Lemma 98.16.2 (07Y2)—The Stacks project

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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

$a$ is injective,
$F$ satisfies axioms [0], [1], [2], [4], and [5],
$\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$ ,
$G$ 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 applies ¹. $\square$

[1] The set theoretic condition [-1] holds for

$F$ as it holds for

$G$ . Details omitted.

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