Definition 111.28.1. A Noetherian local ring $A$ is said to be *Cohen-Macaulay* of dimension $d$ if it has dimension $d$ and there exists a system of parameters $x_1, \ldots , x_ d$ for $A$ such that $x_ i$ is a nonzerodivisor in $A/(x_1, \ldots , x_{i-1})$ for $i = 1, \ldots , d$.

## 111.28 Cohen-Macaulay rings of dimension 1

Exercise 111.28.2. Cohen-Macaulay rings of dimension 1. Part I: Theory.

Let $(A, {\mathfrak m})$ be a local Noetherian with $\dim A = 1$. Show that if $x\in {\mathfrak m}$ is not a zerodivisor then

$\dim A/xA = 0$, in other words $A/xA$ is Artinian, in other words $\{ x\} $ is a system of parameters for $A$.

$A$ is has no embedded prime.

Conversely, let $(A, {\mathfrak m})$ be a local Noetherian ring of dimension $1$. Show that if $A$ has no embedded prime then there exists a nonzerodivisor in ${\mathfrak m}$.

Exercise 111.28.3. Cohen-Macaulay rings of dimension 1. Part II: Examples.

Let $A$ be the local ring at $(x, y)$ of $k[x, y]/(x^2, xy)$.

Show that $A$ has dimension 1.

Prove that every element of ${\mathfrak m}\subset A$ is a zerodivisor.

Find $z\in {\mathfrak m}$ such that $\dim A/zA = 0$ (no proof required).

Let $A$ be the local ring at $(x, y)$ of $k[x, y]/(x^2)$. Find a nonzerodivisor in ${\mathfrak m}$ (no proof required).

Exercise 111.28.4. Local rings of embedding dimension $1$. Suppose that $(A, {\mathfrak m}, k)$ is a Noetherian local ring of embedding dimension $1$, i.e.,

Show that the function $f(n) = \dim _ k {\mathfrak m}^ n/{\mathfrak m}^{n + 1}$ is either constant with value $1$, or its values are

Exercise 111.28.5. Regular local rings of dimension $1$. Suppose that $(A, {\mathfrak m}, k)$ is a regular Noetherian local ring of dimension $1$. Recall that this means that $A$ has dimension $1$ and embedding dimension $1$, i.e.,

Let $x\in {\mathfrak m}$ be any element whose class in ${\mathfrak m}/{\mathfrak m}^2$ is not zero.

Show that for every element $y$ of ${\mathfrak m}$ there exists an integer $n$ such that $y$ can be written as $y = ux^ n$ with $u\in A^\ast $ a unit.

Show that $x$ is a nonzerodivisor in $A$.

Conclude that $A$ is a domain.

Exercise 111.28.6. Let $(A, {\mathfrak m}, k)$ be a Noetherian local ring with associated graded $Gr_{\mathfrak m}(A)$.

Suppose that $x\in {\mathfrak m}^ d$ maps to a nonzerodivisor $\bar x \in {\mathfrak m}^ d/{\mathfrak m}^{d + 1}$ in degree $d$ of $Gr_{\mathfrak m}(A)$. Show that $x$ is a nonzerodivisor.

Suppose the depth of $A$ is at least $1$. Namely, suppose that there exists a nonzerodivisor $y \in {\mathfrak m}$. In this case we can do better: assume just that $x\in {\mathfrak m}^ d$ maps to the element $\bar x \in {\mathfrak m}^ d/{\mathfrak m}^{d + 1}$ in degree $d$ of $Gr_{\mathfrak m}(A)$ which is a nonzerodivisor on sufficiently high degrees: $\exists N$ such that for all $n \geq N$ the map of multiplication by $\bar x$

\[ {\mathfrak m}^ n/{\mathfrak m}^{n + 1} \longrightarrow {\mathfrak m}^{n + d}/{\mathfrak m}^{n + d + 1} \]is injective. Then show that $x$ is a nonzerodivisor.

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

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 $I$.

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.

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

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

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

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

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.

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

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

Remark 111.28.9. This suggests that a local Noetherian Cohen-Macaulay ring of dimension 1 and embedding dimension 2 is of the form $B/FB$, where $B$ is a 2-dimensional regular local ring. This is more or less true (under suitable “niceness” properties of the ring).

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