nLab
etale cover

Definition

Definition

An étale cover of an algebraic scheme $X$ is a set $\{p_i:U_i\to X\}$ of étale morphisms locally of finite type which are jointly surjective in the sense that $X$ equals the union of set-theoretic images:

$$X={\cup }_i{p}_i(U_i).$$X = \union_i p_i(U_i).

Remark

The condition of being locally of finite type is just strengthening the variant of the notion of étale: in the case of non-Noetherian schemes Grothendieck requires instead that étale morphisms be locally of finite presentation; for the purpose of étale topology locally of finite type is required.

Remark

The étale site has coverings given by the étale covers.

Properties

Proposition

Every étale cover is a cover in the fpqc topology.

This appears for instance as (de Jong, lemma 3.3).

References

• Aise Johan de Jong, Étale cohomology (pdf)