An étale morphism of schemes is a morphism that is flat, unramified, and locally of finite presentation.
It is an algebraic analog of a local isomorphism.
An étale morphism need not be a globally injective map. For a field 
, the morphism 
from a finite extension
is etale precisely when the extension
is separable. A finite etale surjective morphism is called
a finite etale cover.
Finite etale covers define the etale fundamental group and enter the 
-adic section conjecture.
