An etale morphism \(U\to X\) is:

• A smooth morphism (or flat morphism) and an unramified morphism.
• A [smooth morphism](smooth%20morphism) of relative dimension zero.
• formally etale and locally of finite presentation.

The idea: like local diffeomorphisms of manifolds, so inducing isomorphisms on tangent spaces at every point. Thus some version of the implicit function theorem holds in the analytic setting. The analog of a covering space is a finite etale morphism

An etale cover is a family of morphisms \(\ts{U_i \to X}\) which are etale and [locally of finite type](locally%20of%20finite%20type) such that \(X \subseteq \Union U_i\).

Etale descent