Definition 37.59.2. Let $f : X \to S$ be a morphism of schemes.

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.We say $f$ is a

*Koszul morphism*, or that $f$ is a*local complete intersection morphism*if $f$ is Koszul at every point.

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