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

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

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

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

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

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

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

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

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 $A$,

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

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

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

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.

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