Definition 37.60.2. 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.

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