Theorem 41.12.2. Let f : A \to B be an unramified morphism 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 quotient of 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 41.3.2 we see that A \to B' is unramified at \mathfrak q. Hence we may apply Algebra, Proposition 10.152.1 to see that a principal localization of B' is a quotient of a standard étale A-algebra. \square
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