The Stacks project

Lemma 15.30.7. Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $M$ be a nonzero finite $R$-module. Let $f_1, \ldots , f_ r \in \mathfrak m$. The following are equivalent

  1. $f_1, \ldots , f_ r$ is an $M$-regular sequence,

  2. $f_1, \ldots , f_ r$ is a $M$-Koszul-regular sequence,

  3. $f_1, \ldots , f_ r$ is an $M$-$H_1$-regular sequence,

  4. $f_1, \ldots , f_ r$ is an $M$-quasi-regular sequence.

In particular the sequence $f_1, \ldots , f_ r$ is a regular sequence in $R$ if and only if it is a Koszul regular sequence, if and only if it is a $H_1$-regular sequence, if and only if it is a quasi-regular sequence.

Proof. The implication (1) $\Rightarrow $ (2) is Lemma 15.30.2. The implication (2) $\Rightarrow $ (3) is Lemma 15.30.3. The implication (3) $\Rightarrow $ (4) is Lemma 15.30.6. The implication (4) $\Rightarrow $ (1) is Algebra, Lemma 10.69.6. $\square$

Comments (0)

There are also:

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

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09CC. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 15.30.7 as pdf