étale morphism


one way

Definition 1

A morphism of schemes f:XY is étale if it is flat and unramified.

This is the appropriate generalizationPlanetmathPlanetmath of “local homeomorphism” from topologyMathworldPlanetmathPlanetmath or “local isomorphism” from real differential geometry. Equivalently, f is étale if and only if any of the following conditions hold:

  • f is locally of finite type and formally étale.

  • f is flat and the relative sheaf of differentialsMathworldPlanetmath vanishes.

  • f is smooth of relative dimension zero.

  • f locally looks like A[x1,,xn]/(p1,,pn) where the Jacobian vanishes.

A morphism f:XY of varietiesMathworldPlanetmathPlanetmath over an algebraically closed field is étale at a point xX if it induces an isomorphismPlanetmathPlanetmathPlanetmath between the completed local rings 𝒪^x and 𝒪^f(x). If X and Y are over an arbitrary field k, then the required condition becomes that k(x) is a separablePlanetmathPlanetmathPlanetmath algebraic extensionMathworldPlanetmath of k(y), where y=f(x), and f induces an isomorphism between 𝒪^yk(y)k(x) and 𝒪^x.

A morphism f of nonsingular varieties over an algebraically closed field is étale if and only if f induces an isomorphism on the tangent spacesMathworldPlanetmath. In the differentiableMathworldPlanetmath category, the implicit function theorem implies that such a function is actually an isomorphism on some small neighborhood. On schemes, of course, the Zariski topologyMathworldPlanetmath is too coarse for this to be the case. One way to define a finer “topology”, making the scheme into a site, is by using étale maps.

The word étale comes from French, where it can be used to describe a calm or slack sea.


  • 1 Jean Dieudonné, A Panorama of Pure Mathematics, Academic Press, 1982.
  • 2 Robin Hartshorne, Algebraic GeometryMathworldPlanetmath, Springer–Verlag, 1977 (GTM 52).
Title étale morphism
Canonical name etaleMorphism
Date of creation 2013-03-22 14:08:40
Last modified on 2013-03-22 14:08:40
Owner mps (409)
Last modified by mps (409)
Numerical id 14
Author mps (409)
Entry type Definition
Classification msc 14F20
Classification msc 14A15
Synonym étale
Related topic site
Related topic Site
Related topic FlatMorphism
Related topic EtaleFundamentalGroup
Related topic EtaleCohomology
Related topic CoveringSpace