Lemma 10.124.2. Let $(R, \mathfrak m_ R) \to (S, \mathfrak m_ S)$ be a local homomorphism of local rings. Assume

1. $R \to S$ is essentially of finite type,

2. $\kappa (\mathfrak m_ R) \subset \kappa (\mathfrak m_ S)$ is finite, and

3. $\dim (S/\mathfrak m_ RS) = 0$.

Then $S$ is the localization of a finite $R$-algebra.

Proof. Let $S'$ be a finite type $R$-algebra such that $S = S'_{\mathfrak q'}$ for some prime $\mathfrak q'$ of $S'$. By Definition 10.122.3 we see that $R \to S'$ is quasi-finite at $\mathfrak q'$. After replacing $S'$ by $S'_{g'}$ for some $g' \in S'$, $g' \not\in \mathfrak q'$ we may assume that $R \to S'$ is quasi-finite, see Lemma 10.123.13. Then by Lemma 10.123.14 there exists a finite $R$-algebra $S''$ and elements $g' \in S'$, $g' \not\in \mathfrak q'$ and $g'' \in S''$ such that $S'_{g'} \cong S''_{g''}$ as $R$-algebras. This proves the lemma. $\square$

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