The Stacks project

Remark 37.40.3. In terms of topologies Lemmas 37.40.1 and 37.40.2 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].