Lemma 10.127.16. Suppose $R \to S$ is a ring map. Assume that $S$ is of finite type over $R$. Then there exists a directed set $(\Lambda , \leq )$, and a system of ring maps $R_\lambda \to S_\lambda$ such that

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

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

3. Each $S_\lambda$ is 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 a quotient of $S_\lambda \otimes _{R_\lambda } R_\mu$.

Proof. This is the non-local version of Lemma 10.127.10. Proof is similar and left to the reader. $\square$

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