Lemma 10.127.18. Suppose $R \to S$ is a ring map. Assume that $S$ is of finite presentation over $R$. Let $M$ be a finitely presented $S$-module. Then there exists a directed set $(\Lambda , \leq )$, and a system of ring maps $R_\lambda \to S_\lambda$ together with $S_\lambda$-modules $M_\lambda$, such that

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

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

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

4. Each $M_\lambda$ is finite over $S_\lambda$.

5. For each $\lambda \leq \mu$ the map $S_\lambda \otimes _{R_\lambda } R_\mu \to S_\mu$ is an isomorphism.

6. For each $\lambda \leq \mu$ the map $M_\lambda \otimes _{S_\lambda } S_\mu \to M_\mu$ is an isomorphism.

In particular, for every $\lambda \in \Lambda$ we have

$M = M_\lambda \otimes _{S_\lambda } S = M_\lambda \otimes _{R_\lambda } R.$

Proof. This is the non-local version of Lemma 10.127.13. Proof is similar and left to the reader. $\square$

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