The Stacks project

Exercise 111.28.7. Suppose that $(A, {\mathfrak m}, k)$ is a Noetherian local ring of dimension $1$. Assume also that the embedding dimension of $A$ is $2$, i.e., assume that

\[ \dim _ k {\mathfrak m}/{\mathfrak m}^2 = 2. \]

Notation: $f(n) = \dim _ k {\mathfrak m}^ n/{\mathfrak m}^{n + 1}$. Pick generators $x, y \in {\mathfrak m}$ and write $Gr_{\mathfrak m}(A) = k[\bar x, \bar y]/I$ for some homogeneous ideal $I$.

  1. Show that there exists a homogeneous element $F\in k[\bar x, \bar y]$ such that $I \subset (F)$ with equality in all sufficiently high degrees.

  2. Show that $f(n) \leq n + 1$.

  3. Show that if $f(n) < n + 1$ then $n \geq \deg (F)$.

  4. Show that if $f(n) < n + 1$, then $f(n + 1) \leq f(n)$.

  5. Show that $f(n) = \deg (F)$ for all $n >> 0$.

Comments (0)

There are also:

  • 1 comment(s) on Section 111.28: Cohen-Macaulay rings of dimension 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Your email address will not be published.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 02EL. Beware of the difference between the letter 'O' and the digit '0'.



View Exercise 111.28.7 as pdf