## 41.16 The functorial characterization

We finally present the promised functorial characterization. Thus there are four ways to think about étale morphisms of schemes:

1. as a smooth morphism of relative dimension $0$,

2. as locally finitely presented, flat, and unramified morphisms,

3. using the structure theorem, and

4. 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

1. $f$ is étale,

2. for all affine $S$-schemes $Y$, and closed subschemes $Y_0 \subset Y$ defined by square-zero ideals, the natural map

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

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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