Lemma 15.30.4 (062G)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.30: Koszul regular sequences
Lemma 15.30.4 (cite)

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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.

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.
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 July 03, 2020 at 06:27

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

Comment #5600 by Johan on November 12, 2020 at 19:57

THanks and fixed here.

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