The Stacks project

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

  1. $R \to S$ is faithfully flat,

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

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

Proof. This follows quickly from Lemma 10.39.15, because we saw in Remark 10.18.5 that $\mathfrak p$ is in the image if and only if the ring $S \otimes _ R \kappa (\mathfrak p)$ is nonzero. $\square$

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