Lemma 10.148.3. Let $R \to S$ be a ring map. The following are equivalent:

1. $R \to S$ is formally unramified,

2. $R \to S_{\mathfrak q}$ is formally unramified for all primes $\mathfrak q$ of $S$, and

3. $R_{\mathfrak p} \to S_{\mathfrak q}$ is formally unramified for all primes $\mathfrak q$ of $S$ with $\mathfrak p = R \cap \mathfrak q$.

Proof. We have seen in Lemma 10.148.2 that (1) is equivalent to $\Omega _{S/R} = 0$. Similarly, by Lemma 10.131.8 we see that (2) and (3) are equivalent to $(\Omega _{S/R})_{\mathfrak q} = 0$ for all $\mathfrak q$. Hence the equivalence follows from Lemma 10.23.1. $\square$

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).