---
tag: H84VHC8PX2
kind: concept
name: Étale morphism
msc: [14B25, 14A15]
specializes: [EXKGG9X6J0, 1JQX6HD8E5]
sources:
  - url: https://en.wikipedia.org/wiki/%C3%89tale_morphism
same_as:
  wikidata: Q8078428
  wikipedia: Étale morphism
---

## Summary

An étale morphism of schemes is a morphism that is flat, unramified and locally of finite presentation.
It is the algebraic analogue of a local diffeomorphism, or of a covering map in topology.
Étale morphisms are the open sets of the étale topology, which gives étale cohomology and the étale fundamental group.

## Explanations

### L1 human

The Zariski topology has very few open sets, so the inverse function theorem fails for it.
Étale morphisms replace the missing small open sets: they are maps that look locally like an isomorphism on tangent spaces.
For example, over a field of characteristic not $2$, the squaring map $t \mapsto t^2$ from the punctured line to itself is étale.
It is a two-sheeted covering, like the map $z \mapsto z^2$ on the punctured complex plane.
At the origin, the full squaring map of the line is not étale, because there the two sheets meet.