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The Stacks project

98.16 Algebraic spaces

The following is our first main result on algebraic spaces.

Proposition 98.16.1. 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 98.15), 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 98.13.8 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 80.12.3 (this is where we use axiom [0]). By Criteria for Representability, Lemma 97.5.6 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 29.16.7. 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

Lemma 98.16.2. Let S be a locally Noetherian scheme. Let a : F \to G be a transformation of functors (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}. Assume that

  1. a is injective,

  2. F satisfies axioms [0], [1], [2], [4], and [5],

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

  4. G is an algebraic space locally of finite type over S,

Then F is an algebraic space.

Proof. By Lemma 98.8.1 the functor G satisfies [3]. As F \to G is injective, we conclude that F also satisfies [3]. Moreover, as F \to G is injective, we see that given schemes U, V and morphisms U \to F and V \to F, then U \times _ F V = U \times _ G V. Hence \Delta : F \to F \times F is representable (by schemes) as this holds for G by assumption. Thus Proposition 98.16.1 applies2. \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.
[2] The set theoretic condition [-1] holds for F as it holds for G. Details omitted.

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