Lemma 37.12.3. Let $f : X \to S$ be a finite type morphism of locally Noetherian schemes. Let $Z \subset S$ be a closed subscheme with $n$th infinitesimal neighbourhood $Z_ n \subset S$. Set $X_ n = Z_ n \times _ S X$.

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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