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 $f^{-1}(Z)$.

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