Lemma 31.22.3. Let f : X \to S be a morphism of schemes. Let Z \to X be a relative quasi-regular immersion. If x \in Z and \mathcal{O}_{X, x} is Noetherian, then f is flat at x.
Proof. Let f_1, \ldots , f_ r \in \mathcal{O}_{X, x} be a quasi-regular sequence cutting out the ideal of Z at x. By Algebra, Lemma 10.69.6 we know that f_1, \ldots , f_ r is a regular sequence. Hence f_ r is a nonzerodivisor on \mathcal{O}_{X, x}/(f_1, \ldots , f_{r - 1}) such that the quotient is a flat \mathcal{O}_{S, f(x)}-module. By Lemma 31.18.5 we conclude that \mathcal{O}_{X, x}/(f_1, \ldots , f_{r - 1}) is a flat \mathcal{O}_{S, f(x)}-module. Continuing by induction we find that \mathcal{O}_{X, x} is a flat \mathcal{O}_{S, s}-module. \square
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