# The Stacks Project

## Tag 02IN

### 27.8. Cohen-Macaulay schemes

Recall, see Algebra, Definition 10.103.1, that a local Noetherian ring $(R, \mathfrak m)$ is said to be Cohen-Macaulay if $\text{depth}_{\mathfrak m}(R) = \dim(R)$. Recall that a Noetherian ring $R$ is said to be Cohen-Macaulay if every local ring $R_{\mathfrak p}$ of $R$ is Cohen-Macaulay, see Algebra, Definition 10.103.6.

Definition 27.8.1. Let $X$ be a scheme. We say $X$ is Cohen-Macaulay if for every $x \in X$ there exists an affine open neighbourhood $U \subset X$ of $x$ such that the ring $\mathcal{O}_X(U)$ is Noetherian and Cohen-Macaulay.

Lemma 27.8.2. Let $X$ be a scheme. The following are equivalent:

1. $X$ is Cohen-Macaulay,
2. $X$ is locally Noetherian and all of its local rings are Cohen-Macaulay, and
3. $X$ is locally Noetherian and for any closed point $x \in X$ the local ring $\mathcal{O}_{X, x}$ is Cohen-Macaulay.

Proof. Algebra, Lemma 10.103.5 says that the localization of a Cohen-Macaulay local ring is Cohen-Macaulay. The lemma follows by combining this with Lemma 27.5.2, with the existence of closed points on locally Noetherian schemes (Lemma 27.5.9), and the definitions. $\square$

Lemma 27.8.3. Let $X$ be a scheme. The following are equivalent:

1. The scheme $X$ is Cohen-Macaulay.
2. For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is Noetherian and Cohen-Macaulay.
3. There exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ is Noetherian and Cohen-Macaulay.
4. There exists an open covering $X = \bigcup X_j$ such that each open subscheme $X_j$ is Cohen-Macaulay.

Moreover, if $X$ is Cohen-Macaulay then every open subscheme is Cohen-Macaulay.

Proof. Combine Lemmas 27.5.2 and 27.8.2. $\square$

More information on Cohen-Macaulay schemes and depth can be found in Cohomology of Schemes, Section 29.11.

The code snippet corresponding to this tag is a part of the file properties.tex and is located in lines 923–995 (see updates for more information).

\section{Cohen-Macaulay schemes}
\label{section-Cohen-Macaulay}

\noindent
Recall, see Algebra, Definition \ref{algebra-definition-local-ring-CM},
that a local Noetherian ring $(R, \mathfrak m)$ is
said to be Cohen-Macaulay if $\text{depth}_{\mathfrak m}(R) = \dim(R)$.
Recall that a Noetherian ring $R$ is said to be Cohen-Macaulay if
every local ring $R_{\mathfrak p}$ of $R$ is Cohen-Macaulay,
see Algebra, Definition \ref{algebra-definition-ring-CM}.

\begin{definition}
\label{definition-Cohen-Macaulay}
Let $X$ be a scheme. We say $X$ is {\it Cohen-Macaulay} if
for every $x \in X$ there exists an affine open neighbourhood
$U \subset X$ of $x$ such that the ring $\mathcal{O}_X(U)$ is
Noetherian and Cohen-Macaulay.
\end{definition}

\begin{lemma}
\label{lemma-characterize-Cohen-Macaulay}
Let $X$ be a scheme. The following are equivalent:
\begin{enumerate}
\item $X$ is Cohen-Macaulay,
\item $X$ is locally Noetherian and all of its local rings are Cohen-Macaulay,
and
\item $X$ is locally Noetherian and for any closed point $x \in X$
the local ring $\mathcal{O}_{X, x}$ is Cohen-Macaulay.
\end{enumerate}
\end{lemma}

\begin{proof}
Algebra, Lemma \ref{algebra-lemma-localize-CM} says that the localization of
a Cohen-Macaulay local ring is Cohen-Macaulay. The lemma follows
by combining this with Lemma \ref{lemma-locally-Noetherian},
with the existence of closed
points on locally Noetherian schemes
(Lemma \ref{lemma-locally-Noetherian-closed-point}), and
the definitions.
\end{proof}

\begin{lemma}
\label{lemma-locally-Cohen-Macaulay}
Let $X$ be a scheme. The following are equivalent:
\begin{enumerate}
\item The scheme $X$ is Cohen-Macaulay.
\item For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$
is Noetherian and Cohen-Macaulay.
\item There exists an affine open covering $X = \bigcup U_i$ such that
each $\mathcal{O}_X(U_i)$ is Noetherian and Cohen-Macaulay.
\item There exists an open covering $X = \bigcup X_j$
such that each open subscheme $X_j$ is Cohen-Macaulay.
\end{enumerate}
Moreover, if $X$ is Cohen-Macaulay then every open subscheme
is Cohen-Macaulay.
\end{lemma}

\begin{proof}
Combine Lemmas \ref{lemma-locally-Noetherian}
and \ref{lemma-characterize-Cohen-Macaulay}.
\end{proof}

\noindent
Cohomology of Schemes, Section \ref{coherent-section-depth}.

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