Proposition 92.16.1. Let $S$ be a locally Noetherian scheme. Let $F : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. Assume that

1. $\Delta : F \to F \times F$ is representable by algebraic spaces,

2. $F$ satisfies axioms [-1], , , , , ,  (see Section 92.15), and

3. $\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$.

Then $F$ is an algebraic space.

Proof. Lemma 92.13.8 applies to $F$. Using this we choose, for every finite type field $k$ over $S$ and $x_0 \in F(\mathop{\mathrm{Spec}}(k))$, an affine scheme $U_{k, x_0}$ of finite type over $S$ and a smooth morphism $U_{k, x_0} \to F$ such that there exists a finite type point $u_{k, x_0} \in U_{k, x_0}$ with residue field $k$ such that $x_0$ is the image of $u_{k, x_0}$. Then

$U = \coprod \nolimits _{k, x_0} U_{k, x_0} \longrightarrow F$

is smooth1. To finish the proof it suffices to show this map is surjective, see Bootstrap, Lemma 74.12.3 (this is where we use axiom ). By Criteria for Representability, Lemma 91.5.6 it suffices to show that $U \times _ F V \to V$ is surjective for those $V \to F$ where $V$ is an affine scheme locally of finite presentation over $S$. Since $U \times _ F V \to V$ is smooth the image is open. Hence it suffices to show that the image of $U \times _ F V \to V$ contains all finite type points of $V$, see Morphisms, Lemma 28.15.7. Let $v_0 \in V$ be a finite type point. Then $k = \kappa (v_0)$ is a finite type field over $S$. Denote $x_0$ the composition $\mathop{\mathrm{Spec}}(k) \xrightarrow {v_0} V \to F$. Then $(u_{k, x_0}, v_0) : \mathop{\mathrm{Spec}}(k) \to U \times _ F V$ is a point mapping to $v_0$ and we win. $\square$

 Set theoretical remark: This coproduct is (isomorphic) to an object of $(\mathit{Sch}/S)_{fppf}$ as we have a bound on the index set by axiom [-1], see Sets, Lemma 3.9.9.

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