it size} of an algebraic space $X$ over $S$. Namely, given a cardinal
$\kappa$ we will say $X$ has $\text{size}(X) \leq \kappa$ if and only
if there exists a scheme $U$ with $\text{size}(U) \leq \kappa$ (see
Sets, Section \ref{sets-section-categories-schemes}) and a surjective
\'etale morphism $U \to X$.
\begin{lemma}
\label{lemma-check-representable-limit-preserving}
Let $S$ be a scheme.
Let $\kappa = \text{size}(T)$ for some $T \in \Ob((\Sch/S)_{fppf})$.
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism
of categories fibred in groupoids over $(\Sch/S)_{fppf}$
such that
\begin{enumerate}
\item $\mathcal{Y} \to (\Sch/S)_{fppf}$ is limit preserving on objects,
\item for an affine scheme $V$ locally of finite presentation over $S$ and
$y \in \Ob(\mathcal{Y}_V)$ the fibre product
$(\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}$ is representable
by an algebraic space of size $\leq \kappa$\footnote{The condition on
size can be dropped by those ignoring set theoretic issues.},
\item $\mathcal{X}$ and $\mathcal{Y}$ are stacks for the Zariski topology.
\end{enumerate}
Then $f$ is representable by algebraic spaces.
\end{lemma}
\begin{proof}
Let $V$ be a scheme over $S$ and $y \in \mathcal{Y}_V$. We have to prove
$(\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}$ is representable
by an algebraic space.
\medskip\noindent
Case I: $V$ is affine and maps into an affine open $\Spec(\Lambda) \subset S$.
Then we can write $V = \lim V_i$ with each $V_i$ affine and of finite
presentation over $\Spec(\Lambda)$, see
Algebra, Lemma \ref{algebra-lemma-ring-colimit-fp}.
Then $y$ comes from an object $y_i$ over $V_i$ for some $i$ by assumption (1).
By assumption (3) the fibre product
$(\Sch/V_i)_{fppf} \times_{y_i, \mathcal{Y}} \mathcal{X}$ is representable
by an algebraic space $Z_i$. Then
$(\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}$ is representable
by $Z \times_{V_i} V$.
\medskip\noindent
Case II: $V$ is general. Choose an affine open covering
$V = \bigcup_{i \in I} V_i$ such that each $V_i$ maps into an affine open
of $S$. We first claim
that $\mathcal{Z} = (\Sch/V)_{fppf} \times_{y, \mathcal{Y}} \mathcal{X}$
is a stack in setoids for the Zariski topology. Namely, it is a stack in
groupoids for the Zariski topology by
Stacks, Lemma \ref{stacks-lemma-2-product-stacks-in-groupoids}.
Then suppose that $z$ is an object of $\mathcal{Z}$ over a scheme $T$.
Denote $g : T \to V$ the morphism corresponding to the
projection of $z$ in $(\Sch/V)_{fppf}$. Consider the Zariski sheaf
$\mathit{I} = \mathit{Isom}_{\mathcal{Z}}(z, z)$. By Case I we see that
$\mathit{I}|_{g^{-1}(V_i)} = *$ (the singleton sheaf). Hence
$\mathcal{I} = *$. Thus $\mathcal{Z}$ is fibred in setoids. To finish
the proof we have to show that the Zariski sheaf
$Z : T \mapsto \Ob(\mathcal{Z}_T)/\cong$ is an algebraic space, see
Algebraic Stacks, Lemma
\ref{algebraic-lemma-characterize-representable-by-space}.
There is a map $p : Z \to V$ (transformation of functors) and by Case I
we know that $Z_i = p^{-1}(V_i)$ is an algebraic space. The morphisms
$Z_i \to Z$ are representable by open immersions and
$\coprod Z_i \to Z$ is surjective (in the Zariski topology).
Hence $Z$ is a sheaf for the fppf topology by
Bootstrap, Lemma \ref{bootstrap-lemma-glueing-sheaves}.
Thus Spaces, Lemma \ref{spaces-lemma-glueing-algebraic-spaces}
applies and we conclude that $Z$ is an algebraic space\footnote{
To see that the set theoretic condition of that lemma is satisfied
we argue as follows: First choose the open covering such that
$|I| \leq \text{size}(V)$. Next, choose schemes $U_i$ of size
$\leq \max(\kappa, \text{size}(V))$ and surjective \'etale morphisms
$U_i \to Z_i$; we can do this by assumption (2) and
Sets, Lemma \ref{sets-lemma-bound-size-fibre-product}
(details omitted). Then
Sets, Lemma \ref{sets-lemma-what-is-in-it}
implies that $\coprod U_i$ is an object of $(\Sch/S)_{fppf}$.
Hence $\coprod Z_i$ is an algebraic space by
Spaces, Lemma \ref{spaces-lemma-coproduct-algebraic-spaces}.
}.
\end{proof}
\begin{lemma}
\label{lemma-check-property-limit-preserving}
Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism
of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\mathcal{P}$
be a property of morphisms of algebraic spaces as in
Algebraic Stacks, Definition
\ref{algebraic-definition-relative-representable-property}. If
\begin{enumerate}
\item $f$ is representable by algebraic spaces,
\item $\mathcal{Y} \to (\Sch/S)_{fppf}$ is limit preserving on objects,
\item for an affine scheme $V$ locally of finite presentation over $S$ and
$y \in \mathcal{Y}_V$ the resulting morphism of algebraic spaces
$f_y : F_y \to V$, see Algebraic Stacks, Equation
(\ref{algebraic-equation-representable-by-algebraic-spaces}),
has property $\mathcal{P}$.
\end{enumerate}
Then $f$ has property $\mathcal{P}$.
\end{lemma}
\begin{proof}
Let $V$ be a scheme over $S$ and $y \in \mathcal{Y}_V$. We have to show
that $F_y \to V$ has property $\mathcal{P}$. Since $\mathcal{P}$ is
fppf local on the base we may assume that $V$ is an affine scheme which
maps into an affine open $\Spec(\Lambda) \subset S$. Thus we can write
$V = \lim V_i$ with each $V_i$ affine and of finite presentation over
$\Spec(\Lambda)$, see Algebra, Lemma \ref{algebra-lemma-ring-colimit-fp}.
Then $y$ comes from an object $y_i$ over $V_i$ for some $i$ by assumption (2).
By assumption (3) the morphism $F_{y_i} \to V_i$ has property $\mathcal{P}$.
As $\mathcal{P}$ is stable under arbitrary base change and since
$F_y = F_{y_i} \times_{V_i} V$ we conclude that $F_y \to V$ has property
$\mathcal{P}$ as desired.
\end{proof}
\section{Formally smooth on objects}
\label{section-formally-smooth}
\noindent
Let $S$ be a scheme. Let $p : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism
of categories fibred in groupoids over $(\Sch/S)_{fppf}$. We will say that
$p$ is {\it formally smooth on objects} if the following condition holds:
Given any data consisting of
\begin{enumerate}
\item a first order thickening $U \subset U'$ of affine schemes over $S$,
\item an object $y'$ of $\mathcal{Y}$ over $U'$,
\item an object $x$ of $\mathcal{X}$ over $U$, and
\item an isomorphism $\gamma : p(x) \to y'|_U$,
\end{enumerate}
then there exists an object $x'$ of
$\mathcal{X}$ over $U'$ with an isomorphism
$\beta : x'|_U \to x$ and an isomorphism $\gamma' : p(x') \to y'$
such that
\begin{equation}
\label{equation-formally-smooth}
\vcenter{
\xymatrix{
p(x'|_U) \ar[d]_{p(\beta)} \ar[rr]_{\gamma'|_U} & &
y'|_U \ar@{=}[d] \\
p(x) \ar[rr]^\gamma & & y'|_U
}
}
\end{equation}
commutes. In this situation we say that ``$(x', \beta, \gamma')$
is a {\it solution} to the problem posed by our data (1), (2), (3), (4)''.
\begin{lemma}
\label{lemma-base-change-formally-smooth}
Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$
be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$.
If $p : \mathcal{X} \to \mathcal{Y}$ is formally smooth on objects, then so
is the base change
$p' : \mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to \mathcal{Z}$
of $p$ by $q$.
\end{lemma}
\begin{proof}
This is formal. Let $U \subset U'$ be a first order thickening
of affine schemes over $S$, let $z'$ be an object of $\mathcal{Z}$
over $U'$, let $w$ be an object of
$\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$ over $U$, and let
$\delta : p'(w) \to z'|_U$ be an isomorphism.
We may write
$w = (U, x, z, \alpha)$ for some object $x$ of $\mathcal{X}$ over $U$
and object $z$ of $\mathcal{Z}$ over $U$ and isomorphism
$\alpha : p(x) \to q(z)$. Note that $p'(w) = z$ hence
$\delta : z \to z|_U$. Set $y' = q(z')$ and
$\gamma = q(\delta) \circ \alpha : p(x) \to y'|_U$.
As $p$ is formally smooth on objects there exists an
object $x'$ of $\mathcal{X}$ over $U'$ as well as
isomorphisms $\beta : x'|_U \to x$ and $\gamma' : p(x') \to y'$ such that
(\ref{equation-formally-smooth}) commutes. Then we consider the object
$w = (U', x', z', \gamma')$ of $\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$
over $U'$ and define isomorphisms
$$
w'|_U = (U, x'|_U, z'|_U, \gamma'|_U)
\xrightarrow{(\beta, \delta^{-1})}
(U, x, z, \alpha) = w
$$
and
$$
p'(w') = z' \xrightarrow{\text{id}} z'.
$$
These combine to give a solution to the problem.
\end{proof}
\begin{lemma}
\label{lemma-composition-formally-smooth}
Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Y} \to \mathcal{Z}$
be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$.
If $p$ and $q$ are formally smooth on objects, then so is the composition
$q \circ p$.
\end{lemma}
\begin{proof}
This is formal. Let $U \subset U'$ be a first order thickening
of affine schemes over $S$, let $z'$ be an object of $\mathcal{Z}$
over $U'$, let $x$ be an object of $\mathcal{X}$ over $U$,
and let $\gamma : q(p(x)) \to z'|_U$ be an isomorphism. As $q$ is
formally smooth on objects there exist an object
$y'$ of $\mathcal{Y}$ over $U'$, an isomorphism
$\beta : y'|_U \to p(x)$, and an isomorphism $\gamma' : q(y') \to z'$
such that (\ref{equation-formally-smooth}) is commutative. As $p$ is
formally smooth on objects there exist an object
$x'$ of $\mathcal{X}$ over $U'$, an isomorphism
$\beta' : x'|_U \to x$, and an isomorphism $\gamma'' : p(x') \to y'$
such that (\ref{equation-formally-smooth}) is commutative.
The solution is to take $x'$ over $U'$ with isomorphism
$$
q(p(x')) \xrightarrow{q(\gamma'')} q(y') \xrightarrow{\gamma'} z'
$$
and isomorphism $\beta' : x'|_U \to x$. We omit the verification
that (\ref{equation-formally-smooth}) is commutative.
\end{proof}
\noindent
Note that the class of formally smooth morphisms of algebraic spaces is
stable under arbitrary base change and local on the target in the
fpqc topology, see
More on Morphisms of Spaces,
Lemma \ref{spaces-more-morphisms-lemma-base-change-formally-smooth} and
\ref{spaces-more-morphisms-lemma-descending-property-formally-smooth}.
Hence condition (2) in the lemma below makes sense.
\begin{lemma}
\label{lemma-representable-by-spaces-formally-smooth}
Let $p : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories
fibred in groupoids over $(\Sch/S)_{fppf}$. If $p$ is
representable by algebraic spaces, then the following are equivalent:
\begin{enumerate}
\item $p$ is formally smooth on objects, and
\item $p$ is formally smooth (see
Algebraic Stacks,
Definition \ref{algebraic-definition-relative-representable-property}).
\end{enumerate}
\end{lemma}
\begin{proof}
Assume (2). Let $U \subset U'$ be a first order thickening
of affine schemes over $S$, let $y'$ be an object of $\mathcal{Y}$
over $U'$, let $x$ be an object of $\mathcal{X}$ over $U$,
and let $\gamma : p(x) \to y'|_U$ be an isomorphism. Let
$X_{y'}$ denote an algebraic space over $U'$ representing the $2$-fibre
product
$$
(\Sch/U')_{fppf} \times_{y', \mathcal{Y}, p} \mathcal{X}.
$$
Note that $\xi = (U, U \to U', x, \gamma^{-1})$ defines an object of
this $2$-fibre product over $U$. Via the $2$-Yoneda lemma $\xi$ corresponds
to a morphism $f_\xi : U \to X_{y'}$ over $U'$. As $X_{y'} \to U'$ is
formally smooth by assumption there exists a morphism
$f' : U' \to X_{y'}$ such that $f_\xi$ is the composition of $f'$
and the morphism $U \to U'$. Also, the $2$-Yoneda lemma tells us that
$f'$ corresponds to an object $\xi' = (U', U' \to U', x', \alpha)$ of
the displayed $2$-fibre product over $U'$ whose restriction to
$U$ recovers $\xi$. In particular we obtain an isomorphism
$\gamma : x'|U \to x$. Note that $\alpha : y' \to p(x')$.
Hence we see that taking $x'$, the isomorphism
$\gamma : x'|U \to x$, and the isomorphism
$\beta = \alpha^{-1} : p(x') \to y'$
is a solution to the problem.
\medskip\noindent
Assume (1). Choose a scheme $T$ and a $1$-morphism
$y : (\Sch/T)_{fppf} \to \mathcal{Y}$. Let
$X_y$ be an algebraic space over $T$ representing the $2$-fibre product
$(\Sch/T)_{fppf} \times_{y, \mathcal{Y}, p} \mathcal{X}$.
We have to show that $X_y \to T$ is formally smooth.
Hence it suffices to show that given a first order thickening
$U \subset U'$ of affine schemes over $T$, then
$X_y(U') \to X_y(U')$ is surjective (morphisms in the
category of algebraic spaces over $T$). Set $y' = y|_{U'}$.
By the $2$-Yoneda lemma morphisms $U \to X_y$ over $T$ correspond bijectively
to isomorphism classes of pairs $(x, \alpha)$ where $x$ is an object
of $\mathcal{X}$ over $U$ and $\alpha : y|_U \to p(x)$ is an isomorphism.
Of course giving $\alpha$ is, up to an inverse, the same thing as giving
an isomorphism $\gamma : p(x) \to y'|_U$.
Similarly for morphisms $U' \to X_y$ over $T$. Hence (1) guarantees
the surjectivity of $X_y(U') \to X_y(U')$
in this situation and we win.
\end{proof}
\section{Surjective on objects}
\label{section-formally-surjective}
\noindent
Let $S$ be a scheme. Let $p : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism
of categories fibred in groupoids over $(\Sch/S)_{fppf}$. We will say that
$p$ is {\it surjective on objects} if the following condition holds:
Given any data consisting of
\begin{enumerate}
\item a field $k$ over $S$, and
\item an object $y$ of $\mathcal{Y}$ over $\Spec(k)$,
\end{enumerate}
then there exists an extension $k \subset K$ of fields over $S$, an
object $x$ of $\mathcal{X}$ over $\Spec(K)$
such that $p(x) \cong y|_{\Spec(K)}$.
\begin{lemma}
\label{lemma-base-change-surjective}
Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$
be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$.
If $p : \mathcal{X} \to \mathcal{Y}$ is surjective on objects, then so
is the base change
$p' : \mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to \mathcal{Z}$
of $p$ by $q$.
\end{lemma}
\begin{proof}
This is formal. Let $z$ be an object of $\mathcal{Z}$ over a field $k$.
As $p$ is surjective on objects there exists an extension $k \subset K$
and an object $x$ of $\mathcal{X}$ over $K$ and an isomorphism
$\alpha : p(x) \to q(z)|_{\Spec(K)}$. Then
$w = (\Spec(K), x, z|_{\Spec(K)}, \alpha)$ is an object of
$\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$ over $K$ with
$p'(w) = z|_{\Spec(K)}$.
\end{proof}
\begin{lemma}
\label{lemma-composition-surjective}
Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Y} \to \mathcal{Z}$
be $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$.
If $p$ and $q$ are surjective on objects, then so is the composition
$q \circ p$.
\end{lemma}
\begin{proof}
This is formal. Let $z$ be an object of $\mathcal{Z}$ over a field $k$.
As $q$ is surjective on objects there exists a field extension $k \subset K$
and an object $y$ of $\mathcal{Y}$ over $K$ such that
$q(y) \cong x|_{\Spec(K)}$. As $p$ is surjective on objects there
exists a field extension $K \subset L$ and an object $x$ of $\mathcal{X}$
over $L$ such that $p(x) \cong y|_{\Spec(L)}$. Then the field extension
$k \subset L$ and the object $x$ of $\mathcal{X}$ over $L$ satisfy
$q(p(x)) \cong z|_{\Spec(L)}$ as desired.
\end{proof}
\begin{lemma}
\label{lemma-representable-by-spaces-surjective}
Let $p : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories
fibred in groupoids over $(\Sch/S)_{fppf}$. If $p$ is
representable by algebraic spaces, then the following are equivalent:
\begin{enumerate}
\item $p$ is surjective on objects, and
\item $p$ is surjective (see
Algebraic Stacks,
Definition \ref{algebraic-definition-relative-representable-property}).
\end{enumerate}
\end{lemma}
\begin{proof}
Assume (2). Let $k$ be a field and let $y$ be an object of
$\mathcal{Y}$ over $k$. Let $X_y$ denote an algebraic space over $k$
representing the $2$-fibre product
$$
(\Sch/\Spec(k))_{fppf} \times_{y, \mathcal{Y}, p} \mathcal{X}.
$$
As we've assumed that $p$ is surjective we see that $X_y$ is not empty.
Hence we can find a field extension $k \subset K$ and a $K$-valued point
$x$ of $X_y$. Via the $2$-Yoneda lemma this corresponds to an object
$x$ of $\mathcal{X}$ over $K$ together with an isomorphism
$p(x) \cong y|_{\Spec(K)}$ and we see that (1) holds.
\medskip\noindent
Assume (1). Choose a scheme $T$ and a $1$-morphism
$y : (\Sch/T)_{fppf} \to \mathcal{Y}$. Let
$X_y$ be an algebraic space over $T$ representing the $2$-fibre product
$(\Sch/T)_{fppf} \times_{y, \mathcal{Y}, p} \mathcal{X}$.
We have to show that $X_y \to T$ is surjective. By
Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-surjective}
we have to show that $|X_y| \to |T|$ is surjective.
This means exactly that given a field $k$ over $T$ and a
morphism $t : \Spec(k) \to T$ there exists a field extension
$k \subset K$ and a morphism $x : \Spec(K) \to X_y$ such that
$$
\xymatrix{
\Spec(K) \ar[d] \ar[r]_x & X_y \ar[d] \\
\Spec(k) \ar[r]^t & T
}
$$
commutes. By the $2$-Yoneda lemma this means exactly that we have to find
$k \subset K$ and an object $x$ of $\mathcal{X}$ over $K$ such that
$p(x) \cong t^*y|_{\Spec(K)}$. Hence (1) guarantees that this is
the case and we win.
\end{proof}
\section{Algebraic morphisms}
\label{section-algebraic}
\noindent
The following notion is occasionally useful.
\begin{definition}
\label{definition-algebraic}
Let $S$ be a scheme. Let $F : \mathcal{X} \to \mathcal{Y}$ be a
$1$-morphism of stacks in groupoids over $(\Sch/S)_{fppf}$.
We say that $F$ is {\it algebraic} if for every scheme $T$ and every
object $\xi$ of $\mathcal{Y}$ over $T$ the $2$-fibre product
$$
(\Sch/T)_{fppf} \times_{\xi, \mathcal{Y}} \mathcal{X}
$$
is an algebraic stack over $S$.
\end{definition}
\noindent
With this terminology in place we have the following result that generalizes
Algebraic Stacks, Lemma
\ref{algebraic-lemma-representable-morphism-to-algebraic}.
\begin{lemma}
\label{lemma-algebraic-morphism-to-algebraic}
Let $S$ be a scheme.
Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of
stacks in groupoids over $(\Sch/S)_{fppf}$. If
\begin{enumerate}
\item $\mathcal{Y}$ is an algebraic stack, and
\item $F$ is algebraic (see above),
\end{enumerate}
then $\mathcal{X}$ is an algebraic stack.
\end{lemma}
\begin{proof}
By assumption (1) there exists a scheme $T$ and an object
$\xi$ of $\mathcal{Y}$ over $T$ such that the corresponding
$1$-morphism $\xi : (\Sch/T)_{fppf} \to \mathcal{Y}$
is smooth an surjective. Then
$\mathcal{U} = (\Sch/T)_{fppf} \times_{\xi, \mathcal{Y}} \mathcal{X}$
is is an algebraic stack by assumption (2).
Choose a scheme $U$ and a surjective smooth $1$-morphism
$(\Sch/U)_{fppf} \to \mathcal{U}$.
The projection $\mathcal{U} \longrightarrow \mathcal{X}$
is, as the base change of the morphism
$\xi : (\Sch/T)_{fppf} \to \mathcal{Y}$,
surjective and smooth, see
Algebraic Stacks, Lemma
\ref{algebraic-lemma-base-change-representable-transformations-property}.
Then the composition
$(\Sch/U)_{fppf} \to \mathcal{U} \to \mathcal{X}$
is surjective and smooth as a composition of surjective and smooth
morphisms, see
Algebraic Stacks, Lemma
\ref{algebraic-lemma-composition-representable-transformations-property}.
Hence $\mathcal{X}$ is an algebraic stack by
Algebraic Stacks, Lemma
\ref{algebraic-lemma-smooth-surjective-morphism-implies-algebraic}.
\end{proof}