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 10.17.8 that $\mathfrak p$ is in the image if and only if the ring $S \otimes _ R \kappa (\mathfrak p)$ is nonzero. $\square$

There are also:

• 1 comment(s) on Section 10.39: Flat modules and flat ring maps

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).