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Tag 069F

Chapter 36: More on Morphisms > Section 36.51: Local complete intersection morphisms

Definition 36.51.2. Let $f : X \to S$ be a morphism of schemes.

  1. Let $x \in X$. We say that $f$ is Koszul at $x$ if $f$ is of finite type at $x$ and there exists an open neighbourhood and a factorization of $f|_U$ as $\pi \circ i$ where $i : U \to P$ is a Koszul-regular immersion and $\pi : P \to S$ is smooth.
  2. We say $f$ is a Koszul morphism, or that $f$ is a local complete intersection morphism if $f$ is Koszul at every point.

    The code snippet corresponding to this tag is a part of the file more-morphisms.tex and is located in lines 15259–15271 (see updates for more information).

    \begin{definition}
    \label{definition-lci}
    Let $f : X \to S$ be a morphism of schemes.
    \begin{enumerate}
    \item Let $x \in X$. We say that $f$ is {\it Koszul at $x$} if $f$
    is of finite type at $x$ and there exists an open neighbourhood
    and a factorization of $f|_U$ as $\pi \circ i$ where $i : U \to P$
    is a Koszul-regular immersion and $\pi : P \to S$ is smooth.
    \item We say $f$ is a {\it Koszul morphism}, or that
    $f$ is a {\it local complete intersection morphism}
    if $f$ is Koszul at every point.
    \end{enumerate}
    \end{definition}

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