111.28 Cohen-Macaulay rings of dimension 1

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 $i = 1, \ldots , d$.

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

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

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

2. $A$ is has no embedded prime.

2. 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.

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

1. Show that $A$ has dimension 1.

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

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

2. 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.,

$\dim _ k {\mathfrak m}/{\mathfrak m}^2 = 1.$

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

$1, 1, \ldots , 1, 0, 0, 0, 0, 0, \ldots$

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.,

$\dim _ k {\mathfrak m}/{\mathfrak m}^2 = 1.$

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

1. 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.

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

3. Conclude that $A$ is a domain.

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

1. 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.

2. 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

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

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 $n >> 0$.

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

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

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

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

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).

Comment #8810 by Samuel Tiersma on

Exercise 02EM, part (1): the linear form should be coprime to capital F, not little f.

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