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The Stacks project

Lemma 37.16.3. Let S be a scheme. Let f : X \to Y be a morphism of schemes over S. Assume

  1. S, X, Y are locally Noetherian,

  2. X is flat over S,

  3. for every s \in S the morphism f_ s : X_ s \to Y_ s is flat.

Then f is flat. If f is also surjective, then Y is flat over S.

Proof. This is a special case of Theorem 37.16.1. \square


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