(Étale)
\(f : X \to Y\) is étale if it is flat and unramified, equivalently flat, locally of finite presentation, with \(\Omega_{X/Y} = 0\), equivalently smooth of relative dimension \(0\).
An étale morphism is open. Finite étale morphisms form the category of covers used to define the étale fundamental group.
Étale does not imply local isomorphism in the Zariski topology: \(\GG_m \to \GG_m\), \(t \mapsto t^2\), over a field of characteristic not \(2\) is finite étale of degree \(2\) and is not an isomorphism over any nonempty Zariski open. Its base change along itself is trivial: \(\GG_m\times_{\GG_m}\GG_m\cong\GG_m\sqcup\GG_m\), the two components corresponding to the two square roots of \(1\).
A finite étale morphism of degree \(1\) is an isomorphism. Bijectivity on underlying points does not suffice: \(\Spec L\to\Spec k\) is bijective and finite étale for a nontrivial finite separable field extension \(L/k\).
Dependencies
D-MORUNRUnramified morphismsuses
Backlinks
D-HENSELHensel's lemma and henselizationrelated-toD-MORSMSmooth morphisms and relative dimensionusesD-ETFPPFThe étale and fppf topologies, and the small étale siteusesD-IV2ETCOVÉtale covers, trivial covers, and simple connectedness of $\PP^1$uses