Étale morphism

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 of relative dimension $0$. The ring version of this notion is an étale morphism of rings.