Lemma 37.18.2. Let $f : X \to Y$ be a morphism of schemes. Assume all fibres of $f$ are locally Noetherian. The following are equivalent

$f$ is normal, and

$f$ is flat and its fibres are geometrically normal schemes.

Lemma 37.18.2. Let $f : X \to Y$ be a morphism of schemes. Assume all fibres of $f$ are locally Noetherian. The following are equivalent

$f$ is normal, and

$f$ is flat and its fibres are geometrically normal schemes.

**Proof.**
This follows directly from the definitions.
$\square$

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