Lemma 10.138.15. Let $A = \mathop{\mathrm{colim}}\nolimits A_ i$ be a filtered colimit of rings. Let $A \to B$ be a smooth ring map. There exists an $i$ and a smooth ring map $A_ i \to B_ i$ such that $B = B_ i \otimes _{A_ i} A$.

**Proof.**
Follows from Lemma 10.138.14 since $R_0 \to A$ will factor through $A_ i$ for some $i$ by Lemma 10.127.3.
$\square$

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