The Stacks project

Lemma 53.4.2. Let $X$ be a proper scheme over a field $k$ which is Cohen-Macaulay and equidimensional of dimension $1$. The module $\omega _ X$ of Lemma 53.4.1 has the following properties

  1. $\omega _ X$ is a dualizing module on $X$ (Duality for Schemes, Section 48.22),

  2. $\omega _ X$ is a coherent Cohen-Macaulay module whose support is $X$,

  3. there are functorial isomorphisms $\mathop{\mathrm{Ext}}\nolimits ^ i_ X(K, \omega _ X[1]) = \mathop{\mathrm{Hom}}\nolimits _ k(H^{-i}(X, K), k)$ compatible with shifts for $K \in D_\mathit{QCoh}(X)$,

  4. there are functorial isomorphisms $\mathop{\mathrm{Ext}}\nolimits ^{1 + i}(\mathcal{F}, \omega _ X) = \mathop{\mathrm{Hom}}\nolimits _ k(H^{-i}(X, \mathcal{F}), k)$ for $\mathcal{F}$ quasi-coherent on $X$.

Proof. Recall from the proof of Lemma 53.4.1 that $\omega _ X$ is as in Duality for Schemes, Example 48.22.1 and hence is a dualizing module. The other statements follow from Lemma 53.4.1 and the fact that $\omega _ X^\bullet = \omega _ X[1]$ as $X$ is Cohen-Macualay (Duality for Schemes, Lemma 48.23.1). $\square$

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Your email address will not be published.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 0BS3. Beware of the difference between the letter 'O' and the digit '0'.