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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