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

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

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

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

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 $.

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