The Stacks project

Lemma 10.154.3. Let $R$ be a ring. Let $A = \mathop{\mathrm{colim}}\nolimits A_ i$ be a filtered colimit of $R$-algebras such that each $A_ i$ is a filtered colimit of étale $R$-algebras. Then $A$ is a filtered colimit of étale $R$-algebras.

Proof. Write $A_ i = \mathop{\mathrm{colim}}\nolimits _{j \in J_ i} A_ j$ where $J_ i$ is a directed set and $A_ j$ is an étale $R$-algebra. For each $i \leq i'$ and $j \in J_ i$ there exists an $j' \in J_{i'}$ and an $R$-algebra map $\varphi _{jj'} : A_ j \to A_{j'}$ making the diagram

\[ \xymatrix{ A_ i \ar[r] & A_{i'} \\ A_ j \ar[u] \ar[r]^{\varphi _{jj'}} & A_{j'} \ar[u] } \]

commute. This is true because $R \to A_ j$ is of finite presentation so that Lemma 10.127.3 applies. Let $\mathcal{J}$ be the category with objects $\coprod _{i \in I} J_ i$ and morphisms triples $(j, j', \varphi _{jj'})$ as above (and obvious composition law). Then $\mathcal{J}$ is a filtered category and $A = \mathop{\mathrm{colim}}\nolimits _\mathcal {J} A_ j$. Details omitted. $\square$

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