The Stacks project

Lemma 35.33.6. Let $d \in \{ 0, 1, 2, \ldots , \infty \} $. The property of morphisms of germs

\[ \mathcal{P}_ d((X, x) \to (S, s)) = \text{the local ring } \mathcal{O}_{X_ s, x} \text{ of the fibre has dimension }d \]

is étale local on the source-and-target.

Proof. Given a diagram as in Definition 35.33.1 we obtain an étale morphism of fibres $U'_{v'} \to U_ v$ mapping $u'$ to $u$, see Lemma 35.33.5. Hence the result follows from Lemma 35.21.3. $\square$

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