Definition 31.21.1. Let $i : Z \to X$ be an immersion of schemes. Choose an open subscheme $U \subset X$ such that $i$ identifies $Z$ with a closed subscheme of $U$ and denote $\mathcal{I} \subset \mathcal{O}_ U$ the corresponding quasi-coherent sheaf of ideals.

1. We say $i$ is a regular immersion if $\mathcal{I}$ is regular.

2. We say $i$ is a Koszul-regular immersion if $\mathcal{I}$ is Koszul-regular.

3. We say $i$ is a $H_1$-regular immersion if $\mathcal{I}$ is $H_1$-regular.

4. We say $i$ is a quasi-regular immersion if $\mathcal{I}$ is quasi-regular.

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