# 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)