Lemma 10.39.16. Let $R \to S$ be a flat ring map. The following are equivalent:

$R \to S$ is faithfully flat,

the induced map on $\mathop{\mathrm{Spec}}$ is surjective, and

any closed point $x \in \mathop{\mathrm{Spec}}(R)$ is in the image of the map $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$.

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