---
tag: FTRW0679E2
kind: concept
name: Étale morphism of rings
msc: [13B40, 14B25]
specializes: [286G0P1B69, GY3QA6NH0E]
sources:
  - url: https://en.wikipedia.org/wiki/Étale_algebra
---

## Summary

An étale morphism of rings is a map $R \to A$ of commutative rings that is both [[smooth|286G0P1B69]] and [[unramified|GY3QA6NH0E]].
Equivalently, $A$ is finitely presented, flat and unramified over $R$.
It is the algebraic analogue of a local diffeomorphism, or of a covering map.
Over a field $k$, the étale algebras are the finite products of finite separable field extensions of $k$.
Étale maps are the basis of étale cohomology and of the Henselization of a local ring.

## Explanations

### L1 human

An étale map looks, near each point, like an isomorphism to first order, even though it may not be an isomorphism.
A typical example is adjoining a simple root of a polynomial: $R[x]/(f)$ where the derivative $f'$ is a unit.
Over $\Z[\frac12]$, adjoining a square root of $-1$ is étale, while over $\Z$ it is not, because of the prime $2$.
Over a field, an étale algebra is a finite product of separable field extensions, such as $\Q(\sqrt 2) \times \Q$.
Étale maps let one pass to finer "neighbourhoods" than the Zariski topology allows.