Processing math: 100%

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


Comments (0)


Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.