Lemma 41.11.2. Let $A \to B$ be of finite type with $A$ a Noetherian ring. Let $\mathfrak q$ be a prime of $B$ lying over $\mathfrak p \subset A$. Then $A \to B$ is étale at $\mathfrak q$ if and only if $A_{\mathfrak p} \to B_{\mathfrak q}$ is an étale homomorphism of local rings.

**Proof.**
See Algebra, Lemmas 10.143.3 (flatness of étale maps), 10.143.5 (étale maps are unramified) and 10.143.7 (flat and unramified maps are étale).
$\square$

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