Lemma 76.27.5 (0E17)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.27: Gorenstein morphisms
Lemma 76.27.5 (cite)

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Lemma 76.27.5. Let $S$ be a scheme. Let $f : X \to Y$ be a flat morphism of locally Noetherian algebraic spaces over $S$ . If $X$ is Gorenstein, then $f$ is Gorenstein and $\mathcal{O}_{Y, f(\overline{x})}$ is Gorenstein for all $x \in |X|$ .

Proof. After translating into algebra using Lemma 76.27.3 (compare with the proof of Lemma 76.27.4) this follows from Dualizing Complexes, Lemma 47.21.8. $\square$

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