# The Stacks Project

## Tag 069F

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 14988–15000 (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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