Lemma 37.40.3. 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

1. $f$ is locally of finite type, and

2. $x_ i \in X_ s$ is isolated for $i = 1, \ldots , n$.

Then there exist

1. an étale neighbourhood $(U, u) \to (S, s)$,

2. for each $i$ an integer $m_ i$ and open subschemes $V_{i, j} \subset X_ U$, $j = 1, \ldots , m_ i$

such that we have

1. each $V_{i, j} \to U$ is a finite morphism,

2. there is a unique point $v_{i, j}$ of $V_{i, j}$ mapping to $u$ in $U$ with $\kappa (u) \subset \kappa (v_{i, j})$ finite purely inseparable,

3. if $v_{i, j} = v_{i', j'}$, then $i = i'$ and $j = j'$, and

4. the points $v_{i, j}$ map to $x_ i$ in $X$ and no other points of $(X_ U)_ u$ map to $x_ i$.

Proof. This proof is a variant of the proof of Algebra, Lemma 10.145.4 in the language of schemes. By Morphisms, Lemma 29.20.6 the morphism $f$ is quasi-finite at each of the points $x_ i$. Hence $\kappa (s) \subset \kappa (x_ i)$ is finite for each $i$ (Morphisms, Lemma 29.20.5). For each $i$, let $\kappa (s) \subset L_ i \subset \kappa (x_ i)$ be the subfield such that $L_ i/\kappa (s)$ is separable, and $\kappa (x_ i)/L_ i$ is purely inseparable. Choose a finite Galois extension $L/\kappa (s)$ such that there exist $\kappa (s)$-embeddings $L_ i \to L$ for $i = 1, \ldots , n$. Choose an étale neighbourhood $(U, u) \to (S, s)$ such that $L \cong \kappa (u)$ as $\kappa (s)$-extensions (Lemma 37.34.2).

Let $y_{i, j}$, $j = 1, \ldots , m_ i$ be the points of $X_ U$ lying over $x_ i \in X$ and $u \in U$. By Schemes, Lemma 26.17.5 these points $y_{i, j}$ correspond exactly to the primes in the rings $\kappa (u) \otimes _{\kappa (s)} \kappa (x_ i)$. This also explains why there are finitely many; in fact $m_ i = [L_ i : \kappa (s)]$ but we do not need this. By our choice of $L$ (and elementary field theory) we see that $\kappa (u) \subset \kappa (y_{i, j})$ is finite purely inseparable for each pair $i, j$. Also, by Morphisms, Lemma 29.20.13 for example, the morphism $X_ U \to U$ is quasi-finite at the points $y_{i, j}$ for all $i, j$.

Apply Lemma 37.40.2 to the morphism $X_ U \to U$, the point $u \in U$ and the points $y_{i, j} \in (X_ U)_ u$. This gives an étale neighbourhood $(U', u') \to (U, u)$ with $\kappa (u) = \kappa (u')$ and opens $V_{i, j} \subset X_{U'}$ with the properties (i), (ii), and (iii) of that lemma. We claim that the étale neighbourhood $(U', u') \to (S, s)$ and the opens $V_{i, j} \subset X_{U'}$ are a solution to the problem posed by the lemma. We omit the verifications. $\square$

Comment #2993 by Dan Dore on

The numbering in the Lemma says "(\romannumeral1)", etc. instead of "(i)", etc.

Comment #3116 by on

THis is a problem with the website and not with the latex.

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