Lemma 10.148.3. Let $R \to S$ be a ring map. The following are equivalent:
$R \to S$ is formally unramified,
$R \to S_{\mathfrak q}$ is formally unramified for all primes $\mathfrak q$ of $S$, and
$R_{\mathfrak p} \to S_{\mathfrak q}$ 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. $\square$
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