The Stacks project

Lemma 63.5.9. Let $S$ be a scheme. Let $F, G : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ be sheaves. Let $a : F \to G$ be representable, flat, locally of finite presentation, and surjective. Then $a : F \to G$ is surjective as a map of sheaves.

Proof. Let $T$ be a scheme over $S$ and let $g : T \to G$ be a $T$-valued point of $G$. By assumption $T' = F \times _ G T$ is (representable by) a scheme and the morphism $T' \to T$ is a flat, locally of finite presentation, and surjective. Hence $\{ T' \to T\} $ is an fppf covering such that $g|_{T'} \in G(T')$ comes from an element of $F(T')$, namely the map $T' \to F$. This proves the map is surjective as a map of sheaves, see Sites, Definition 7.11.1. $\square$


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