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.

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    $\begingroup$ This error on Wikipedia has now been fixed. $\endgroup$
    – BCnrd
    Dec 28, 2010 at 21:25
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    $\begingroup$ 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. $\endgroup$ Dec 29, 2010 at 7:40

1 Answer 1


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


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