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Since its creation, the entry étale map started out as wanting to be about the abstraction of the notion of local homeomorphism and étale morphism to more general contexts.
I have added references to such axiomatizations now. But I am not sure that it is good to have “étale map” be interpreted so much differently from “étale morphism”. I think both of these should point to the same page, which discusses the general abstract notion, and then what currently is étale morphism should be renamed to “étale morphism of schemes”.
Finally, I changed the wording of the Idea-section at the beginning of étale map: it used to say that an étale map is like a “local isomorphism”. As we have recently seen in discussion, this is a very misleading thing to say, since it is not like a local isomorphism but like a local homeomorphism. These two concepts, maybe unfortunately termed, are not about the same idea.
I wrote:
But I am not sure that it is good to have “étale map” be interpreted so much differently from “étale morphism”. I think both of these should point to the same page, which discusses the general abstract notion, and then what currently is étale morphism should be renamed to “étale morphism of schemes”.
I went ahead and implemented this. Give me a minute or two now to clear the cache to avoid the cache bug kicking in…
So there is now
etale map = etale morphism pointing to the general notion
At etale map I removed the following stubby lines, as they are redundant by the discussion at etale morphism of schemes:
An étale map between commutative rings is usually called a étale morphism of rings: a ring homomorphism with the property that, when viewed as a morphism between affine schemes, it is étale. See this comment by Harry Gindi for a purely ring-theoretic characterisation.
+– {: .query}
Zoran: that is the infinitesimal lifting property for smooth morphisms, need an additional condition in general.
=–
By the way, Zoran has still another old query box sitting at etale map. would be nice if somebody would take care of editing this.
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