The Stacks Project


Tag 03FO

56.15. Change of big site

In this section we briefly discuss what happens when we change big sites. The upshot is that we can always enlarge the big site at will, hence we may assume any set of schemes we want to consider is contained in the big fppf site over which we consider our algebraic space. Here is a precise statement of the result.

Lemma 56.15.1. Suppose given big sites $\textit{Sch}_{fppf}$ and $\textit{Sch}'_{fppf}$. Assume that $\textit{Sch}_{fppf}$ is contained in $\textit{Sch}'_{fppf}$, see Topologies, Section 33.11. Let $S$ be an object of $\textit{Sch}_{fppf}$. Let \begin{align*} g : \mathop{\textit{Sh}}\nolimits((\textit{Sch}/S)_{fppf}) \longrightarrow \mathop{\textit{Sh}}\nolimits((\textit{Sch}'/S)_{fppf}), \\ f : \mathop{\textit{Sh}}\nolimits((\textit{Sch}'/S)_{fppf}) \longrightarrow \mathop{\textit{Sh}}\nolimits((\textit{Sch}/S)_{fppf}) \end{align*} be the morphisms of topoi of Topologies, Lemma 33.11.2. Let $F$ be a sheaf of sets on $(\textit{Sch}/S)_{fppf}$. Then

  1. if $F$ is representable by a scheme $X \in \mathop{\rm Ob}\nolimits((\textit{Sch}/S)_{fppf})$ over $S$, then $f^{-1}F$ is representable too, in fact it is representable by the same scheme $X$, now viewed as an object of $(\textit{Sch}'/S)_{fppf}$, and
  2. if $F$ is an algebraic space over $S$, then $f^{-1}F$ is an algebraic space over $S$ also.

Proof. Let $X \in \mathop{\rm Ob}\nolimits((\textit{Sch}/S)_{fppf})$. Let us write $h_X$ for the representable sheaf on $(\textit{Sch}/S)_{fppf}$ associated to $X$, and $h'_X$ for the representable sheaf on $(\textit{Sch}'/S)_{fppf}$ associated to $X$. By the description of $f^{-1}$ in Topologies, Section 33.11 we see that $f^{-1}h_X = h'_X$. This proves (1).

Next, suppose that $F$ is an algebraic space over $S$. By Lemma 56.9.1 this means that $F = h_U/h_R$ for some étale equivalence relation $R \to U \times_S U$ in $(\textit{Sch}/S)_{fppf}$. Since $f^{-1}$ is an exact functor we conclude that $f^{-1}F = h'_U/h'_R$. Hence $f^{-1}F$ is an algebraic space over $S$ by Theorem 56.10.5. $\square$

Note that this lemma is purely set theoretical and has virtually no content. Moreover, it is not true (in general) that the restriction of an algebraic space over the bigger site is an algebraic space over the smaller site (simply by reasons of cardinality). Hence we can only ever use a simple lemma of this kind to enlarge the base category and never to shrink it.

Lemma 56.15.2. Suppose $\textit{Sch}_{fppf}$ is contained in $\textit{Sch}'_{fppf}$. Let $S$ be an object of $\textit{Sch}_{fppf}$. Denote $\textit{Spaces}/S$ the category of algebraic spaces over $S$ defined using $\textit{Sch}_{fppf}$. Similarly, denote $\textit{Spaces}'/S$ the category of algebraic spaces over $S$ defined using $\textit{Sch}'_{fppf}$. The construction of Lemma 56.15.1 defines a fully faithful functor $$ \textit{Spaces}/S \longrightarrow \textit{Spaces}'/S $$ whose essential image consists of those $X' \in \mathop{\rm Ob}\nolimits(\textit{Spaces}'/S)$ such that there exist $U, R \in \mathop{\rm Ob}\nolimits((\textit{Sch}/S)_{fppf})$1 and morphisms $$ U \longrightarrow X' \quad\text{and}\quad R \longrightarrow U \times_{X'} U $$ in $\mathop{\textit{Sh}}\nolimits((\textit{Sch}'/S)_{fppf})$ which are surjective as maps of sheaves (for example if the displayed morphisms are surjective and étale).

Proof. In Sites, Lemma 7.20.8 we have seen that the functor $f^{-1} : \mathop{\textit{Sh}}\nolimits((\textit{Sch}/S)_{fppf}) \to \mathop{\textit{Sh}}\nolimits((\textit{Sch}'/S)_{fppf})$ is fully faithful (see discussion in Topologies, Section 33.11). Hence we see that the displayed functor of the lemma is fully faithful.

Suppose that $X' \in \mathop{\rm Ob}\nolimits(\textit{Spaces}'/S)$ such that there exists $U \in \mathop{\rm Ob}\nolimits((\textit{Sch}/S)_{fppf})$ and a map $U \to X'$ in $\mathop{\textit{Sh}}\nolimits((\textit{Sch}'/S)_{fppf})$ which is surjective as a map of sheaves. Let $U' \to X'$ be a surjective étale morphism with $U' \in \mathop{\rm Ob}\nolimits((\textit{Sch}'/S)_{fppf})$. Let $\kappa = \text{size}(U)$, see Sets, Section 3.9. Then $U$ has an affine open covering $U = \bigcup_{i \in I} U_i$ with $|I| \leq \kappa$. Observe that $U' \times_{X'} U \to U$ is étale and surjective. For each $i$ we can pick a quasi-compact open $U'_i \subset U'$ such that $U'_i \times_{X'} U_i \to U_i$ is surjective (because the scheme $U' \times_{X'} U_i$ is the union of the Zariski opens $W \times_{X'} U_i$ for $W \subset U'$ affine and because $U' \times_{X'} U_i \to U_i$ is étale hence open). Then $\coprod_{i \in I} U'_i \to X$ is surjective étale because of our assumption that $U \to X$ and hence $\coprod U_i \to X$ is a surjection of sheaves (details omitted). Because $U'_i \times_{X'} U \to U'_i$ is a surjection of sheaves and because $U'_i$ is quasi-compact, we can find a quasi-compact open $W_i \subset U'_i \times_{X'} U$ such that $W_i \to U'_i$ is surjective as a map of sheaves (details omitted). Then $W_i \to U$ is étale and we conclude that $\text{size}(W_i) \leq \text{size}(U)$, see Sets, Lemma 3.9.7. By Sets, Lemma 3.9.11 we conclude that $\text{size}(U'_i) \leq \text{size}(U)$. Hence $\coprod_{i \in I} U'_i$ is isomorphic to an object of $(\textit{Sch}/S)_{fppf}$ by Sets, Lemma 3.9.5.

Now let $X'$, $U \to X'$ and $R \to U \times_{X'} U$ be as in the statement of the lemma. In the previous paragraph we have seen that we can find $U' \in \mathop{\rm Ob}\nolimits((\textit{Sch}/S)_{fppf})$ and a surjective étale morphism $U' \to X'$ in $\mathop{\textit{Sh}}\nolimits((\textit{Sch}'/S)_{fppf})$. Then $U' \times_{X'} U \to U'$ is a surjection of sheaves, i.e., we can find an fppf covering $\{U'_i \to U'\}$ such that $U'_i \to U'$ factors through $U' \times_{X'} U \to U'$. By Sets, Lemma 3.9.12 we can find $\tilde U \to U'$ which is surjective, flat, and locally of finite presentation, with $\text{size}(\tilde U) \leq \text{size}(U')$, such that $\tilde U \to U'$ factors through $U' \times_{X'} U \to U'$. Then we consider $$ \xymatrix{ U' \times_{X'} U' \ar[d] & \tilde U \times_{X'} \tilde U \ar[l] \ar[d] \ar[r] & U \times_{X'} U \ar[d] \\ U' \times_S U' & \tilde U \times_S \tilde U \ar[l] \ar[r] & U \times_S U } $$ The squares are cartesian. We know the objects of the bottom row are represented by objects of $(\textit{Sch}/S)_{fppf}$. By the result of the argument of the previous paragraph, the same is true for $U \times_{X'} U$ (as we have the surjection of sheaves $R \to U \times_{X'} U$ by assumption). Since $(\textit{Sch}/S)_{fppf}$ is closed under fibre products (by construction), we see that $\tilde U \times_{X'} \tilde U$ is represented by an object of $(\textit{Sch}/S)_{fppf}$. Finally, the map $\tilde U \times_{X'} \tilde U \to U' \times_{X'} U'$ is a surjection of fppf sheaves as $\tilde U \to U'$ is so. Thus we can once more apply the result of the previous paragraph to conclude that $R' = U' \times_{X'} U'$ is represented by an object of $(\textit{Sch}/S)_{fppf}$. At this point Lemma 56.9.1 and Theorem 56.10.5 imply that $X = h_{U'}/h_{R'}$ is an object of $\textit{Spaces}/S$ such that $f^{-1}X \cong X'$ as desired. $\square$

  1. Requiring the existence of $R$ is necessary because of our choice of the function $Bound$ in Sets, Equation (3.9.1.1). The size of the fibre product $U \times_{X'} U$ can grow faster than $Bound$ in terms of the size of $U$. We can illustrate this by setting $S = \mathop{\rm Spec}(A)$, $U = \mathop{\rm Spec}(A[x_i, i \in I])$ and $R = \coprod_{(\lambda_i) \in A^I} \mathop{\rm Spec}(A[x_i, y_i]/(x_i - \lambda_i y_i))$. In this case the size of $R$ grows like $\kappa^\kappa$ where $\kappa$ is the size of $U$.

The code snippet corresponding to this tag is a part of the file spaces.tex and is located in lines 2517–2687 (see updates for more information).

\section{Change of big site}
\label{section-change-big-site}

\noindent
In this section we briefly discuss what happens when we change big sites.
The upshot is that we can always enlarge the big site at will, hence we
may assume any set of schemes we want to consider is contained in the big
fppf site over which we consider our algebraic space. Here is a precise
statement of the result.

\begin{lemma}
\label{lemma-change-big-site}
Suppose given big sites $\Sch_{fppf}$ and $\Sch'_{fppf}$.
Assume that $\Sch_{fppf}$ is contained in $\Sch'_{fppf}$,
see Topologies, Section \ref{topologies-section-change-alpha}.
Let $S$ be an object of $\Sch_{fppf}$. Let
\begin{align*}
g : \Sh((\Sch/S)_{fppf})
\longrightarrow
\Sh((\Sch'/S)_{fppf}), \\
f : \Sh((\Sch'/S)_{fppf})
\longrightarrow
\Sh((\Sch/S)_{fppf})
\end{align*}
be the morphisms of topoi of
Topologies, Lemma \ref{topologies-lemma-change-alpha}.
Let $F$ be a sheaf of sets on $(\Sch/S)_{fppf}$. Then
\begin{enumerate}
\item if $F$ is representable by a scheme
$X \in \Ob((\Sch/S)_{fppf})$ over $S$,
then $f^{-1}F$ is representable too, in fact it is representable by the
same scheme $X$, now viewed as an object of $(\Sch'/S)_{fppf}$, and
\item if $F$ is an algebraic space over $S$, then $f^{-1}F$ is an algebraic
space over $S$ also.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $X \in \Ob((\Sch/S)_{fppf})$. Let us write $h_X$ for the
representable sheaf on $(\Sch/S)_{fppf}$ associated to $X$, and
$h'_X$ for the representable sheaf on $(\Sch'/S)_{fppf}$ associated to
$X$. By the description of $f^{-1}$ in
Topologies, Section \ref{topologies-section-change-alpha}
we see that $f^{-1}h_X = h'_X$. This proves (1).

\medskip\noindent
Next, suppose that $F$
is an algebraic space over $S$. By Lemma \ref{lemma-space-presentation}
this means that $F = h_U/h_R$ for some \'etale equivalence relation
$R \to U \times_S U$ in $(\Sch/S)_{fppf}$. Since $f^{-1}$ is an
exact functor we conclude that $f^{-1}F = h'_U/h'_R$. Hence
$f^{-1}F$ is an algebraic space over $S$ by Theorem \ref{theorem-presentation}.
\end{proof}

\noindent
Note that this lemma is purely set theoretical and has virtually no content.
Moreover, it is not true (in general) that the restriction of an algebraic
space over the bigger site is an algebraic space over the smaller site (simply
by reasons of cardinality). Hence we can only ever use a simple lemma of this
kind to enlarge the base category and never to shrink it.

\begin{lemma}
\label{lemma-fully-faithful}
Suppose $\Sch_{fppf}$ is contained in $\Sch'_{fppf}$.
Let $S$ be an object of $\Sch_{fppf}$. Denote
$\textit{Spaces}/S$ the category of algebraic spaces over $S$
defined using $\Sch_{fppf}$. Similarly, denote
$\textit{Spaces}'/S$ the category of algebraic spaces over $S$
defined using $\Sch'_{fppf}$. The construction of
Lemma \ref{lemma-change-big-site}
defines a fully faithful functor
$$
\textit{Spaces}/S \longrightarrow \textit{Spaces}'/S
$$
whose essential image consists of those $X' \in \Ob(\textit{Spaces}'/S)$
such that there exist $U, R \in \Ob((\Sch/S)_{fppf})$\footnote{Requiring the
existence of $R$ is necessary because of our choice of the function $Bound$ in
Sets, Equation (\ref{sets-equation-bound}). The size of the fibre product
$U \times_{X'} U$ can grow faster than $Bound$ in terms of the size of $U$. We
can illustrate this by setting $S = \Spec(A)$, $U = \Spec(A[x_i, i \in I])$ and
$R = \coprod_{(\lambda_i) \in A^I} \Spec(A[x_i, y_i]/(x_i - \lambda_i y_i))$.
In this case the size of $R$ grows like $\kappa^\kappa$ where $\kappa$ is the
size of $U$.} and morphisms
$$
U \longrightarrow X'
\quad\text{and}\quad
R \longrightarrow U \times_{X'} U
$$
in $\Sh((\Sch'/S)_{fppf})$ which are surjective as maps of sheaves
(for example if the displayed morphisms are surjective and \'etale).
\end{lemma}

\begin{proof}
In Sites, Lemma \ref{sites-lemma-bigger-site} we have seen that the functor
$f^{-1} : \Sh((\Sch/S)_{fppf}) \to \Sh((\Sch'/S)_{fppf})$
is fully faithful (see discussion in
Topologies, Section \ref{topologies-section-change-alpha}).
Hence we see that the displayed functor of the lemma is fully faithful.

\medskip\noindent
Suppose that $X' \in \Ob(\textit{Spaces}'/S)$ such that there exists
$U \in \Ob((\Sch/S)_{fppf})$ and a map $U \to X'$ in
$\Sh((\Sch'/S)_{fppf})$ which is surjective as a map of sheaves.
Let $U' \to X'$ be a surjective \'etale morphism with
$U' \in \Ob((\Sch'/S)_{fppf})$. Let $\kappa = \text{size}(U)$, see
Sets, Section \ref{sets-section-categories-schemes}.
Then $U$ has an affine open covering $U = \bigcup_{i \in I} U_i$
with $|I| \leq \kappa$. Observe that $U' \times_{X'} U \to U$ is \'etale
and surjective. For each $i$ we can pick a quasi-compact
open $U'_i \subset U'$ such that $U'_i \times_{X'} U_i \to U_i$
is surjective (because the scheme $U' \times_{X'} U_i$ is the
union of the Zariski opens $W \times_{X'} U_i$ for $W \subset U'$
affine and because $U' \times_{X'} U_i \to U_i$ is \'etale hence open).
Then $\coprod_{i \in I} U'_i \to X$ is surjective \'etale
because of our assumption that $U \to X$ and hence $\coprod U_i \to X$
is a surjection of sheaves (details omitted).
Because $U'_i \times_{X'} U \to U'_i$ is a surjection of sheaves
and because $U'_i$ is quasi-compact,
we can find a quasi-compact open $W_i \subset U'_i \times_{X'} U$
such that $W_i \to U'_i$ is surjective as a map of sheaves
(details omitted). Then $W_i \to U$ is \'etale and we conclude
that $\text{size}(W_i) \leq \text{size}(U)$, see
Sets, Lemma \ref{sets-lemma-bound-finite-type}. By
Sets, Lemma \ref{sets-lemma-bound-by-covering} we conclude
that $\text{size}(U'_i) \leq \text{size}(U)$.
Hence $\coprod_{i \in I} U'_i$ is isomorphic to an object of
$(\Sch/S)_{fppf}$ by Sets, Lemma \ref{sets-lemma-bound-size}.

\medskip\noindent
Now let $X'$, $U \to X'$ and $R \to U \times_{X'} U$ be as in the
statement of the lemma. In the previous paragraph we have seen that
we can find $U' \in \Ob((\Sch/S)_{fppf})$ and a surjective \'etale morphism
$U' \to X'$ in $\Sh((\Sch'/S)_{fppf})$. Then
$U' \times_{X'} U \to U'$ is a surjection of sheaves, i.e., we can find an fppf
covering $\{U'_i \to U'\}$ such that $U'_i \to U'$ factors through
$U' \times_{X'} U \to U'$.
By Sets, Lemma \ref{sets-lemma-bound-fppf-covering}
we can find
$\tilde U \to U'$
which is surjective, flat, and locally of finite presentation,
with $\text{size}(\tilde U) \leq \text{size}(U')$,
such that $\tilde U \to U'$ factors through
$U' \times_{X'} U \to U'$. Then we consider
$$
\xymatrix{
U' \times_{X'} U' \ar[d] &
\tilde U \times_{X'} \tilde U  \ar[l] \ar[d] \ar[r] &
U \times_{X'} U \ar[d] \\
U' \times_S U' & \tilde U \times_S \tilde U \ar[l] \ar[r] & U \times_S U
}
$$
The squares are cartesian. We know the objects of the bottom row
are represented by objects of $(\Sch/S)_{fppf}$. By the result of the
argument of the previous paragraph, the same is true for $U \times_{X'} U$
(as we have the surjection of sheaves $R \to U \times_{X'} U$
by assumption). Since $(\Sch/S)_{fppf}$ is closed under fibre
products (by construction), we see that $\tilde U \times_{X'} \tilde U$
is represented by an object of $(\Sch/S)_{fppf}$. Finally, the
map $\tilde U \times_{X'} \tilde U \to U' \times_{X'} U'$ is
a surjection of fppf sheaves as $\tilde U \to U'$ is so.
Thus we can once more apply the result of the previous paragraph
to conclude that $R' = U' \times_{X'} U'$ is represented by an object
of $(\Sch/S)_{fppf}$. At this point
Lemma \ref{lemma-space-presentation} and
Theorem \ref{theorem-presentation} imply that $X = h_{U'}/h_{R'}$
is an object of $\textit{Spaces}/S$ such that $f^{-1}X \cong X'$
as desired.
\end{proof}

Comments (0)

There are no comments yet for this tag.

Add a comment on tag 03FO

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

This captcha seems more appropriate than the usual illegible gibberish, right?