Lemma 31.20.7. Let $X$ be a scheme. Let $\mathcal{J}$ be a sheaf of ideals. Then $\mathcal{J}$ is regular (resp. Koszul-regular, $H_1$-regular, quasi-regular) if and only if for every $x \in \text{Supp}(\mathcal{O}_ X/\mathcal{J})$ there exists an affine open neighbourhood $x \in U \subset X$, $U = \mathop{\mathrm{Spec}}(A)$ such that $\mathcal{J}|_ U = \widetilde{I}$ and such that $I$ is generated by a regular (resp. Koszul-regular, $H_1$-regular, quasi-regular) sequence $f_1, \ldots , f_ r \in A$.

Proof. By assumption we can find an open neighbourhood $U$ of $x$ over which $\mathcal{J}$ is generated by a regular (resp. Koszul-regular, $H_1$-regular, quasi-regular) sequence $f_1, \ldots , f_ r \in \mathcal{O}_ X(U)$. After shrinking $U$ we may assume that $U$ is affine, say $U = \mathop{\mathrm{Spec}}(A)$. Since $\mathcal{J}$ is quasi-coherent by Lemma 31.20.6 we see that $\mathcal{J}|_ U = \widetilde{I}$ for some ideal $I \subset A$. Now we can use the fact that

$\widetilde{\ } : \text{Mod}_ A \longrightarrow \mathit{QCoh}(\mathcal{O}_ U)$

is an equivalence of categories which preserves exactness. For example the fact that the functions $f_ i$ generate $\mathcal{J}$ means that the $f_ i$, seen as elements of $A$ generate $I$. The fact that (31.20.0.1) is injective (resp. (31.20.0.2) is exact, (31.20.0.2) is exact in degree $1$, (31.20.0.3) is an isomorphism) implies the corresponding property of the map $A/(f_1, \ldots , f_{i - 1}) \to A/(f_1, \ldots , f_{i - 1})$ (resp. the complex $K_\bullet (A, f_1, \ldots , f_ r)$, the map $A/I[T_1, \ldots , T_ r] \to \bigoplus I^ n/I^{n + 1}$). Thus $f_1, \ldots , f_ r \in A$ is a regular (resp. Koszul-regular, $H_1$-regular, quasi-regular) sequence of the ring $A$. $\square$

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