The Stacks project

Lemma 10.127.14. Suppose $R \to S$ is a ring map. 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 $.

Proof. This is the non-local version of Lemma 10.127.9. Proof is similar and left to the reader. $\square$

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