Lemma 31.21.2. Let $i : Z \to X$ be an immersion of schemes. We have the following implications: $i$ is regular $\Rightarrow $ $i$ is Koszul-regular $\Rightarrow $ $i$ is $H_1$-regular $\Rightarrow $ $i$ is quasi-regular.

**Proof.**
The lemma immediately reduces to Lemma 31.20.3.
$\square$

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