Remark 37.35.3. In terms of topologies Lemmas 37.35.1 and 37.35.2 mean the following. Let $S$ be any scheme. Let $\{ f_ i : U_ i \to S\} $ be an étale covering of $S$. There exists a Zariski open covering $S = \bigcup V_ j$, for each $j$ a finite locally free, surjective morphism $W_ j \to V_ j$, and for each $j$ a Zariski open covering $\{ W_{j, k} \to W_ j\} $ such that the family $\{ W_{j, k} \to S\} $ refines the given étale covering $\{ f_ i : U_ i \to S\} $. What does this mean in practice? Well, for example, suppose we have a descent problem which we know how to solve for Zariski coverings and for fppf coverings of the form $\{ \pi : T \to S\} $ with $\pi $ finite locally free and surjective. Then this descent problem has an affirmative answer for étale coverings as well. This trick was used by Gabber in his proof that $\text{Br}(X) = \text{Br}'(X)$ for an affine scheme $X$, see [Hoobler].

## Post a comment

Your email address will not be published.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.

## Comments (1)

Comment #418 by Pieter Belmans on