## Tag `0CMH`

## 86.32. Scheme theoretic image

Here is the definition.

Definition 86.32.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The

scheme theoretic imageof $f$ is the smallest closed substack $\mathcal{Z} \subset \mathcal{Y}$ through which $f$ factors^{1}.We often denote $f : \mathcal{X} \to \mathcal{Z}$ the factorization of $f$. If the morphism $f$ is not quasi-compact, then (in general) the construction of the scheme theoretic image does not commute with restriction to open substacks of $\mathcal{Y}$. However, if $f$ is quasi-compact then the scheme theoretic image commutes with flat base change (Lemma 86.32.5).

Lemma 86.32.2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $g : \mathcal{W} \to \mathcal{X}$ be a morphism of algebraic stacks which is surjective, flat, and locally of finite presentation. Then the scheme theoretic image of $f$ exists if and only if the scheme theoretic image of $f \circ g$ exists and if so then these scheme theoretic images are the same.

Proof.Assume $\mathcal{Z} \subset \mathcal{Y}$ is a closed substack and $f \circ g$ factors through $\mathcal{Z}$. To prove the lemma it suffices to show that $f$ factors through $\mathcal{Z}$. Consider a scheme $T$ and a morphism $T \to \mathcal{X}$ given by an object $x$ of the fibre category of $\mathcal{X}$ over $T$. We will show that $x$ is in fact in the fibre category of $\mathcal{Z}$ over $T$. Namely, the projection $T \times_\mathcal{X} \mathcal{W} \to T$ is a surjective, flat, locally finitely presented morphism. Hence there is an fppf covering $\{T_i \to T\}$ such that $T_i \to T$ factors through $T \times_\mathcal{X} \mathcal{W} \to T$ for all $i$. Then $T_i \to \mathcal{X}$ factors through $\mathcal{W}$ and hence $T_i \to \mathcal{Y}$ factors through $\mathcal{Z}$. Thus $x|_{T_i}$ is an object of $\mathcal{Z}$. Since $\mathcal{Z}$ is a strictly full substack, we conclude that $x$ is an object of $\mathcal{Z}$ as desired. $\square$Lemma 86.32.3. Let $f : \mathcal{Y} \to \mathcal{X}$ be a morphism of algebraic stacks. Then the scheme theoretic image of $f$ exists.

Proof.Choose a scheme $V$ and a surjective smooth morphism $V \to \mathcal{Y}$. By Lemma 86.32.2 we may replace $\mathcal{Y}$ by $V$. Thus it suffices to show that if $X \to \mathcal{X}$ is a morphism from a scheme to an algebraic stack, then the scheme theoretic image exists. Choose a scheme $U$ and a surjective smooth morphism $U \to \mathcal{X}$. Set $R = U \times_\mathcal{X} U$. We have $\mathcal{X} = [U/R]$ by Algebraic Stacks, Lemma 79.16.2. By Properties of Stacks, Lemma 85.9.10 the closed substacks $\mathcal{Z}$ of $\mathcal{X}$ are in $1$-to-$1$ correspondence with $R$-invariant closed subschemes $Z \subset U$. Let $Z_1 \subset U$ be the scheme theoretic image of $X \times_\mathcal{X} U \to U$. Observe that $X \to \mathcal{X}$ factors through $\mathcal{Z}$ if and only if $X \times_\mathcal{X} U \to U$ factors through the corresponding $R$-invariant closed subscheme $Z$ (details omitted; hint: this follows because $X \times_\mathcal{X} U \to X$ is surjective and smooth). Thus we have to show that there exists a smallest $R$-invariant closed subscheme $Z \subset U$ containing $Z_1$.Let $\mathcal{I}_1 \subset \mathcal{O}_U$ be the quasi-coherent ideal sheaf corresponding to the closed subscheme $Z_1 \subset U$. Let $Z_\alpha$, $\alpha \in A$ be the set of all $R$-invariant closed subschemes of $U$ containing $Z_1$. For $\alpha \in A$, let $\mathcal{I}_\alpha \subset \mathcal{O}_U$ be the quasi-coherent ideal sheaf corresponding to the closed subscheme $Z_\alpha \subset U$. The containment $Z_1 \subset Z_\alpha$ means $\mathcal{I}_\alpha \subset \mathcal{I}_1$. The $R$-invariance of $Z_\alpha$ means that $$ s^{-1}\mathcal{I}_\alpha \cdot \mathcal{O}_R = t^{-1}\mathcal{I}_\alpha \cdot \mathcal{O}_R $$ as (quasi-coherent) ideal sheaves on (the algebraic space) $R$. Consider the image $$ \mathcal{I} = \mathop{\rm Im}\left( \bigoplus\nolimits_{\alpha \in A} \mathcal{I}_\alpha \to \mathcal{I}_1 \right) = \mathop{\rm Im}\left( \bigoplus\nolimits_{\alpha \in A} \mathcal{I}_\alpha \to \mathcal{O}_X \right) $$ Since direct sums of quasi-coherent sheaves are quasi-coherent and since images of maps between quasi-coherent sheaves are quasi-coherent, we find that $\mathcal{I}$ is quasi-coherent. Since pull back is exact and commutes with direct sums we find $$ s^{-1}\mathcal{I} \cdot \mathcal{O}_R = t^{-1}\mathcal{I} \cdot \mathcal{O}_R $$ Hence $\mathcal{I}$ defines an $R$-invariant closed subscheme $Z \subset U$ which is contained in every $Z_\alpha$ and containes $Z_1$ as desired. $\square$

Lemma 86.32.4. Let $$ \xymatrix{ \mathcal{X}_1 \ar[d] \ar[r]_{f_1} & \mathcal{Y}_1 \ar[d] \\ \mathcal{X}_2 \ar[r]^{f_2} & \mathcal{Y}_2 } $$ be a commutative diagram of algebraic stacks. Let $\mathcal{Z}_i \subset \mathcal{Y}_i$, $i = 1, 2$ be the scheme theoretic image of $f_i$. Then the morphism $\mathcal{Y}_1 \to \mathcal{Y}_2$ induces a morphism $\mathcal{Z}_1 \to \mathcal{Z}_2$ and a commutative diagram $$ \xymatrix{ \mathcal{X}_1 \ar[r] \ar[d] & \mathcal{Z}_1 \ar[d] \ar[r] & \mathcal{Y}_1 \ar[d] \\ \mathcal{X}_2 \ar[r] & \mathcal{Z}_2 \ar[r] & \mathcal{Y}_2 } $$

Proof.The scheme theoretic inverse image of $\mathcal{Z}_2$ in $\mathcal{Y}_1$ is a closed substack of $\mathcal{Y}_1$ through which $f_1$ factors. Hence $\mathcal{Z}_1$ is contained in this. This proves the lemma. $\square$Lemma 86.32.5. Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact morphism of algebraic stacks. Then formation of the scheme theoretic image commutes with flat base change.

Proof.Let $\mathcal{Y}' \to \mathcal{Y}$ be a flat morphism of algebraic stacks. Choose a scheme $V$ and a surjective smooth morphism $V \to \mathcal{Y}$. Choose a scheme $V'$ and a surjective smooth morphism $V' \to \mathcal{Y}' \times_\mathcal{Y} V$. We may and do assume that $V = \coprod_{i \in I} V_i$ is a disjoint union of affine schemes and that $V' = \coprod_{i \in I} \coprod_{j \in J_i} V_{i, j}$ is a disjoint union of affine schemes with each $V_{i, j}$ mapping into $V_i$. Let

- $\mathcal{Z} \subset \mathcal{Y}$ be the scheme theoretic image of $f$,
- $\mathcal{Z}' \subset \mathcal{Y}'$ be the scheme theoretic image of the base change of $f$ by $\mathcal{Y}' \to \mathcal{Y}$,
- $Z \subset V$ be the scheme theoretic image of the base change of $f$ by $V \to \mathcal{Y}$,
- $Z' \subset V'$ be the scheme theoretic image of the base change of $f$ by $V' \to \mathcal{Y}$.
If we can show that (a) $Z = V \times_\mathcal{Y} \mathcal{Z}$, (b) $Z' = V' \times_{\mathcal{Y}'} \mathcal{Z}'$, and (c) $Z' = V' \times_V Z$ then the lemma follows: the inclusion $\mathcal{Z}' \to \mathcal{Z} \times_\mathcal{Y} \mathcal{Y}'$ (Lemma 86.32.4) has to be an isomorphism because after base change by the surjective smooth morphism $V' \to \mathcal{Y}'$ it is.

Proof of (a). Set $R = V \times_\mathcal{Y} V$. By Properties of Stacks, Lemma 85.9.10 the rule $\mathcal{Z} \mapsto \mathcal{Z} \times_\mathcal{Y} V$ defines a $1$-to-$1$ correspondence between closed substacks of $\mathcal{Y}$ and $R$-invariant closed subspaces of $V$. Moreover, $f : \mathcal{X} \to \mathcal{Y}$ factors through $\mathcal{Z}$ if and only if the base change $g : \mathcal{X} \times_\mathcal{Y} V \to V$ factors through $\mathcal{Z} \times_\mathcal{Y} V$. We claim: the scheme theoretic image $Z \subset V$ of $g$ is $R$-invariant. The claim implies (a) by what we just said.

For each $i$ the morphism $\mathcal{X} \times_\mathcal{Y} V_i \to V_i$ is quasi-compact and hence $\mathcal{X} \times_\mathcal{Y} V_i$ is quasi-compact. Thus we can choose an affine scheme $W_i$ and a surjective smooth morphism $W_i \to \mathcal{X} \times_\mathcal{Y} V_i$. Observe that $W = \coprod W_i$ is a scheme endowed with a smooth and surjective morphism $W \to \mathcal{X} \times_\mathcal{Y} V$ such that the composition $W \to V$ with $g$ is quasi-compact. Let $Z \to V$ be the scheme theoretic image of $W \to V$, see Morphisms, Section 28.6 and Morphisms of Spaces, Section 55.16. It follows from Lemma 86.32.2 that $Z \subset V$ is the scheme theoretic image of $g$. To show that $Z$ is $R$-invariant we claim that both $$ \text{pr}_0^{-1}(Z), \text{pr}_1^{-1}(Z) \subset R = V \times_\mathcal{Y} V $$ are the scheme theoretic image of $\mathcal{X} \times_\mathcal{Y} R \to R$. Namely, we first use Morphisms of Spaces, Lemma 55.29.12 to see that $\text{pr}_0^{-1}(Z)$ is the scheme theoretic image of the composition $$ W \times_{V, \text{pr}_0} R = W \times_\mathcal{Y} V \to \mathcal{X} \times_\mathcal{Y} R \to R $$ Since the first arrow here is surjective and smooth we see that $\text{pr}_0^{-1}(Z)$ is the scheme theoretic image of $\mathcal{X} \times_\mathcal{Y} R \to R$. The same argument applies that $\text{pr}_1^{-1}(Z)$. Hence $Z$ is $R$-invariant.

Statement (b) is proved in exactly the same way as one proves (a).

Proof of (c). Let $Z_i \subset V_i$ be the scheme theoretic image of $\mathcal{X} \times_\mathcal{Y} V_i \to V_i$ and let $Z_{i, j} \subset V_{i, j}$ be the scheme theoretic image of $\mathcal{X} \times_\mathcal{Y} V_{i, j} \to V_{i, j}$. Clearly it suffices to show that the inverse image of $Z_i$ in $V_{i, j}$ is $Z_{i, j}$. Above we've seen that $Z_i$ is the scheme theoretic image of $W_i \to V_i$ and by the same token $Z_{i, j}$ is the scheme theoretic image of $W_i \times_{V_i} V_{i, j} \to V_{i, j}$. Hence the equality follows from the case of schemes (Morphisms, Lemma 28.24.15) and the fact that $V_{i, j} \to V_i$ is flat. $\square$

Lemma 86.32.6. Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact morphism of algebraic stacks. Let $\mathcal{Z} \subset \mathcal{Y}$ be the scheme theoretic image of $f$. Then $|\mathcal{Z}|$ is the closure of the image of $|f|$.

Proof.Let $z \in |\mathcal{Z}|$ be a point. Choose an affine scheme $V$, a point $v \in V$, and a smooth morphism $V \to \mathcal{Y}$ mapping $v$ to $z$. Then $\mathcal{X} \times_\mathcal{Y} V$ is a quasi-compact algebraic stack. Hence we can find an affine scheme $W$ and a surjective smooth morphism $W \to \mathcal{X} \times_\mathcal{Y} V$. By Lemma 86.32.5 the scheme theoretic image of $\mathcal{X} \times_\mathcal{Y} V \to V$ is $Z = \mathcal{Z} \times_\mathcal{Y} V$. Hence the inverse image of $|\mathcal{Z}|$ in $|V|$ is $|Z|$ by Properties of Stacks, Lemma 85.4.3. By Lemma 86.32.2 $Z$ is the scheme theoretic image of $W \to V$. By Morphisms of Spaces, Lemma 55.16.3 we see that the image of $|W| \to |Z|$ is dense. Hence the image of $|\mathcal{X} \times_\mathcal{Y} V| \to |Z|$ is dense. Observe that $v \in Z$. Since $|V| \to |\mathcal{Y}|$ is open, a topology argument tells us that $z$ is in the closure of the image of $|f|$ as desired. $\square$Lemma 86.32.7. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces and separated. Let $\mathcal{V} \subset \mathcal{Y}$ be an open substack such that $\mathcal{V} \to \mathcal{Y}$ is quasi-compact. Let $s : \mathcal{V} \to \mathcal{X}$ be a morphism such that $f \circ s = \text{id}_\mathcal{V}$. Let $\mathcal{Y}'$ be the scheme theoretic image of $s$. Then $\mathcal{Y}' \to \mathcal{Y}$ is an isomorphism over $\mathcal{V}$.

Proof.By Lemma 86.7.7 the morphism $s : \mathcal{V} \to \mathcal{Y}$ is quasi-compact. Hence the construction of the scheme theoretic image $\mathcal{Y}'$ of $s$ commutes with flat base change by Lemma 86.32.5. Thus to prove the lemma we may assume $\mathcal{Y}$ is representable by an algebraic space and we reduce to the case of algebraic spaces which is Morphisms of Spaces, Lemma 55.16.7. $\square$

The code snippet corresponding to this tag is a part of the file `stacks-morphisms.tex` and is located in lines 7024–7345 (see updates for more information).

```
\section{Scheme theoretic image}
\label{section-scheme-theoretic-image}
\noindent
Here is the definition.
\begin{definition}
\label{definition-scheme-theoretic-image}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
The {\it scheme theoretic image} of $f$ is the smallest closed substack
$\mathcal{Z} \subset \mathcal{Y}$ through which $f$
factors\footnote{We will see in
Lemma \ref{lemma-scheme-theoretic-image-existence}
that the scheme theoretic image always exists.}.
\end{definition}
\noindent
We often denote $f : \mathcal{X} \to \mathcal{Z}$ the factorization of $f$.
If the morphism $f$ is not quasi-compact, then (in general) the
construction of the scheme theoretic image does not commute with
restriction to open substacks of $\mathcal{Y}$. However, if $f$ is
quasi-compact then the scheme theoretic image commutes with flat base change
(Lemma \ref{lemma-existence-plus-flat-base-change}).
\begin{lemma}
\label{lemma-cover-upstairs}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
Let $g : \mathcal{W} \to \mathcal{X}$ be a morphism of algebraic stacks
which is surjective, flat, and locally of finite presentation.
Then the scheme theoretic image of $f$ exists if and only if the
scheme theoretic image of $f \circ g$ exists and if so then these
scheme theoretic images are the same.
\end{lemma}
\begin{proof}
Assume $\mathcal{Z} \subset \mathcal{Y}$
is a closed substack and $f \circ g$ factors through $\mathcal{Z}$.
To prove the lemma it suffices to show
that $f$ factors through $\mathcal{Z}$.
Consider a scheme $T$ and a morphism $T \to \mathcal{X}$
given by an object $x$ of the fibre category of $\mathcal{X}$ over $T$.
We will show that $x$ is in fact in the fibre category of $\mathcal{Z}$
over $T$. Namely, the projection $T \times_\mathcal{X} \mathcal{W} \to T$
is a surjective, flat, locally finitely presented morphism.
Hence there is an fppf covering $\{T_i \to T\}$ such that
$T_i \to T$ factors through $T \times_\mathcal{X} \mathcal{W} \to T$
for all $i$. Then $T_i \to \mathcal{X}$ factors through $\mathcal{W}$
and hence $T_i \to \mathcal{Y}$ factors through $\mathcal{Z}$.
Thus $x|_{T_i}$ is an object of $\mathcal{Z}$.
Since $\mathcal{Z}$ is a strictly full substack, we conclude
that $x$ is an object of $\mathcal{Z}$ as desired.
\end{proof}
\begin{lemma}
\label{lemma-scheme-theoretic-image-existence}
Let $f : \mathcal{Y} \to \mathcal{X}$ be a morphism of algebraic stacks.
Then the scheme theoretic image of $f$ exists.
\end{lemma}
\begin{proof}
Choose a scheme $V$ and a surjective smooth morphism $V \to \mathcal{Y}$.
By Lemma \ref{lemma-cover-upstairs} we may replace $\mathcal{Y}$ by $V$.
Thus it suffices to show that if $X \to \mathcal{X}$ is a morphism from
a scheme to an algebraic stack, then the scheme theoretic image exists.
Choose a scheme $U$ and a surjective smooth morphism $U \to \mathcal{X}$.
Set $R = U \times_\mathcal{X} U$.
We have $\mathcal{X} = [U/R]$ by
Algebraic Stacks, Lemma \ref{algebraic-lemma-stack-presentation}.
By Properties of Stacks, Lemma
\ref{stacks-properties-lemma-substacks-presentation}
the closed substacks $\mathcal{Z}$ of $\mathcal{X}$
are in $1$-to-$1$ correspondence with $R$-invariant
closed subschemes $Z \subset U$.
Let $Z_1 \subset U$ be the scheme theoretic image of
$X \times_\mathcal{X} U \to U$.
Observe that $X \to \mathcal{X}$ factors through $\mathcal{Z}$
if and only if $X \times_\mathcal{X} U \to U$ factors through
the corresponding $R$-invariant closed subscheme $Z$
(details omitted; hint: this follows because
$X \times_\mathcal{X} U \to X$ is surjective and smooth).
Thus we have to show that there exists a smallest $R$-invariant
closed subscheme $Z \subset U$ containing $Z_1$.
\medskip\noindent
Let $\mathcal{I}_1 \subset \mathcal{O}_U$ be the quasi-coherent
ideal sheaf corresponding to the closed subscheme $Z_1 \subset U$.
Let $Z_\alpha$, $\alpha \in A$ be the set of all $R$-invariant
closed subschemes of $U$ containing $Z_1$.
For $\alpha \in A$, let $\mathcal{I}_\alpha \subset \mathcal{O}_U$
be the quasi-coherent ideal sheaf corresponding to the closed subscheme
$Z_\alpha \subset U$. The containment $Z_1 \subset Z_\alpha$
means $\mathcal{I}_\alpha \subset \mathcal{I}_1$.
The $R$-invariance of $Z_\alpha$ means that
$$
s^{-1}\mathcal{I}_\alpha \cdot \mathcal{O}_R =
t^{-1}\mathcal{I}_\alpha \cdot \mathcal{O}_R
$$
as (quasi-coherent) ideal sheaves on (the algebraic space) $R$.
Consider the image
$$
\mathcal{I} =
\Im\left(
\bigoplus\nolimits_{\alpha \in A} \mathcal{I}_\alpha \to \mathcal{I}_1
\right) =
\Im\left(
\bigoplus\nolimits_{\alpha \in A} \mathcal{I}_\alpha \to \mathcal{O}_X
\right)
$$
Since direct sums of quasi-coherent sheaves are quasi-coherent
and since images of maps between quasi-coherent sheaves are
quasi-coherent, we find that $\mathcal{I}$ is quasi-coherent.
Since pull back is exact and commutes with direct sums we find
$$
s^{-1}\mathcal{I} \cdot \mathcal{O}_R =
t^{-1}\mathcal{I} \cdot \mathcal{O}_R
$$
Hence $\mathcal{I}$ defines an $R$-invariant closed subscheme
$Z \subset U$ which is contained in every $Z_\alpha$ and containes
$Z_1$ as desired.
\end{proof}
\begin{lemma}
\label{lemma-factor-factor}
Let
$$
\xymatrix{
\mathcal{X}_1 \ar[d] \ar[r]_{f_1} & \mathcal{Y}_1 \ar[d] \\
\mathcal{X}_2 \ar[r]^{f_2} & \mathcal{Y}_2
}
$$
be a commutative diagram of algebraic stacks.
Let $\mathcal{Z}_i \subset \mathcal{Y}_i$, $i = 1, 2$ be
the scheme theoretic image of $f_i$. Then the morphism
$\mathcal{Y}_1 \to \mathcal{Y}_2$ induces a morphism
$\mathcal{Z}_1 \to \mathcal{Z}_2$ and a
commutative diagram
$$
\xymatrix{
\mathcal{X}_1 \ar[r] \ar[d] &
\mathcal{Z}_1 \ar[d] \ar[r] &
\mathcal{Y}_1 \ar[d] \\
\mathcal{X}_2 \ar[r] &
\mathcal{Z}_2 \ar[r] &
\mathcal{Y}_2
}
$$
\end{lemma}
\begin{proof}
The scheme theoretic inverse image of $\mathcal{Z}_2$ in $\mathcal{Y}_1$
is a closed substack of $\mathcal{Y}_1$ through
which $f_1$ factors. Hence $\mathcal{Z}_1$ is contained in this.
This proves the lemma.
\end{proof}
\begin{lemma}
\label{lemma-existence-plus-flat-base-change}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact
morphism of algebraic stacks. Then formation of the scheme theoretic image
commutes with flat base change.
\end{lemma}
\begin{proof}
Let $\mathcal{Y}' \to \mathcal{Y}$ be a flat morphism of algebraic stacks.
Choose a scheme $V$ and a surjective smooth morphism $V \to \mathcal{Y}$.
Choose a scheme $V'$ and a
surjective smooth morphism $V' \to \mathcal{Y}' \times_\mathcal{Y} V$.
We may and do assume that $V = \coprod_{i \in I} V_i$ is a disjoint
union of affine schemes and that
$V' = \coprod_{i \in I} \coprod_{j \in J_i} V_{i, j}$
is a disjoint union of affine schemes with each $V_{i, j}$ mapping into $V_i$.
Let
\begin{enumerate}
\item $\mathcal{Z} \subset \mathcal{Y}$ be the scheme theoretic image of $f$,
\item $\mathcal{Z}' \subset \mathcal{Y}'$ be the scheme theoretic image
of the base change of $f$ by $\mathcal{Y}' \to \mathcal{Y}$,
\item $Z \subset V$ be the scheme theoretic image
of the base change of $f$ by $V \to \mathcal{Y}$,
\item $Z' \subset V'$ be the scheme theoretic image
of the base change of $f$ by $V' \to \mathcal{Y}$.
\end{enumerate}
If we can show that
(a) $Z = V \times_\mathcal{Y} \mathcal{Z}$,
(b) $Z' = V' \times_{\mathcal{Y}'} \mathcal{Z}'$, and
(c) $Z' = V' \times_V Z$
then the lemma follows: the inclusion
$\mathcal{Z}' \to \mathcal{Z} \times_\mathcal{Y} \mathcal{Y}'$
(Lemma \ref{lemma-factor-factor})
has to be an isomorphism because after base change by the surjective
smooth morphism $V' \to \mathcal{Y}'$ it is.
\medskip\noindent
Proof of (a). Set $R = V \times_\mathcal{Y} V$.
By Properties of Stacks, Lemma
\ref{stacks-properties-lemma-substacks-presentation}
the rule $\mathcal{Z} \mapsto \mathcal{Z} \times_\mathcal{Y} V$
defines a $1$-to-$1$ correspondence between closed substacks
of $\mathcal{Y}$ and $R$-invariant closed subspaces of $V$.
Moreover, $f : \mathcal{X} \to \mathcal{Y}$ factors through $\mathcal{Z}$
if and only if the base change
$g : \mathcal{X} \times_\mathcal{Y} V \to V$ factors through
$\mathcal{Z} \times_\mathcal{Y} V$.
We claim: the scheme theoretic image $Z \subset V$ of $g$
is $R$-invariant. The claim implies (a) by what we just said.
\medskip\noindent
For each $i$ the morphism $\mathcal{X} \times_\mathcal{Y} V_i \to V_i$
is quasi-compact and hence $\mathcal{X} \times_\mathcal{Y} V_i$ is
quasi-compact. Thus we can choose an affine scheme $W_i$ and a
surjective smooth morphism $W_i \to \mathcal{X} \times_\mathcal{Y} V_i$.
Observe that $W = \coprod W_i$ is a scheme endowed with
a smooth and surjective morphism $W \to \mathcal{X} \times_\mathcal{Y} V$
such that the composition $W \to V$ with $g$ is quasi-compact.
Let $Z \to V$ be the scheme theoretic image of $W \to V$, see
Morphisms, Section
\ref{morphisms-section-scheme-theoretic-image} and
Morphisms of Spaces, Section
\ref{spaces-morphisms-section-scheme-theoretic-image}.
It follows from Lemma \ref{lemma-cover-upstairs}
that $Z \subset V$ is the scheme theoretic image of $g$.
To show that $Z$ is $R$-invariant we claim that both
$$
\text{pr}_0^{-1}(Z), \text{pr}_1^{-1}(Z) \subset R = V \times_\mathcal{Y} V
$$
are the scheme theoretic image of $\mathcal{X} \times_\mathcal{Y} R \to R$.
Namely, we first use Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-flat-base-change-scheme-theoretic-image}
to see that $\text{pr}_0^{-1}(Z)$ is the scheme theoretic image
of the composition
$$
W \times_{V, \text{pr}_0} R = W \times_\mathcal{Y} V \to
\mathcal{X} \times_\mathcal{Y} R \to R
$$
Since the first arrow here is surjective and smooth we see that
$\text{pr}_0^{-1}(Z)$ is the scheme theoretic image of
$\mathcal{X} \times_\mathcal{Y} R \to R$. The same argument applies
that $\text{pr}_1^{-1}(Z)$. Hence $Z$ is $R$-invariant.
\medskip\noindent
Statement (b) is proved in exactly the same way as one proves (a).
\medskip\noindent
Proof of (c). Let $Z_i \subset V_i$ be the scheme theoretic image
of $\mathcal{X} \times_\mathcal{Y} V_i \to V_i$ and let
$Z_{i, j} \subset V_{i, j}$ be the scheme theoretic image of
$\mathcal{X} \times_\mathcal{Y} V_{i, j} \to V_{i, j}$.
Clearly it suffices to show that the inverse image of $Z_i$
in $V_{i, j}$ is $Z_{i, j}$. Above we've seen that
$Z_i$ is the scheme theoretic image of $W_i \to V_i$
and by the same token $Z_{i, j}$ is the scheme theoretic
image of $W_i \times_{V_i} V_{i, j} \to V_{i, j}$.
Hence the equality follows from the case of schemes
(Morphisms, Lemma
\ref{morphisms-lemma-flat-base-change-scheme-theoretic-image})
and the fact that $V_{i, j} \to V_i$ is flat.
\end{proof}
\begin{lemma}
\label{lemma-topology-scheme-theoretic-image}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact
morphism of algebraic stacks. Let $\mathcal{Z} \subset \mathcal{Y}$
be the scheme theoretic image of $f$. Then $|\mathcal{Z}|$
is the closure of the image of $|f|$.
\end{lemma}
\begin{proof}
Let $z \in |\mathcal{Z}|$ be a point.
Choose an affine scheme $V$, a point $v \in V$, and a smooth morphism
$V \to \mathcal{Y}$ mapping $v$ to $z$.
Then $\mathcal{X} \times_\mathcal{Y} V$ is a quasi-compact algebraic stack.
Hence we can find an affine scheme $W$ and a surjective smooth
morphism $W \to \mathcal{X} \times_\mathcal{Y} V$.
By Lemma \ref{lemma-existence-plus-flat-base-change}
the scheme theoretic image of
$\mathcal{X} \times_\mathcal{Y} V \to V$ is
$Z = \mathcal{Z} \times_\mathcal{Y} V$.
Hence the inverse image of $|\mathcal{Z}|$ in $|V|$ is $|Z|$ by
Properties of Stacks, Lemma \ref{stacks-properties-lemma-points-cartesian}.
By Lemma \ref{lemma-cover-upstairs} $Z$ is
the scheme theoretic image of $W \to V$.
By Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-quasi-compact-scheme-theoretic-image}
we see that the image of $|W| \to |Z|$ is dense.
Hence the image of $|\mathcal{X} \times_\mathcal{Y} V| \to |Z|$
is dense. Observe that $v \in Z$.
Since $|V| \to |\mathcal{Y}|$ is open, a topology argument
tells us that $z$ is in the closure of the image of $|f|$ as desired.
\end{proof}
\begin{lemma}
\label{lemma-scheme-theoretic-image-of-partial-section}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks
which is representable by algebraic spaces and separated.
Let $\mathcal{V} \subset \mathcal{Y}$ be an open substack such that
$\mathcal{V} \to \mathcal{Y}$ is quasi-compact.
Let $s : \mathcal{V} \to \mathcal{X}$ be a morphism such that
$f \circ s = \text{id}_\mathcal{V}$.
Let $\mathcal{Y}'$ be the scheme theoretic image of $s$.
Then $\mathcal{Y}' \to \mathcal{Y}$ is an isomorphism over $\mathcal{V}$.
\end{lemma}
\begin{proof}
By Lemma \ref{lemma-quasi-compact-permanence}
the morphism $s : \mathcal{V} \to \mathcal{Y}$ is quasi-compact.
Hence the construction of the scheme theoretic image $\mathcal{Y}'$
of $s$ commutes with flat base change by
Lemma \ref{lemma-existence-plus-flat-base-change}.
Thus to prove the lemma
we may assume $\mathcal{Y}$ is representable by an algebraic space
and we reduce to the case of algebraic spaces which is
Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-scheme-theoretic-image-of-partial-section}.
\end{proof}
```

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