Lemma 10.127.10. Suppose $R \to S$ is a local homomorphism of local rings. Assume that $S$ is essentially of finite type over $R$. Then 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 $R_\lambda$.

4. For each $\lambda \leq \mu$ the map $S_\lambda \otimes _{R_\lambda } R_\mu \to S_\mu$ presents $S_\mu$ as the localization of a quotient of $S_\lambda \otimes _{R_\lambda } R_\mu$.

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 $x_1, \ldots , x_ n \in S$ be elements such that $S$ is a localization of the sub $R$-algebra of $S$ generated by $x_1, \ldots , x_ n$. In other words, $S$ is a quotient of a localization of the polynomial ring $R[x_1, \ldots , x_ n]$.

Let $\Lambda = \{ A \subset R \mid \# A < \infty \}$ be the set of finite subsets of $R$. As partial ordering we take the inclusion relation. For each $\lambda = A \in \Lambda$ 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 $\varphi (a)$, $a \in A$ and the elements $x_1, \ldots , x_ n$. 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

$\xymatrix{ \varphi (A) \amalg \{ x_ i\} \ar[r] & S'_\lambda \ar[r] & S_\lambda \ar[r] & S \\ A \ar[r] \ar[u] & R'_\lambda \ar[r] \ar[u] & R_\lambda \ar[r] \ar[u] & R \ar[u] }$

It is clear that if $A \subset B$ corresponds to $\lambda \leq \mu$ in $\Lambda$, then there are canonical maps $R_\lambda \to R_\mu$, and $S_\lambda \to S_\mu$ and we obtain a system over the directed set $\Lambda$.

The assertion that $R = \mathop{\mathrm{colim}}\nolimits R_\lambda$ is clear because all the maps $R_\lambda \to R$ are injective and any element of $R$ eventually is in the image. The same argument works for $S = \mathop{\mathrm{colim}}\nolimits S_\lambda$. Assertions (2), (3) are true by construction. The final assertion holds because clearly the maps $S'_\lambda \otimes _{R'_\lambda } R'_\mu \to S'_\mu$ are surjective. $\square$

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