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Tag 02G4

Chapter 28: Morphisms of Schemes > Section 28.36: Unramified morphisms

Definition 28.36.1. Let $f : X \to S$ be a morphism of schemes.

  1. We say that $f$ is unramified at $x \in X$ if there exists a affine open neighbourhood $\mathop{\rm Spec}(A) = U \subset X$ of $x$ and affine open $\mathop{\rm Spec}(R) = V \subset S$ with $f(U) \subset V$ such that the induced ring map $R \to A$ is unramified.
  2. We say that $f$ is G-unramified at $x \in X$ if there exists a affine open neighbourhood $\mathop{\rm Spec}(A) = U \subset X$ of $x$ and affine open $\mathop{\rm Spec}(R) = V \subset S$ with $f(U) \subset V$ such that the induced ring map $R \to A$ is G-unramified.
  3. We say that $f$ is unramified if it is unramified at every point of $X$.
  4. We say that $f$ is G-unramified if it is G-unramified at every point of $X$.

    The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 7140–7159 (see updates for more information).

    \begin{definition}
    \label{definition-unramified}
    Let $f : X \to S$ be a morphism of schemes.
    \begin{enumerate}
    \item We say that $f$ is {\it unramified at $x \in X$} if
    there exists a affine open neighbourhood $\Spec(A) = U \subset X$
    of $x$ and affine open $\Spec(R) = V \subset S$
    with $f(U) \subset V$ such that the induced ring map
    $R \to A$ is unramified.
    \item We say that $f$ is {\it G-unramified at $x \in X$} if
    there exists a affine open neighbourhood $\Spec(A) = U \subset X$
    of $x$ and affine open $\Spec(R) = V \subset S$
    with $f(U) \subset V$ such that the induced ring map
    $R \to A$ is G-unramified.
    \item We say that $f$ is {\it unramified} if it is unramified
    at every point of $X$.
    \item We say that $f$ is {\it G-unramified} if it is G-unramified
    at every point of $X$.
    \end{enumerate}
    \end{definition}

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