The Stacks project

Lemma 48.23.1. Let $X$ be a locally Noetherian scheme with dualizing complex $\omega _ X^\bullet $.

  1. $X$ is Cohen-Macaulay $\Leftrightarrow $ $\omega _ X^\bullet $ locally has a unique nonzero cohomology sheaf,

  2. $\mathcal{O}_{X, x}$ is Cohen-Macaulay $\Leftrightarrow $ $\omega _{X, x}^\bullet $ has a unique nonzero cohomology,

  3. $U = \{ x \in X \mid \mathcal{O}_{X, x}\text{ is Cohen-Macaulay}\} $ is open and Cohen-Macaulay.

If $X$ is connected and Cohen-Macaulay, then there is an integer $n$ and a coherent Cohen-Macaulay $\mathcal{O}_ X$-module $\omega _ X$ such that $\omega _ X^\bullet = \omega _ X[-n]$.

Proof. By definition and Dualizing Complexes, Lemma 47.15.6 for every $x \in X$ the complex $\omega _{X, x}^\bullet $ is a dualizing complex over $\mathcal{O}_{X, x}$. By Dualizing Complexes, Lemma 47.20.2 we see that (2) holds.

To see (3) assume that $\mathcal{O}_{X, x}$ is Cohen-Macaulay. Let $n_ x$ be the unique integer such that $H^{n_{x}}(\omega _{X, x}^\bullet )$ is nonzero. For an affine neighbourhood $V \subset X$ of $x$ we have $\omega _ X^\bullet |_ V$ is in $D^ b_{\textit{Coh}}(\mathcal{O}_ V)$ hence there are finitely many nonzero coherent modules $H^ i(\omega _ X^\bullet )|_ V$. Thus after shrinking $V$ we may assume only $H^{n_ x}$ is nonzero, see Modules, Lemma 17.9.5. In this way we see that $\mathcal{O}_{X, v}$ is Cohen-Macaulay for every $v \in V$. This proves that $U$ is open as well as a Cohen-Macaulay scheme.

Proof of (1). The implication $\Leftarrow $ follows from (2). The implication $\Rightarrow $ follows from the discussion in the previous paragraph, where we showed that if $\mathcal{O}_{X, x}$ is Cohen-Macaulay, then in a neighbourhood of $x$ the complex $\omega _ X^\bullet $ has only one nonzero cohomology sheaf.

Assume $X$ is connected and Cohen-Macaulay. The above shows that the map $x \mapsto n_ x$ is locally constant. Since $X$ is connected it is constant, say equal to $n$. Setting $\omega _ X = H^ n(\omega _ X^\bullet )$ we see that the lemma holds because $\omega _ X$ is Cohen-Macaulay by Dualizing Complexes, Lemma 47.20.2 (and Cohomology of Schemes, Definition 30.11.4). $\square$

Comments (0)

Post a comment

Your email address will not be published.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 toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

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 input field. As a reminder, this is tag 0AWT. Beware of the difference between the letter 'O' and the digit '0'.