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Gorensteinnes of the total space of a flat fibration implies same for base and fibres

Lemma 48.25.7. Let $f : X \to Y$ be a flat morphism of locally Noetherian schemes. If $X$ is Gorenstein, then $f$ is Gorenstein and $\mathcal{O}_{Y, f(x)}$ is Gorenstein for all $x \in X$.

Proof. After translating into algebra this follows from Dualizing Complexes, Lemma 47.21.8. $\square$

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