The Stacks project

Proof. Given a diagram as in Descent, Definition 35.33.1 we obtain an étale morphism of fibres $U'_{v'} \to U_ v$ mapping $u'$ to $u$, see Descent, Lemma 35.33.5. Thus $\mathcal{O}_{U_ v, u} \to \mathcal{O}_{U'_{v'}, u'}$ is the localization of an étale ring map. Hence the first is Noetherian if and only if the second is Noetherian, see More on Algebra, Lemma 15.44.1. Then, since $\mathcal{O}_{U'_{v'}, u'}/\mathfrak m_ u \mathcal{O}_{U'_{v'}, u'} = \kappa (u')$ (Algebra, Lemma 10.143.5) is a Gorenstein ring, we see that $\mathcal{O}_{U_ v, u}$ is Gorenstein if and only if $\mathcal{O}_{U'_{v'}, u'}$ is Gorenstein by Dualizing Complexes, Lemma 47.21.8. $\square$