Lemma 31.18.5. Let f : X \to S be a morphism of schemes. Let D \subset X be a relative effective Cartier divisor on X/S. If x \in D and \mathcal{O}_{X, x} is Noetherian, then f is flat at x.
Proof. Set A = \mathcal{O}_{S, f(x)} and B = \mathcal{O}_{X, x}. Let h \in B be an element which generates the ideal of D. Then h is a nonzerodivisor in B such that B/hB is a flat local A-algebra. Let I \subset A be a finitely generated ideal. Consider the commutative diagram
\xymatrix{ 0 \ar[r] & B \ar[r]_ h & B \ar[r] & B/hB \ar[r] & 0 \\ 0 \ar[r] & B \otimes _ A I \ar[r]^ h \ar[u] & B \otimes _ A I \ar[r] \ar[u] & B/hB \otimes _ A I \ar[r] \ar[u] & 0 }
The lower sequence is short exact as B/hB is flat over A, see Algebra, Lemma 10.39.12. The right vertical arrow is injective as B/hB is flat over A, see Algebra, Lemma 10.39.5. Hence multiplication by h is surjective on the kernel K of the middle vertical arrow. By Nakayama's lemma, see Algebra, Lemma 10.20.1 we conclude that K= 0. Hence B is flat over A, see Algebra, Lemma 10.39.5. \square
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