We finally present the promised functorial characterization. Thus there are four ways to think about étale morphisms of schemes:
as a smooth morphism of relative dimension 0,
as locally finitely presented, flat, and unramified morphisms,
using the structure theorem, and
using the functorial characterization.
Theorem 41.16.1. Let f : X \to S be a morphism that is locally of finite presentation. The following are equivalent
f is étale,
for all affine S-schemes Y, and closed subschemes Y_0 \subset Y defined by square-zero ideals, the natural map
is bijective.
Proof. This is More on Morphisms, Lemma 37.8.9. \square
This characterization says that solutions to the equations defining X can be lifted uniquely through nilpotent thickenings.
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