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
Comments (0)