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