The Stacks project

Lemma 76.26.1. The property of morphisms of germs of schemes

\begin{align*} & \mathcal{P}((X, x) \to (S, s)) = \\ & \text{the local ring } \mathcal{O}_{X_ s, x} \text{ of the fibre is Noetherian and Cohen-Macaulay} \end{align*}

is étale local on the source-and-target (Descent, Definition 35.33.1).

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 the strict henselizations of the local rings $\mathcal{O}_{U'_{v'}, u'}$ and $\mathcal{O}_{U_ v, u}$ are the same. We conclude by More on Algebra, Lemma 15.45.9. $\square$

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