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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