The Stacks project

Lemma 31.21.3. Let $i : Z \to X$ be an immersion of schemes. Assume $X$ is locally Noetherian. Then $i$ is regular $\Leftrightarrow $ $i$ is Koszul-regular $\Leftrightarrow $ $i$ is $H_1$-regular $\Leftrightarrow $ $i$ is quasi-regular.

Proof. Follows immediately from Lemma 31.21.2 and Lemma 31.20.8. $\square$

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