Lemma 35.33.7. Let $r \in \{ 0, 1, 2, \ldots , \infty \} $. The property of morphisms of germs
is étale local on the source-and-target.
Lemma 35.33.7. Let $r \in \{ 0, 1, 2, \ldots , \infty \} $. The property of morphisms of germs
is étale 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
Note that the vertical arrows are localizations of étale 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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