Lemma 37.12.4 (0D4G)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.12: Smoothness over a Noetherian base
Lemma 37.12.4 (cite)

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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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