The Stacks project

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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