Contents

Idea

The notion of étale map is an abstraction of that of local homeomorphism in topology. The concept is usually found in places with a geometric or topological flavour.

Examples

Between topological spaces

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).

Between smooth manifolds

An étale map between smooth spaces is a local diffeomorphisms, a special case of a local homeomorphism.

Between schemes (affine schemes / rings)

For an étale map between schemes see étale morphism of schemes.

Restricted to affine schemes, this yields, dually, a notion of étale morphisms between rings.

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.

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.

Between analytic spaces

• 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.

Axiomatization

The idea of étale morphisms can be axiomatized in any topos. This idea goes back to lectures by André Joyal in the 1970. See (Joyal-Moerdijk 1994) and (Dubuc 2000).

References

Axiomatizations of the notion of étale maps in general toposes are discussed in

• Eduardo Dubuc, Axiomatic étale maps and a theorem of spectrum, Journal of pure and applied algebra, 149 (2000)

Revised on March 19, 2012 12:14:30 by Urs Schreiber (89.204.155.155)