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Tag 02LL

Chapter 36: More on Morphisms > Section 36.36: Étale localization of quasi-finite morphisms

Lemma 36.36.2. 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

  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. (a)    an elementary étale neighbourhood $(U, u) \to (S, s)$,
  2. (b)    for each $i$ an open subscheme $V_i \subset X_U$,

such that for each $i$ we have

  1. (\romannumeral1)    $V_i \to U$ is a finite morphism,
  2. (\romannumeral2)    there is a unique point $v_i$ of $V_i$ mapping to $u$ in $U$, and
  3. (\romannumeral3)    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 36.36.1 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 36.36.1 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. $\square$

    The code snippet corresponding to this tag is a part of the file more-morphisms.tex and is located in lines 10102–10124 (see updates for more information).

    \begin{lemma}
    \label{lemma-etale-makes-quasi-finite-finite-multiple-points}
    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
    \begin{enumerate}
    \item $f$ is locally of finite type, and
    \item $x_i \in X_s$ is isolated for $i = 1, \ldots, n$.
    \end{enumerate}
    Then there exist
    \begin{enumerate}
    \item[(a)] an elementary \'etale neighbourhood $(U, u) \to (S, s)$,
    \item[(b)] for each $i$ an open subscheme $V_i \subset X_U$,
    \end{enumerate}
    such that for each $i$ we have
    \begin{enumerate}
    \item[(\romannumeral1)] $V_i \to U$ is a finite morphism,
    \item[(\romannumeral2)] there is a unique point $v_i$ of $V_i$
    mapping to $u$ in $U$, and
    \item[(\romannumeral3)] the point $v_i$ maps to $x_i$ in $X$ and
    $\kappa(x_i) = \kappa(v_i)$.
    \end{enumerate}
    \end{lemma}
    
    \begin{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 \ref{lemma-etale-makes-quasi-finite-finite-at-point} there
    exists an elementary \'etale 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 \ref{lemma-etale-makes-quasi-finite-finite-at-point}
    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.
    \end{proof}

    Comments (2)

    Comment #2348 by Eric Ahlqvist on January 11, 2017 a 9:06 am UTC

    space after \romannumeral

    Comment #2417 by Johan (site) on February 17, 2017 a 1:51 pm UTC

    This is a problem with the website rendering of the LaTeX...

    There are also 2 comments on Section 36.36: More on Morphisms.

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