Lemma 76.44.4. Let $S$ be a scheme. Let $i : Z \to X$ be an immersion of algebraic spaces over $S$. Assume $X$ is locally Noetherian. Then $i$ is Koszul-regular $\Leftrightarrow $ $i$ is $H_1$-regular $\Leftrightarrow $ $i$ is quasi-regular.
Proof. Via Definition 76.44.2 (and the definition of a locally Noetherian algebraic space in Properties of Spaces, Section 66.7) this immediately translates to the case of schemes which is Divisors, Lemma 31.21.3. $\square$
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