The Stacks project

Lemma 110.15.1. There exists a local ring $R$ and a regular sequence $x, y, z$ (in the maximal ideal) such that there exists a nonzero element $\delta \in R/zR$ with $x\delta = y\delta = 0$.

Proof. Let $R = k[x, y, z] \oplus E$ where $E$ is the module above considered as a square zero ideal. Then it is clear that $x, y, z$ is a regular sequence in $R$, and that the element $\delta \in E/zE \subset R/zR$ gives an element with the desired properties. To get a local example we may localize $R$ at the maximal ideal $\mathfrak m = (x, y, z, E)$. The sequence $x, y, z$ remains a regular sequence (as localization is exact), and the element $\delta $ remains nonzero as it is supported at $\mathfrak m$. $\square$


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Your email address will not be published.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 0640. Beware of the difference between the letter 'O' and the digit '0'.