Lemma 15.30.4. Let $f_1, \ldots , f_{r - 1} \in R$ be a sequence and $f, g \in R$. Let $M$ be an $R$-module.

1. If $f_1, \ldots , f_{r - 1}, f$ and $f_1, \ldots , f_{r - 1}, g$ are $M$-$H_1$-regular then $f_1, \ldots , f_{r - 1}, fg$ is $M$-$H_1$-regular too.

2. If $f_1, \ldots , f_{r - 1}, f$ and $f_1, \ldots , f_{r - 1}, g$ are $M$-Koszul-regular then $f_1, \ldots , f_{r - 1}, fg$ is $M$-Koszul-regular too.

Proof. By Lemma 15.28.11 we have exact sequences

$H_ i(K_\bullet (f_1, \ldots , f_{r - 1}, f) \otimes M) \to H_ i(K_\bullet (f_1, \ldots , f_{r - 1}, fg) \otimes M) \to H_ i(K_\bullet (f_1, \ldots , f_{r - 1}, g) \otimes M)$

for all $i$. $\square$

Comment #5363 by MAO Zhouhang on

A typo in the hypothesis of the second statement: the second sequence should be $f_1,\dots,f_{r-1},g$, not $f_1,\dots,f_{r-1},f$.

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