Lemma 31.20.8. Let $X$ be a locally Noetherian scheme. Let $\mathcal{J} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $x$ be a point of the support of $\mathcal{O}_ X/\mathcal{J}$. The following are equivalent

1. $\mathcal{J}_ x$ is generated by a regular sequence in $\mathcal{O}_{X, x}$,

2. $\mathcal{J}_ x$ is generated by a Koszul-regular sequence in $\mathcal{O}_{X, x}$,

3. $\mathcal{J}_ x$ is generated by an $H_1$-regular sequence in $\mathcal{O}_{X, x}$,

4. $\mathcal{J}_ x$ is generated by a quasi-regular sequence in $\mathcal{O}_{X, x}$,

5. there exists an affine neighbourhood $U = \mathop{\mathrm{Spec}}(A)$ of $x$ such that $\mathcal{J}|_ U = \widetilde{I}$ and $I$ is generated by a regular sequence in $A$, and

6. there exists an affine neighbourhood $U = \mathop{\mathrm{Spec}}(A)$ of $x$ such that $\mathcal{J}|_ U = \widetilde{I}$ and $I$ is generated by a Koszul-regular sequence in $A$, and

7. there exists an affine neighbourhood $U = \mathop{\mathrm{Spec}}(A)$ of $x$ such that $\mathcal{J}|_ U = \widetilde{I}$ and $I$ is generated by an $H_1$-regular sequence in $A$, and

8. there exists an affine neighbourhood $U = \mathop{\mathrm{Spec}}(A)$ of $x$ such that $\mathcal{J}|_ U = \widetilde{I}$ and $I$ is generated by a quasi-regular sequence in $A$,

9. there exists a neighbourhood $U$ of $x$ such that $\mathcal{J}|_ U$ is regular, and

10. there exists a neighbourhood $U$ of $x$ such that $\mathcal{J}|_ U$ is Koszul-regular, and

11. there exists a neighbourhood $U$ of $x$ such that $\mathcal{J}|_ U$ is $H_1$-regular, and

12. there exists a neighbourhood $U$ of $x$ such that $\mathcal{J}|_ U$ is quasi-regular.

In particular, on a locally Noetherian scheme the notions of regular, Koszul-regular, $H_1$-regular, or quasi-regular ideal sheaf all agree.

Proof. It follows from Lemma 31.20.7 that (5) $\Leftrightarrow$ (9), (6) $\Leftrightarrow$ (10), (7) $\Leftrightarrow$ (11), and (8) $\Leftrightarrow$ (12). It is clear that (5) $\Rightarrow$ (1), (6) $\Rightarrow$ (2), (7) $\Rightarrow$ (3), and (8) $\Rightarrow$ (4). We have (1) $\Rightarrow$ (5) by Algebra, Lemma 10.68.6. We have (9) $\Rightarrow$ (10) $\Rightarrow$ (11) $\Rightarrow$ (12) by Lemma 31.20.3. Finally, (4) $\Rightarrow$ (1) by Algebra, Lemma 10.69.6. Now all 12 statements are equivalent. $\square$

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