The Stacks project

Lemma 97.16.2. 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 97.8.1 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 97.16.1 applies1. $\square$

[1] The set theoretic condition [-1] holds for $F$ as it holds for $G$. Details omitted.

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