The Stacks project

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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View Lemma 10.127.18 as pdf