The Stacks project

Lemma 15.33.4. Let $R$ be a ring. If $R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ is a relative global complete intersection, then $f_1, \ldots , f_ c$ is a Koszul regular sequence.

Proof. Recall that the homology groups $H_ i(K_\bullet (f_\bullet ))$ are annihilated by the ideal $(f_1, \ldots , f_ c)$. Hence it suffices to show that $H_ i(K_\bullet (f_\bullet ))_\mathfrak q$ is zero for all primes $\mathfrak q \subset R[x_1, \ldots , x_ n]$ containing $(f_1, \ldots , f_ c)$. This follows from Algebra, Lemma 10.136.13 and the fact that a regular sequence is Koszul regular (Lemma 15.30.2). $\square$


Comments (0)


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Your email address will not be published.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 07D2. Beware of the difference between the letter 'O' and the digit '0'.