Definition 15.31.1. Let $R$ be a ring and let $I \subset R$ be an ideal.

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 $f_1, \ldots , f_ r$.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 $f_1, \ldots , f_ r$.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 $f_1, \ldots , f_ r$.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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