Lemma 48.25.1 (0C03)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 48: Duality for Schemes
Section 48.25: Gorenstein morphisms
Lemma 48.25.1 (cite)

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Lemma 48.25.1. Let $X$ be a locally Noetherian scheme over the field $k$ . Let $k'/k$ be a finitely generated field extension. Let $x \in X$ be a point, and let $x' \in X_{k'}$ be a point lying over $x$ . Then we have

$\mathcal{O}_{X, x}\text{ is Gorenstein} \Leftrightarrow \mathcal{O}_{X_{k'}, x'}\text{ is Gorenstein}$

If $X$ is locally of finite type over $k$ , the same holds for any field extension $k'/k$ .

Proof. In both cases the ring map $\mathcal{O}_{X, x} \to \mathcal{O}_{X_{k'}, x'}$ is a faithfully flat local homomorphism of Noetherian local rings. Thus if $\mathcal{O}_{X_{k'}, x'}$ is Gorenstein, then so is $\mathcal{O}_{X, x}$ by Dualizing Complexes, Lemma 47.21.8. To go up, we use Dualizing Complexes, Lemma 47.21.8 as well. Thus we have to show that

$\mathcal{O}_{X_{k'}, x'}/\mathfrak m_ x \mathcal{O}_{X_{k'}, x'} = \kappa (x) \otimes _ k k'$

is Gorenstein. Note that in the first case $k \to k'$ is finitely generated and in the second case $k \to \kappa (x)$ is finitely generated. Hence this follows as property (A) holds for Gorenstein, see Dualizing Complexes, Lemma 47.23.1. $\square$

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