Lemma 41.11.3. Let $A$, $B$ be Noetherian local rings. Let $A \to B$ be a local homomorphism such that $B$ is essentially of finite type over $A$. The following are equivalent
$A \to B$ is an étale homomorphism of local rings
$A^\wedge \to B^\wedge $ is an étale homomorphism of local rings, and
$A^\wedge \to B^\wedge $ is étale.
Moreover, in this case $B^\wedge \cong (A^\wedge )^{\oplus n}$ as $A^\wedge $-modules for some $n \geq 1$.
Proof. To see the equivalences of (1), (2) and (3), as we have the corresponding results for unramified ring maps (Lemma 41.3.4) it suffices to prove that $A \to B$ is flat if and only if $A^\wedge \to B^\wedge $ is flat. This is clear from our lists of properties of flat maps since the ring maps $A \to A^\wedge $ and $B \to B^\wedge $ are faithfully flat. For the final statement, by Lemma 41.3.3 we see that $B^\wedge $ is a finite flat $A^\wedge $ module. Hence it is finite free by our list of properties on flat modules in Section 41.9. $\square$
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