The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Proposition 90.16.1. 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], [0], [1], [2], [3], [4], [5] (see Section 90.15), 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 90.13.8 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 72.12.3 (this is where we use axiom [0]). By Criteria for Representability, Lemma 89.5.6 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. $\square$

[1] 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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