nLab étale 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:

$$X = \union_i p_i(U_i).$$

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 (tag 03PH of the stacks project).

References

• The Stacks project