Lemma 10.127.6. Suppose that $R = \mathop{\mathrm{colim}}\nolimits _{\lambda \in \Lambda } R_\lambda $ is a directed colimit of rings. Then the category of finitely presented $R$-modules is the colimit of the categories of finitely presented $R_\lambda $-modules. More precisely

Given a finitely presented $R$-module $M$ there exists a $\lambda \in \Lambda $ and a finitely presented $R_\lambda $-module $M_\lambda $ such that $M \cong M_\lambda \otimes _{R_\lambda } R$.

Given a $\lambda \in \Lambda $, finitely presented $R_\lambda $-modules $M_\lambda , N_\lambda $, and an $R$-module map $\varphi : M_\lambda \otimes _{R_\lambda } R \to N_\lambda \otimes _{R_\lambda } R$, then there exists a $\mu \geq \lambda $ and an $R_\mu $-module map $\varphi _\mu : M_\lambda \otimes _{R_\lambda } R_\mu \to N_\lambda \otimes _{R_\lambda } R_\mu $ such that $\varphi = \varphi _\mu \otimes 1_ R$.

Given a $\lambda \in \Lambda $, finitely presented $R_\lambda $-modules $M_\lambda , N_\lambda $, and $R$-module maps $\varphi _\lambda , \psi _\lambda : M_\lambda \to N_\lambda $ such that $\varphi \otimes 1_ R = \psi \otimes 1_ R$, then $\varphi \otimes 1_{R_\mu } = \psi \otimes 1_{R_\mu }$ for some $\mu \geq \lambda $.

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