Étale topology - Imported Knowlpedia Content

Definition

The étale topology is the Grothendieck topology on schemes in which a family

\{U_i\to U\}_{i\in I}

is a covering family exactly when every $U_i\to U$ is étale and the family is jointly surjective on underlying points:

|U|=\bigcup_{i\in I}\operatorname{im}(|U_i|\to |U|).

Unlike the Zariski topology, this is not merely a point-set topology on one scheme. It permits étale morphisms as local charts. Restricting it to schemes over a fixed scheme $X$ that are étale gives the small étale site of $X$ .

Intuition

An étale morphism is the algebraic-geometric analogue of a local diffeomorphism. Allowing such maps as local charts lets algebraic geometry see covers that the Zariski topology cannot detect.

The word “topology” here refers to a rule for covering objects in a category. It does not put a new collection of open subsets on the point set of a single scheme.

Basic example

For a finite separable field extension $K/F$ , $\operatorname{Spec}K\to\operatorname{Spec}F$ is a one-map étale cover.

This is already nontrivial: $\operatorname{Spec}F$ has only one Zariski point, so ordinary Zariski open covers cannot reveal the extension. Étale localization can.