Lemma 15.29.7. Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $M$ be a nonzero finite $R$-module. Let $f_1, \ldots , f_ r \in \mathfrak m$. The following are equivalent

$f_1, \ldots , f_ r$ is an $M$-regular sequence,

$f_1, \ldots , f_ r$ is a $M$-Koszul-regular sequence,

$f_1, \ldots , f_ r$ is an $M$-$H_1$-regular sequence,

$f_1, \ldots , f_ r$ is an $M$-quasi-regular sequence.

In particular the sequence $f_1, \ldots , f_ r$ is a regular sequence in $R$ if and only if it is a Koszul regular sequence, if and only if it is a $H_1$-regular sequence, if and only if it is a quasi-regular sequence.

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