Theorem 41.12.1. Let $f : A \to B$ be an étale homomorphism of local rings. Then there exist $f, g \in A[t]$ such that

$B' = A[t]_ g/(f)$ is standard étale – see (a) and (b) above, and

$B$ is isomorphic to a localization of $B'$ at a prime.

Theorem 41.12.1. Let $f : A \to B$ be an étale homomorphism of local rings. Then there exist $f, g \in A[t]$ such that

$B' = A[t]_ g/(f)$ is standard étale – see (a) and (b) above, and

$B$ is isomorphic to a localization of $B'$ at a prime.

**Proof.**
Write $B = B'_{\mathfrak q}$ for some finite type $A$-algebra $B'$ (we can do this because $B$ is essentially of finite type over $A$). By Lemma 41.11.2 we see that $A \to B'$ is étale at $\mathfrak q$. Hence we may apply Algebra, Proposition 10.144.4 to see that a principal localization of $B'$ is standard étale.
$\square$

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