Definition 10.103.1. A Noetherian local ring $R$ is called *Cohen-Macaulay* if it is Cohen-Macaulay as a module over itself.

## 10.103 Cohen-Macaulay rings

Most of the results of this section are special cases of the results in Section 10.102.

Note that this is equivalent to requiring the existence of a $R$-regular sequence $x_1, \ldots , x_ d$ of the maximal ideal such that $R/(x_1, \ldots , x_ d)$ has dimension $0$. We will usually just say “regular sequence” and not “$R$-regular sequence”.

Lemma 10.103.2. Let $R$ be a Noetherian local Cohen-Macaulay ring with maximal ideal $\mathfrak m $. Let $x_1, \ldots , x_ c \in \mathfrak m$ be elements. Then

If so $x_1, \ldots , x_ c$ can be extended to a regular sequence of length $\dim (R)$ and each quotient $R/(x_1, \ldots , x_ i)$ is a Cohen-Macaulay ring of dimension $\dim (R) - i$.

**Proof.**
Special case of Proposition 10.102.4.
$\square$

Lemma 10.103.3. Let $R$ be Noetherian local. Suppose $R$ is Cohen-Macaulay of dimension $d$. Any maximal chain of ideals $\mathfrak p_0 \subset \mathfrak p_1 \subset \ldots \subset \mathfrak p_ n$ has length $n = d$.

**Proof.**
Special case of Lemma 10.102.9.
$\square$

Lemma 10.103.4. Suppose $R$ is a Noetherian local Cohen-Macaulay ring of dimension $d$. For any prime $\mathfrak p \subset R$ we have

**Proof.**
Follows immediately from Lemma 10.103.3. (Also, this is a special case of Lemma 10.102.10.)
$\square$

Lemma 10.103.5. Suppose $R$ is a Cohen-Macaulay local ring. For any prime $\mathfrak p \subset R$ the ring $R_{\mathfrak p}$ is Cohen-Macaulay as well.

**Proof.**
Special case of Lemma 10.102.11.
$\square$

Definition 10.103.6. A Noetherian ring $R$ is called *Cohen-Macaulay* if all its local rings are Cohen-Macaulay.

Lemma 10.103.7. Suppose $R$ is a Noetherian Cohen-Macaulay ring. Any polynomial algebra over $R$ is Cohen-Macaulay.

**Proof.**
Special case of Lemma 10.102.13.
$\square$

Lemma 10.103.8. Let $R$ be a Noetherian local Cohen-Macaulay ring of dimension $d$. Let $0 \to K \to R^{\oplus n} \to M \to 0$ be an exact sequence of $R$-modules. Then either $M = 0$, or $\text{depth}(K) > \text{depth}(M)$, or $\text{depth}(K) = \text{depth}(M) = d$.

**Proof.**
If $d = 0$, then every nonzero $R$-module has depth $0$ and the lemma is true. Assume $d > 0$. Then $\text{depth}(K) > 0$ as $K$ is a submodule of a module of depth $> 0$. Hence the lemma holds if $\text{depth}(M) = 0$. Assume both $\text{depth}(M) > 0$ and $d > 0$. Then we choose $x \in \mathfrak m$ which is a nonzerodivisor on $M$ and on $R$. Then $x$ is a nonzerodivisor on $M$ and on $K$ and it follows by an easy diagram chase that $0 \to K/xK \to (R/xR)^{\oplus n} \to M/xM \to 0$ is exact. Using Lemmas 10.71.7 and 10.103.2 we find the result follows from the result for $K/xK$ over $R/xR$ which has smaller dimension.
$\square$

Lemma 10.103.9. Let $R$ be a local Noetherian Cohen-Macaulay ring of dimension $d$. Let $M$ be a finite $R$ module of depth $e$. There exists an exact complex

with each $F_ i$ finite free and $K$ maximal Cohen-Macaulay.

**Proof.**
Immediate from the definition and Lemma 10.103.8.
$\square$

Lemma 10.103.10. Let $\varphi : A \to B$ be a map of local rings. Assume that $B$ is Noetherian and Cohen-Macaulay and that $\mathfrak m_ B = \sqrt{\varphi (\mathfrak m_ A) B}$. Then there exists a sequence of elements $f_1, \ldots , f_{\dim (B)}$ in $A$ such that $\varphi (f_1), \ldots , \varphi (f_{\dim (B)})$ is a regular sequence in $B$.

**Proof.**
By induction on $\dim (B)$ it suffices to prove: If $\dim (B) \geq 1$, then we can find an element $f$ of $A$ which maps to a nonzerodivisor in $B$. By Lemma 10.103.2 it suffices to find $f \in A$ whose image in $B$ is not contained in any of the finitely many minimal primes $\mathfrak q_1, \ldots , \mathfrak q_ r$ of $B$. By the assumption that $\mathfrak m_ B = \sqrt{\varphi (\mathfrak m_ A) B}$ we see that $\mathfrak m_ A \not\subset \varphi ^{-1}(\mathfrak q_ i)$. Hence we can find $f$ by Lemma 10.14.2.
$\square$

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