Definition 15.32.1. Let $R$ be a ring and let $I \subset R$ be an ideal.

1. We say $I$ is a regular ideal if for every $\mathfrak p \in V(I)$ there exists a $g \in R$, $g \not\in \mathfrak p$ and a regular sequence $f_1, \ldots , f_ r \in R_ g$ such that $I_ g$ is generated by $f_1, \ldots , f_ r$.

2. We say $I$ is a Koszul-regular ideal if for every $\mathfrak p \in V(I)$ there exists a $g \in R$, $g \not\in \mathfrak p$ and a Koszul-regular sequence $f_1, \ldots , f_ r \in R_ g$ such that $I_ g$ is generated by $f_1, \ldots , f_ r$.

3. We say $I$ is a $H_1$-regular ideal if for every $\mathfrak p \in V(I)$ there exists a $g \in R$, $g \not\in \mathfrak p$ and an $H_1$-regular sequence $f_1, \ldots , f_ r \in R_ g$ such that $I_ g$ is generated by $f_1, \ldots , f_ r$.

4. We say $I$ is a quasi-regular ideal if for every $\mathfrak p \in V(I)$ there exists a $g \in R$, $g \not\in \mathfrak p$ and a quasi-regular sequence $f_1, \ldots , f_ r \in R_ g$ such that $I_ g$ is generated by $f_1, \ldots , f_ r$.

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