Weil conjectures 7: What is an etale morphism?
Channel: Richard E Borcherds
Date: 2020-10-29

This talk explains what etale morphisms are in algebraic geometry. We first review etale morphisms in the usual topology of complex manifolds, where they are just local homeomorphism, and explain why this does not work in algebraic geometry. We give a provisional definition of etale morphisms as morphisms that induce isomorphisms of complete local rings, which works for varieties over algebraically closed fields. Then we give the definition of etale morphisms as formally etale morphisms that are locally of finite presentation, where formally etale morphisms are those satisfying a certain lifting property. Finally we give a few examples of etale morphisms over fields that are not algebraically closed.