Lemma 48.25.3. Let $f : X \to Y$ be a morphism of schemes. Assume all fibres of $f$ are locally Noetherian. The following are equivalent

$f$ is Gorenstein, and

$f$ is flat and its fibres are Gorenstein schemes.

Lemma 48.25.3. Let $f : X \to Y$ be a morphism of schemes. Assume all fibres of $f$ are locally Noetherian. The following are equivalent

$f$ is Gorenstein, and

$f$ is flat and its fibres are Gorenstein schemes.

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

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