The Stacks project

Lemma 35.33.8. Let $d \in \{ 0, 1, 2, \ldots , \infty \} $. The property of morphisms of germs

\[ \mathcal{P}_ d((X, x) \to (S, s)) \Leftrightarrow \dim _ x (X_ s) = d \]

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 now the equality $\dim _ u(U_ v) = \dim _{u'}(U'_{v'})$ follows from Lemma 35.21.2. $\square$

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