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. 



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


