The Stacks project

Exercise 111.28.8. Cohen-Macaulay rings of dimension 1 and embedding dimension 2. Suppose that $(A, {\mathfrak m}, k)$ is a Noetherian local ring which is Cohen-Macaulay of dimension $1$. Assume also that the embedding dimension of $A$ is $2$, i.e., assume that

Notations: $f$, $F$, $x, y\in {\mathfrak m}$, $I$ as in Exercise 111.28.7. Please use any results from the problems above.

1. Suppose that $z\in {\mathfrak m}$ is an element whose class in ${\mathfrak m}/{\mathfrak m}^2$ is a linear form $\alpha \bar x + \beta \bar y \in k[\bar x, \bar y]$ which is coprime with $F$.

   1. Show that $z$ is a nonzerodivisor on $A$.

   2. Let $d = \deg (F)$. Show that ${\mathfrak m}^ n = z^{n + 1-d}{\mathfrak m}^{d-1}$ for all sufficiently large $n$. (Hint: First show $z^{n + 1-d}{\mathfrak m}^{d-1} \to {\mathfrak m}^ n/{\mathfrak m}^{n + 1}$ is surjective by what you know about $Gr_{\mathfrak m}(A)$. Then use NAK.)

2. What condition on $k$ guarantees the existence of such a $z$? (No proof required; it's too easy.)

   Now we are going to assume there exists a $z$ as above. This turns out to be a harmless assumption (in the sense that you can reduce to the situation where it holds in order to obtain the results in parts (d) and (e) below).

3. Now show that ${\mathfrak m}^\ell = z^{\ell - d + 1} {\mathfrak m}^{d-1}$ for all $\ell \geq d$.

4. Conclude that $I = (F)$.

5. Conclude that the function $f$ has values

   $$2, 3, 4, \ldots , d-1, d, d, d, d, d, d, d, \ldots$$