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Tag 01TD

Chapter 28: Morphisms of Schemes > Section 28.19: Quasi-finite morphisms

Definition 28.19.1. Let $f : X \to S$ be a morphism of schemes.

  1. We say that $f$ is quasi-finite at a point $x \in X$ if there exist an affine neighbourhood $\mathop{\rm Spec}(A) = U \subset X$ of $x$ and an affine open $\mathop{\rm Spec}(R) = V \subset S$ such that $f(U) \subset V$, the ring map $R \to A$ is of finite type, and $R \to A$ is quasi-finite at the prime of $A$ corresponding to $x$ (see above).
  2. We say $f$ is locally quasi-finite if $f$ is quasi-finite at every point $x$ of $X$.
  3. We say that $f$ is quasi-finite if $f$ is of finite type and every point $x$ is an isolated point of its fibre.

    The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 3132–3150 (see updates for more information).

    \begin{definition}
    \label{definition-quasi-finite}
    \begin{reference}
    \cite[II Definition 6.2.3]{EGA}
    \end{reference}
    Let $f : X \to S$ be a morphism of schemes.
    \begin{enumerate}
    \item We say that $f$ is {\it quasi-finite at a point $x \in X$}
    if there exist an affine neighbourhood $\Spec(A) = U \subset X$
    of $x$ and an affine open $\Spec(R) = V \subset S$ such that
    $f(U) \subset V$, the ring map $R \to A$ is of finite type,
    and $R \to A$ is quasi-finite at the prime of $A$ corresponding to $x$
    (see above).
    \item We say $f$ is {\it locally quasi-finite} if $f$ is
    quasi-finite at every point $x$ of $X$.
    \item We say that $f$ is {\it quasi-finite} if $f$ is of finite type
    and every point $x$ is an isolated point of its fibre.
    \end{enumerate}
    \end{definition}

    References

    [EGA, II Definition 6.2.3]

    Comments (2)

    Comment #2714 by Ariyan Javanpeykar on August 1, 2017 a 11:36 pm UTC

    A reference: This definition is made in EGA II, Definition 6.2.3

    Comment #2751 by Takumi Murayama (site) on August 2, 2017 a 12:14 am UTC

    Added the reference; thanks!

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