Definition 15.30.1. Let $R$ be a ring. Let $r \geq 0$ and let $f_1, \ldots , f_ r \in R$ be a sequence of elements. Let $M$ be an $R$-module. The sequence $f_1, \ldots , f_ r$ is called

1. $M$-Koszul-regular if $H_ i(K_\bullet (f_1, \ldots , f_ r) \otimes _ R M) = 0$ for all $i \not= 0$,

2. $M$-$H_1$-regular if $H_1(K_\bullet (f_1, \ldots , f_ r) \otimes _ R M) = 0$,

3. Koszul-regular if $H_ i(K_\bullet (f_1, \ldots , f_ r)) = 0$ for all $i \not= 0$, and

4. $H_1$-regular if $H_1(K_\bullet (f_1, \ldots , f_ r)) = 0$.

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