Exercise 111.55.3. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring whose residue field has characteristic not $2$. Suppose that $\mathfrak m$ is generated by three elements $x, y, z$ and that $x^2 + y^2 + z^2 = 0$ in $A$.

1. What are the possible values of $\dim (A)$?

2. Give an example to show that each value is possible.

3. Show that $A$ is a domain if $\dim (A) = 2$. (Hint: look at $\bigoplus _{n \geq 0} \mathfrak m^ n/\mathfrak m^{n + 1}$.)

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Your email address will not be published.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