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Tag 02LM

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

Lemma 36.36.3. 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. (a)    an étale neighbourhood $(U, u) \to (S, s)$,
  2. (b)    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. (\romannumeral1)    each $V_{i, j} \to U$ is a finite morphism,
  2. (\romannumeral2)    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. (\romannumeral4)    if $v_{i, j} = v_{i', j'}$, then $i = i'$ and $j = j'$, and
  4. (\romannumeral3)    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 10.141.23 in the language of schemes. By Morphisms, Lemma 28.19.6 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 28.19.5). 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 $\kappa(s) \subset L$ 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 36.31.2).

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 25.17.5 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 28.19.13 for example, the morphism $X_U \to U$ is quasi-finite at the points $y_{i, j}$ for all $i, j$.

Apply Lemma 36.36.2 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 (\romannumeral1), (\romannumeral2), and (\romannumeral3) 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. $\square$

    The code snippet corresponding to this tag is a part of the file more-morphisms.tex and is located in lines 10376–10402 (see updates for more information).

    \begin{lemma}
    \label{lemma-etale-makes-quasi-finite-finite-multiple-points-var}
    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 \'etale neighbourhood $(U, u) \to (S, s)$,
    \item[(b)] for each $i$ an integer $m_i$ and
    open subschemes $V_{i, j} \subset X_U$, $j = 1, \ldots, m_i$
    \end{enumerate}
    such that we have
    \begin{enumerate}
    \item[(\romannumeral1)] each $V_{i, j} \to U$ is a finite morphism,
    \item[(\romannumeral2)] 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,
    \item[(\romannumeral4)] if $v_{i, j} = v_{i', j'}$, then $i = i'$ and
    $j = j'$, and
    \item[(\romannumeral3)] the points $v_{i, j}$ map to $x_i$ in $X$ and
    no other points of $(X_U)_u$ map to $x_i$.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    This proof is a variant of the proof of
    Algebra, Lemma \ref{algebra-lemma-etale-makes-quasi-finite-finite-variant}
    in the language of schemes.
    By Morphisms, Lemma \ref{morphisms-lemma-quasi-finite-at-point-characterize}
    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 \ref{morphisms-lemma-residue-field-quasi-finite}).
    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 $\kappa(s) \subset L$ such that there exist
    $\kappa(s)$-embeddings $L_i \to L$ for $i = 1, \ldots, n$.
    Choose an \'etale neighbourhood $(U, u) \to (S, s)$ such that
    $L \cong \kappa(u)$ as $\kappa(s)$-extensions
    (Lemma \ref{lemma-realize-prescribed-residue-field-extension-etale}).
    
    \medskip\noindent
    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 \ref{schemes-lemma-points-fibre-product}
    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 \ref{morphisms-lemma-base-change-quasi-finite}
    for example, the morphism
    $X_U \to U$ is quasi-finite at the points $y_{i, j}$ for
    all $i, j$.
    
    \medskip\noindent
    Apply Lemma \ref{lemma-etale-makes-quasi-finite-finite-multiple-points}
    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 \'etale neighbourhood
    $(U', u') \to (U, u)$ with $\kappa(u) = \kappa(u')$ and
    opens $V_{i, j} \subset X_{U'}$ with the properties
    (\romannumeral1), (\romannumeral2), and (\romannumeral3)
    of that lemma. We claim that the \'etale 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.
    \end{proof}

    Comments (1)

    Comment #2993 by Dan Dore on November 8, 2017 a 7:45 pm UTC

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

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

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