Lemma 37.16.3. Let S be a scheme. Let f : X \to Y be a morphism of schemes over S. Assume
S, X, Y are locally Noetherian,
X is flat over S,
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.
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