6
$\begingroup$

Is the projection map $\amalg_{i \in \mathbb{Z}} X \to X$ from an infinite disjoint union of copies of a scheme $X$ to $X$ an etale map, using the definition in EGA IV 17? According to EGA, an etale map of schemes $f:X \to Y$ is locally of finite presentation and formally etale. Hence I don't think etale maps need to be quasicompact or quasifinite. However, wikipedia claims in property #5 under "Etale morphism" that etale maps are quasifinite, so either I'm missing something or wikipedia is using a different definition. I'm sorry to ask such a dumb question but this has come up repeatedly when I'm trying to read Champs Algebriques.

Improve this question
$\endgroup$
2
  • 7
    $\begingroup$ This error on Wikipedia has now been fixed. $\endgroup$
    – BCnrd
    Commented Dec 28, 2010 at 21:25
  • 3
    $\begingroup$ A question which leads to an improvement of a wikipedia article, especially when it involves mathematics beyond the ken of most editors, is certainly a good and useful question. No apologies necessary. Bonne chance avec les champs. $\endgroup$
    – Pete L. Clark
    Commented Dec 29, 2010 at 7:40

1 Answer 1

Reset to default
7
$\begingroup$

No, quasi-compact is not part of the definition of etale. Yes, your map from the infinite disjoint union to $X$ is etale.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .