Lemma 10.127.9. 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
The colimit of the system $R_\lambda \to S_\lambda $ is equal to $R \to S$.
Each $R_\lambda $ is essentially of finite type over $\mathbf{Z}$.
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
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
The transition maps are clear. We leave the proofs of the other assertions to the reader. $\square$
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