Remark 98.25.1 (Warning). The site $(\textit{Noetherian}/S)_\tau $ does not have fibre products. Hence we have to be careful in working with sheaves. For example, the continuous inclusion functor $(\textit{Noetherian}/S)_\tau \to (\mathit{Sch}/S)_\tau $ does not define a morphism of sites. See Examples, Section 110.59 for an example in case $\tau = fppf$.
98.25 Limit preserving functors on Noetherian schemes
It is sometimes convenient to consider functors or stacks defined only on the full subcategory of (locally) Noetherian schemes. In this section we discuss this in the case of algebraic spaces.
Let $S$ be a locally Noetherian scheme. Let us be a bit pedantic in order to line up our categories correctly; people who are ignoring set theoretical issues can just replace the sets of schemes we choose by the collection of all schemes in what follows. As in Topologies, Remark 34.11.1 we choose a category $\mathit{Sch}_\alpha $ of schemes containing $S$ such that we obtain big sites $(\mathit{Sch}/S)_{Zar}$, $(\mathit{Sch}/S)_{\acute{e}tale}$, $(\mathit{Sch}/S)_{smooth}$, $(\mathit{Sch}/S)_{syntomic}$, and $(\mathit{Sch}/S)_{fppf}$ all with the same underlying category $\mathit{Sch}_\alpha /S$. Denote
\[ \textit{Noetherian}_\alpha \subset \mathit{Sch}_\alpha \]
the full subcategory consisting of locally Noetherian schemes. This determines a full subcategory
\[ \textit{Noetherian}_\alpha /S \subset \mathit{Sch}_\alpha /S \]
For $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} $ we have
if $f : X \to Y$ is a morphism of $\mathit{Sch}_\alpha /S$ with $Y$ in $\textit{Noetherian}_\alpha /S$ and $f$ locally of finite type, then $X$ is in $\textit{Noetherian}_\alpha /S$,
for morphisms $f : X \to Y$ and $g : Z \to Y$ of $\textit{Noetherian}_\alpha /S$ with $f$ locally of finite type the fibre product $X \times _ Y Z$ in $\textit{Noetherian}_\alpha /S$ exists and agrees with the fibre product in $\mathit{Sch}_\alpha /S$,
if $\{ X_ i \to X\} _{i \in I}$ is a covering of $(\mathit{Sch}/S)_\tau $ and $X$ is in $\textit{Noetherian}_\alpha /S$, then the objects $X_ i$ are in $\textit{Noetherian}_\alpha /S$
the category $\textit{Noetherian}_\alpha /S$ endowed with the set of coverings of $(\mathit{Sch}/S)_\tau $ whose objects are in $\textit{Noetherian}_\alpha /S$ is a site we will denote $(\textit{Noetherian}/S)_\tau $,
the inclusion functor $(\textit{Noetherian}/S)_\tau \to (\mathit{Sch}/S)_\tau $ is fully faithful, continuous, and cocontinuous.
By Sites, Lemmas 7.21.1 and 7.21.5 we obtain a morphism of topoi
\[ g_\tau : \mathop{\mathit{Sh}}\nolimits ((\textit{Noetherian}/S)_\tau ) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_\tau ) \]
whose pullback functor is the restriction of sheaves along the inclusion functor $(\textit{Noetherian}/S)_\tau \to (\mathit{Sch}/S)_\tau $.
Let $F : (\textit{Noetherian}/S)_\tau ^{opp} \to \textit{Sets}$ be a functor. We say $F$ is limit preserving if for any directed limit of affine schemes $X = \mathop{\mathrm{lim}}\nolimits X_ i$ of $(\textit{Noetherian}/S)_\tau $ we have $F(X) = \mathop{\mathrm{colim}}\nolimits F(X_ i)$.
Lemma 98.25.2. Let $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} $. Restricting along the inclusion functor $(\textit{Noetherian}/S)_\tau \to (\mathit{Sch}/S)_\tau $ defines an equivalence of categories between
the category of limit preserving sheaves on $(\mathit{Sch}/S)_\tau $ and
the category of limit preserving sheaves on $(\textit{Noetherian}/S)_\tau $
Proof. Let $F : (\textit{Noetherian}/S)_\tau ^{opp} \to \textit{Sets}$ be a functor which is both limit preserving and a sheaf. By Topologies, Lemmas 34.13.1 and 34.13.3 there exists a unique functor $F' : (\mathit{Sch}/S)_\tau ^{opp} \to \textit{Sets}$ which is limit preserving, a sheaf, and restricts to $F$. In fact, the construction of $F'$ in Topologies, Lemma 34.13.1 is functorial in $F$ and this construction is a quasi-inverse to restriction. Some details omitted. $\square$
Lemma 98.25.3. Let $X$ be an object of $(\textit{Noetherian}/S)_\tau $. If the functor of points $h_ X : (\textit{Noetherian}/S)_\tau ^{opp} \to \textit{Sets}$ is limit preserving, then $X$ is locally of finite presentation over $S$.
Proof. Let $V \subset X$ be an affine open subscheme which maps into an affine open $U \subset S$. We may write $V = \mathop{\mathrm{lim}}\nolimits V_ i$ as a directed limit of affine schemes $V_ i$ of finite presentation over $U$, see Algebra, Lemma 10.127.2. By assumption, the arrow $V \to X$ factors as $V \to V_ i \to X$ for some $i$. After increasing $i$ we may assume $V_ i \to X$ factors through $V$ as the inverse image of $V \subset X$ in $V_ i$ eventually becomes equal to $V_ i$ by Limits, Lemma 32.4.11. Then the identity morphism $V \to V$ factors through $V_ i$ for some $i$ in the category of schemes over $U$. Thus $V \to U$ is of finite presentation; the corresponding algebra fact is that if $B$ is an $A$-algebra such that $\text{id} : B \to B$ factors through a finitely presented $A$-algebra, then $B$ is of finite presentation over $A$ (nice exercise). Hence $X$ is locally of finite presentation over $S$. $\square$
The following lemma has a variant for transformations representable by algebraic spaces.
Lemma 98.25.4. Let $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} $. Let $F', G' : (\mathit{Sch}/S)_\tau ^{opp} \to \textit{Sets}$ be limit preserving and sheaves. Let $a' : F' \to G'$ be a transformation of functors. Denote $a : F \to G$ the restriction of $a' : F' \to G'$ to $(\textit{Noetherian}/S)_\tau $. The following are equivalent
$a'$ is representable (as a transformation of functors, see Categories, Definition 4.6.4), and
for every object $V$ of $(\textit{Noetherian}/S)_\tau $ and every map $V \to G$ the fibre product $F \times _ G V : (\textit{Noetherian}/S)_\tau ^{opp} \to \textit{Sets}$ is a representable functor, and
same as in (2) but only for $V$ affine finite type over $S$ mapping into an affine open of $S$.
Proof. Assume (1). By Limits of Spaces, Lemma 70.3.4 the transformation $a'$ is limit preserving1. Take $\xi : V \to G$ as in (2). Denote $V' = V$ but viewed as an object of $(\mathit{Sch}/S)_\tau $. Since $G$ is the restriction of $G'$ to $(\textit{Noetherian}/S)_\tau $ we see that $\xi \in G(V)$ corresponds to $\xi ' \in G'(V')$. By assumption $V' \times _{\xi ', G'} F'$ is representable by a scheme $U'$. The morphism of schemes $U' \to V'$ corresponding to the projection $V' \times _{\xi ', G'} F' \to V'$ is locally of finite presentation by Limits of Spaces, Lemma 70.3.5 and Limits, Proposition 32.6.1. Hence $U'$ is a locally Noetherian scheme and therefore $U'$ is isomorphic to an object $U$ of $(\textit{Noetherian}/S)_\tau $. Then $U$ represents $F \times _ G V$ as desired.
The implication (2) $\Rightarrow $ (3) is immediate. Assume (3). We will prove (1). Let $T$ be an object of $(\mathit{Sch}/S)_\tau $ and let $T \to G'$ be a morphism. We have to show the functor $F' \times _{G'} T$ is representable by a scheme $X$ over $T$. Let $\mathcal{B}$ be the set of affine opens of $T$ which map into an affine open of $S$. This is a basis for the topology of $T$. Below we will show that for $W \in \mathcal{B}$ the fibre product $F' \times _{G'} W$ is representable by a scheme $X_ W$ over $W$. If $W_1 \subset W_2$ in $\mathcal{B}$, then we obtain an isomorphism $X_{W_1} \to X_{W_2} \times _{W_2} W_1$ because both $X_{W_1}$ and $X_{W_2} \times _{W_2} W_1$ represent the functor $F' \times _{G'} W_1$. These isomorphisms are canonical and satisfy the cocycle condition mentioned in Constructions, Lemma 27.2.1. Hence we can glue the schemes $X_ W$ to a scheme $X$ over $T$. Compatibility of the glueing maps with the maps $X_ W \to F'$ provide us with a map $X \to F'$. The resulting map $X \to F' \times _{G'} T$ is an isomorphism as we may check this locally on $T$ (as source and target of this arrow are sheaves for the Zariski topology).
Let $W$ be an affine scheme which maps into an affine open $U \subset S$. Let $W \to G'$ be a map. Still assuming (3) we have to show that $F' \times _{G'} W$ is representable by a scheme. We may write $W = \mathop{\mathrm{lim}}\nolimits V'_ i$ as a directed limit of affine schemes $V'_ i$ of finite presentation over $U$, see Algebra, Lemma 10.127.2. Since $V'_ i$ is of finite type over an Noetherian scheme, we see that $V'_ i$ is a Noetherian scheme. Denote $V_ i = V'_ i$ but viewed as an object of $(\textit{Noetherian}/S)_\tau $. As $G'$ is limit preserving can choose an $i$ and a map $V'_ i \to G'$ such that $W \to G'$ is the composition $W \to V'_ i \to G'$. Since $G$ is the restriction of $G'$ to $(\textit{Noetherian}/S)_\tau $ the morphism $V'_ i \to G'$ is the same thing as a morphism $V_ i \to G$ (see above). By assumption (3) the functor $F \times _ G V_ i$ is representable by an object $X_ i$ of $(\textit{Noetherian}/S)_\tau $. The functor $F \times _ G V_ i$ is limit preserving as it is the restriction of $F' \times _{G'} V'_ i$ and this functor is limit preserving by Limits of Spaces, Lemma 70.3.6, the assumption that $F'$ and $G'$ are limit preserving, and Limits, Remark 32.6.2 which tells us that the functor of points of $V'_ i$ is limit preserving. By Lemma 98.25.3 we conclude that $X_ i$ is locally of finite presentation over $S$. Denote $X'_ i = X_ i$ but viewed as an object of $(\mathit{Sch}/S)_\tau $. Then we see that $F' \times _{G'} V'_ i$ and the functors of points $h_{X'_ i}$ are both extensions of $h_{X_ i} : (\textit{Noetherian}/S)_\tau ^{opp} \to \textit{Sets}$ to limit preserving sheaves on $(\mathit{Sch}/S)_\tau $. By the equivalence of categories of Lemma 98.25.2 we deduce that $X'_ i$ represents $F' \times _{G'} V'_ i$. Then finally
\[ F' \times _{G'} W = F' \times _{G'} V'_ i \times _{V'_ i} W = X'_ i \times _{V'_ i} W \]
is representable as desired. $\square$
[1] This makes sense even if $\tau \not= fppf$ as the underlying category of $(\mathit{Sch}/S)_\tau $ equals the underlying category of $(\mathit{Sch}/S)_{fppf}$ and the statement doesn't refer to the topology.
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