10.146 Local homomorphisms

Lemma 10.146.1. 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

$R \to S \to S'_{\mathfrak m'}.$

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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