The Stacks project

Exercise 111.12.8. Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $x, y \in \mathfrak m$ be a regular sequence of length $2$. For any $n \geq 2$ show that there do not exist $a, b \in R$ with

\[ x^{n - 1}y^{n - 1} = a x^ n + b y^ n \]

Suggestion: First try for $n = 2$ to see how to argue. Remark: There is a vast generalization of this result called the monomial conjecture.

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Your email address will not be published.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 0CT2. Beware of the difference between the letter 'O' and the digit '0'.



View Exercise 111.12.8 as pdf