Lemma 31.20.3. Let $X$ be a ringed space. Let $\mathcal{J}$ be a sheaf of ideals. We have the following implications: $\mathcal{J}$ is regular $\Rightarrow $ $\mathcal{J}$ is Koszul-regular $\Rightarrow $ $\mathcal{J}$ is $H_1$-regular $\Rightarrow $ $\mathcal{J}$ is quasi-regular.

**Proof.**
The lemma immediately reduces to Lemma 31.20.1.
$\square$

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