nLab
etale map

An étale map is a local isomorphism. Exactly what this means depends on what ‘local’ means. Presumably (?) it makes sense in any site; the concept is usually found in places with a geometric or topological flavour:

• An étale map between topological spaces is a local homeomorphism; see étalé space (which is the total space of such a map viewed as a bundle).
• An étale map between smooth spaces is a local diffeomorphisms, very similar to (and a special case of) the above.
• For an étale map between schemes, see étale morphism.
• Étale maps between analytic spaces are closely related to étale maps between schemes, while also a special case of an étale map between smooth spaces.

Zoran: I do not understand this statement. Analytic spaces have a different structure sheaf; in general nilpotent elements are allowed. This is additional structure not present in theory of smooth spaces.

However, the term is also used in a dual sense in ring theory:

• An étale map between commutative rings is 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.

  Zoran: that is the infinitesimal lifting property for smooth morphisms, need an additional condition in general.

• Étale maps between noncommutative rings have also been considered.