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
    More information on Cohen-Macaulay schemes and depth can be found in
    Cohomology of Schemes, Section \ref{coherent-section-depth}.

    Comments (0)

    There are no comments yet for this tag.

    Add a comment on tag 02IN

    Your email address will not be published. Required fields are marked.

    In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

    All contributions are licensed under the GNU Free Documentation License.




    In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?