nLab
etale cover

Contents

Definition

Definition

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

X= ip 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

Revised on November 22, 2013 04:18:33 by Urs Schreiber (82.169.114.243)