The Stacks project

Lemma 35.33.7. Let $r \in \{ 0, 1, 2, \ldots , \infty \} $. The property of morphisms of germs

\[ \mathcal{P}_ r((X, x) \to (S, s)) \Leftrightarrow \text{trdeg}_{\kappa (s)} \kappa (x) = r \]

is ├ętale local on the source-and-target.

Proof. Given a diagram as in Definition 35.33.1 we obtain the following diagram of local homomorphisms of local rings

\[ \xymatrix{ \mathcal{O}_{U', u'} & \mathcal{O}_{V', v'} \ar[l] \\ \mathcal{O}_{U, u} \ar[u] & \mathcal{O}_{V, v} \ar[l] \ar[u] } \]

Note that the vertical arrows are localizations of ├ętale ring maps, in particular they are unramified (see Algebra, Section 10.143). Hence $\kappa (u')/\kappa (u)$ and $\kappa (v')/\kappa (v)$ are finite separable field extensions. Thus we have $\text{trdeg}_{\kappa (v)} \kappa (u) = \text{trdeg}_{\kappa (v')} \kappa (u)$ which proves the lemma. $\square$

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