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

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

2. $B$ 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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