Lemma 37.41.2. Let f : X \to S be a morphism of schemes. Let x_1, \ldots , x_ n \in X be points having the same image s in S. Assume that
f is locally of finite type, and
x_ i \in X_ s is isolated for i = 1, \ldots , n.
Then there exist
an elementary étale neighbourhood (U, u) \to (S, s),
for each i an open subscheme V_ i \subset X_ U,
such that for each i we have
V_ i \to U is a finite morphism,
there is a unique point v_ i of V_ i mapping to u in U, and
the point v_ i maps to x_ i in X and \kappa (x_ i) = \kappa (v_ i).
Proof.
We will use induction on n. Namely, suppose (U, u) \to (S, s) and V_ i \subset X_ U, i = 1, \ldots , n - 1 work for x_1, \ldots , x_{n - 1}. Since \kappa (s) = \kappa (u) the fibre (X_ U)_ u = X_ s. Hence there exists a unique point x'_ n \in X_ u \subset X_ U corresponding to x_ n \in X_ s. Also x'_ n is isolated in X_ u. Hence by Lemma 37.41.1 there exists an elementary étale neighbourhood (U', u') \to (U, u) and an open V_ n \subset X_{U'} which works for x'_ n and hence for x_ n. By the final assertion of Lemma 37.41.1 the open subschemes V'_ i = U'\times _ U V_ i for i = 1, \ldots , n - 1 still work with respect to x_1, \ldots , x_{n - 1}. Hence we win.
\square
Comments (2)
Comment #2348 by Eric Ahlqvist on
Comment #2417 by Johan on
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