The Stacks project

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. $\square$

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 $S$. $\square$

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

is representable as desired. $\square$