---
tag: XWCAS1ESAA
kind: definition
name: Étale morphism
msc: [14B25, 14A15]
defines: H84VHC8PX2
uses: [JXRKH6P1Z4, 5M6KS8F47R, 1JQX6HD8E5, EXKGG9X6J0, 764GZHTGV8]
sources:
  - url: https://en.wikipedia.org/wiki/%C3%89tale_morphism
---

## Definition

Let $f \colon X \to Y$ be a [[morphism of schemes|JXRKH6P1Z4]].
The morphism $f$ is *étale* if it satisfies the following three conditions.

1. $f$ is locally of finite presentation: over affine opens $\operatorname{Spec} B \to \operatorname{Spec} A$, the $A$-algebra $B$ is [[finitely presented|764GZHTGV8]].
2. $f$ is [[flat|5M6KS8F47R]].
3. $f$ is [[unramified|1JQX6HD8E5]].

## Remarks

Let $k$ be a field of characteristic not $2$, and let $A = k[t, t^{-1}]$.
The morphism $\operatorname{Spec} A \to \operatorname{Spec} A$ given by $t \mapsto t^2$ is étale.
The ring $A$ is free of rank $2$ over the subring $k[t^2, t^{-2}]$ with basis $1, t$, so the map is flat and finitely presented.
Its module of relative differentials is generated by $dt$ with the relation $2t \, dt = 0$, and $2t$ is a unit, so it is zero and the map is unramified.
The same map on $\operatorname{Spec} k[t]$ is not étale: at $t = 0$ the relation $2t \, dt = 0$ does not kill $dt$, so clause 3 fails.
Equivalently, an étale morphism is a [[smooth morphism|EXKGG9X6J0]] of relative dimension $0$.
The ring version of this notion is an [[étale morphism of rings|FTRW0679E2]].