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

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

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

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

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

Proof. The implication (1) $\Rightarrow$ (2) is Lemma 15.29.2. The implication (2) $\Rightarrow$ (3) is Lemma 15.29.3. The implication (3) $\Rightarrow$ (4) is Lemma 15.29.6. The implication (4) $\Rightarrow$ (1) is Algebra, Lemma 10.68.6. $\square$

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