The Stacks project

Lemma 15.30.2. An $M$-regular sequence is $M$-Koszul-regular. A regular sequence is Koszul-regular.

Proof. Let $R$ be a ring and let $M$ be an $R$-module. It is immediate that an $M$-regular sequence of length $1$ is $M$-Koszul-regular. Let $f_1, \ldots , f_ r$ be an $M$-regular sequence. Then $f_1$ is a nonzerodivisor on $M$. Hence

\[ 0 \to K_\bullet (f_2, \ldots , f_ r) \otimes M \xrightarrow {f_1} K_\bullet (f_2, \ldots , f_ r) \otimes M \to K_\bullet (\overline{f}_2, \ldots , \overline{f}_ r) \otimes M/f_1M \to 0 \]

is a short exact sequence of complexes where $\overline{f}_ i$ is the image of $f_ i$ in $R/(f_1)$. By Lemma 15.28.8 the complex $K_\bullet (R, f_1, \ldots , f_ r)$ is isomorphic to the cone of multiplication by $f_1$ on $K_\bullet (f_2, \ldots , f_ r)$. Thus $K_\bullet (R, f_1, \ldots , f_ r) \otimes M$ is isomorphic to the cone on the first map. Hence $K_\bullet (\overline{f}_2, \ldots , \overline{f}_ r) \otimes M/f_1M$ is quasi-isomorphic to $K_\bullet (f_1, \ldots , f_ r) \otimes M$. As $\overline{f}_2, \ldots , \overline{f}_ r$ is an $M/f_1M$-regular sequence in $R/(f_1)$ the result follows from the case $r = 1$ and induction. $\square$


Comments (2)

Comment #5755 by Kestutis Cesnavicius on

It may be worthwhile mentioning that this holds even without "the ultimate quotient is nonzero" condition in the definition of a regular sequence https://stacks.math.columbia.edu/tag/00LF, which is a mildly more general statement. I guess it is debatable whether that condition should be a part of the definition of a regular sequence or not, for instance, in EGA it is not (IV, 0.15.1.7, 0.15.2.2).

Comment #5765 by on

Of course, you are right!

It is going to haunt me for the rest of my life that I decided to use the classical notion of a regular sequence in Definition 10.68.1. It already caused me a good deal of pain. But I refuse to capitulate! Honestly it would be a lot of work to change it now. (What was worse is that we originally defined the depth of the zero module to be based on looking at the supremum of the set of lengths of regular sequences. Just ask Burt. Argh!)

Anyway, I am going to leave this alone for now.

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

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