Lemma 90.3.4. Let $A_ i \to B_ i$ be a system of ring maps over a directed index set $I$. Then $\mathop{\mathrm{colim}}\nolimits L_{B_ i/A_ i} = L_{\mathop{\mathrm{colim}}\nolimits B_ i/\mathop{\mathrm{colim}}\nolimits A_ i}$.

**Proof.**
This is true because the forgetful functor $V : A\textit{-Alg} \to \textit{Sets}$ and its adjoint $U : \textit{Sets} \to A\textit{-Alg}$ commute with filtered colimits. Moreover, the functor $B/A \mapsto \Omega _{B/A}$ does as well (Algebra, Lemma 10.130.4).
$\square$

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