
Lemma 10.126.9. Suppose $R \to S$ is a local homomorphism of local rings. There exists a directed set $(\Lambda , \leq )$, and a system of local homomorphisms $R_\lambda \to S_\lambda$ of local rings such that

1. The colimit of the system $R_\lambda \to S_\lambda$ is equal to $R \to S$.

2. Each $R_\lambda$ is essentially of finite type over $\mathbf{Z}$.

3. Each $S_\lambda$ is essentially of finite type over $R_\lambda$.

Proof. Denote $\varphi : R \to S$ the ring map. Let $\mathfrak m \subset R$ be the maximal ideal of $R$ and let $\mathfrak n \subset S$ be the maximal ideal of $S$. Let

$\Lambda = \{ (A, B) \mid A \subset R, B \subset S, \# A < \infty , \# B < \infty , \varphi (A) \subset B \} .$

As partial ordering we take the inclusion relation. For each $\lambda = (A, B) \in \Lambda$ we let $R'_\lambda$ be the sub $\mathbf{Z}$-algebra generated by $a \in A$, and we let $S'_\lambda$ be the sub $\mathbf{Z}$-algebra generated by $b$, $b \in B$. Let $R_\lambda$ be the localization of $R'_\lambda$ at the prime ideal $R'_\lambda \cap \mathfrak m$ and let $S_\lambda$ be the localization of $S'_\lambda$ at the prime ideal $S'_\lambda \cap \mathfrak n$. In a picture

$\xymatrix{ B \ar[r] & S'_\lambda \ar[r] & S_\lambda \ar[r] & S \\ A \ar[r] \ar[u] & R'_\lambda \ar[r] \ar[u] & R_\lambda \ar[r] \ar[u] & R \ar[u] }.$

The transition maps are clear. We leave the proofs of the other assertions to the reader. $\square$

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