Lemma 10.168.5. 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 unramified,
C_0 is of finite type over B_0.
Then for some i \geq 0 the map A_ i \otimes _{A_0} B_0 \to A_ i \otimes _{A_0} C_0 is unramified.
Proof. Set B_ i = A_ i \otimes _{A_0} B_0, C_ i = A_ i \otimes _{A_0} C_0, B = A \otimes _{A_0} B_0, and C = A \otimes _{A_0} C_0. Let x_1, \ldots , x_ m be generators for C_0 over B_0. Then \text{d}x_1, \ldots , \text{d}x_ m generate \Omega _{C_0/B_0} over C_0 and their images generate \Omega _{C_ i/B_ i} over C_ i (Lemmas 10.131.14 and 10.131.9). Observe that 0 = \Omega _{C/B} = \mathop{\mathrm{colim}}\nolimits \Omega _{C_ i/B_ i} (Lemma 10.131.5). Thus there is an i such that \text{d}x_1, \ldots , \text{d}x_ m map to zero and hence \Omega _{C_ i/B_ i} = 0 as desired. \square
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