The Stacks project

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 $i$. $\square$

Comments (2)

Comment #5363 by MAO Zhouhang on

A typo in the hypothesis of the second statement: the second sequence should be , not .

There are also:

  • 6 comment(s) on Section 15.30: Koszul regular sequences

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