The Stacks project

The concept of a Koszul-regular ideal sheaf was introduced in [Expose VII, Definition 1.4, SGA6] where it was called a regular ideal sheaf.

Definition 31.20.2. Let $X$ be a ringed space. Let $\mathcal{J} \subset \mathcal{O}_ X$ be a sheaf of ideals.

  1. We say $\mathcal{J}$ is regular if for every $x \in \text{Supp}(\mathcal{O}_ X/\mathcal{J})$ there exists an open neighbourhood $x \in U \subset X$ and a regular sequence $f_1, \ldots , f_ r \in \mathcal{O}_ X(U)$ such that $\mathcal{J}|_ U$ is generated by $f_1, \ldots , f_ r$.

  2. We say $\mathcal{J}$ is Koszul-regular if for every $x \in \text{Supp}(\mathcal{O}_ X/\mathcal{J})$ there exists an open neighbourhood $x \in U \subset X$ and a Koszul-regular sequence $f_1, \ldots , f_ r \in \mathcal{O}_ X(U)$ such that $\mathcal{J}|_ U$ is generated by $f_1, \ldots , f_ r$.

  3. We say $\mathcal{J}$ is $H_1$-regular if for every $x \in \text{Supp}(\mathcal{O}_ X/\mathcal{J})$ there exists an open neighbourhood $x \in U \subset X$ and a $H_1$-regular sequence $f_1, \ldots , f_ r \in \mathcal{O}_ X(U)$ such that $\mathcal{J}|_ U$ is generated by $f_1, \ldots , f_ r$.

  4. We say $\mathcal{J}$ is quasi-regular if for every $x \in \text{Supp}(\mathcal{O}_ X/\mathcal{J})$ there exists an open neighbourhood $x \in U \subset X$ and a quasi-regular sequence $f_1, \ldots , f_ r \in \mathcal{O}_ X(U)$ such that $\mathcal{J}|_ U$ is generated by $f_1, \ldots , f_ r$.

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