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The Stacks project

Lemma 10.127.17. Suppose R \to S is a ring map. Assume that S is of finite presentation 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 is an isomorphism.

Proof. This is the non-local version of Lemma 10.127.11. Proof is similar and left to the reader. \square


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