Exercise 109.28.8. 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 Ex. 6 above. Please use any results from the problems above.

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 $f$.

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

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.)

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).

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

Conclude that $I = (F)$.

Conclude that the function $f$ has values

\[ 2, 3, 4, \ldots , d-1, d, d, d, d, d, d, d, \ldots \]

## Comments (2)

Comment #31 by David Zureick-Brown on

Comment #34 by Johan on