Lemma 10.146.2. Let (R, \mathfrak m_ R) \to (S, \mathfrak m_ S) be a local homomorphism of local rings. Assume S is the localization of an étale ring extension of R. Then there exists a finite, finitely presented, faithfully flat ring map R \to S' such that for every maximal ideal \mathfrak m' of S' there is a factorization
of the ring map R \to S'_{\mathfrak m'}.
Proof. Write S = T_{\mathfrak q} for some étale R-algebra T. By Proposition 10.144.4 we may assume T is standard étale. Apply Lemma 10.144.5 to the ring map R \to T to get R \to S'. Then in particular for every maximal ideal \mathfrak m' of S' we get a factorization \varphi : T \to S'_{g'} for some g' \not\in \mathfrak m' such that \mathfrak q = \varphi ^{-1}(\mathfrak m'S'_{g'}). Thus \varphi induces the desired local ring map S \to S'_{\mathfrak m'}. \square
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