Lemma 37.12.4. Let $f : X \to S$ be a 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 flat for all $n$, then $f$ is flat at every point of $f^{-1}(Z)$.

**Proof.**
This is a translation of Algebra, Lemma 10.99.11 into the language of schemes.
$\square$

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