The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.149.3. 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 10.126.3 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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