Lemma 76.20.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is locally of finite type and $Y$ locally Noetherian. Let $Z \subset Y$ be a closed subspace with $n$th infinitesimal neighbourhood $Z_ n \subset Y$. Set $X_ n = Z_ n \times _ Y X$.
If $X_ n \to Z_ n$ is smooth for all $n$, then $f$ is smooth at every point of $f^{-1}(Z)$.
If $X_ n \to Z_ n$ is étale for all $n$, then $f$ is étale at every point of $f^{-1}(Z)$.
Proof. Assume $X_ n \to Z_ n$ is smooth for all $n$. Let $x \in X$ be a point lying over a point of $Z$. Given a small extension $B' \to B$ and morphisms $\alpha $, $\beta $ as in Lemma 76.20.1 part (3) the maximal ideal of $B'$ is nilpotent (as $B'$ is Artinian) and hence the morphism $\beta $ factors through $Z_ n$ and $\alpha $ factors through $X_ n$ for a suitable $n$. Thus the lifting property for $X_ n \to Z_ n$ kicks in to get the desired dotted arrow in the diagram. This proves (1). Part (2) follows from (1) and the fact that a morphism is étale if and only if it is smooth of relative dimension $0$. $\square$
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