The Stacks project

Lemma 10.168.6. Let $A = \mathop{\mathrm{colim}}\nolimits _{i \in I} A_ i$ be a directed colimit of rings. Let $0 \in I$ and $\varphi _0 : B_0 \to C_0$ a map of $A_0$-algebras. Assume

  1. $A \otimes _{A_0} B_0 \to A \otimes _{A_0} C_0$ is an isomorphism,

  2. $B_0 \to C_0$ is of finite presentation.

Then for some $i \geq 0$ the map $A_ i \otimes _{A_0} B_0 \to A_ i \otimes _{A_0} C_0$ is an isomorphism.

Proof. By Lemma 10.168.4 there exists an $i$ such that $A_ i \otimes _{A_0} B_0 \to A_ i \otimes _{A_0} C_0$ is surjective. Since the map is of finite presentation the kernel is a finitely generated ideal. Let $g_1, \ldots , g_ r \in A_ i \otimes _{A_0} B_0$ generate the kernel. Then we may pick $i' \geq i$ such that $g_ j$ map to zero in $A_{i'} \otimes _{A_0} B_0$. Then $A_{i'} \otimes _{A_0} B_0 \to A_{i'} \otimes _{A_0} C_0$ is an isomorphism. $\square$

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