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.
\[ \mathop{\mathrm{Mor}}\nolimits _ S(Y, X) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _ S(Y_0, X) \]
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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