The Stacks project

Lemma 48.22.3. Let $X$ be a connected Noetherian scheme and let $\omega _ X$ be a dualizing module on $X$. The support of $\omega _ X$ is the union of the irreducible components of maximal dimension with respect to any dimension function and $\omega _ X$ is a coherent $\mathcal{O}_ X$-module having property $(S_2)$.

Proof. By our conventions discussed above there exists a dualizing complex $\omega _ X^\bullet $ such that $\omega _ X$ is the leftmost nonvanishing cohomology sheaf. Since $X$ is connected, any two dimension functions differ by a constant (Topology, Lemma 5.20.3). Hence we may use the dimension function associated to $\omega _ X^\bullet $ (Lemma 48.2.7). With these remarks in place, the lemma now follows from Dualizing Complexes, Lemma 47.17.5 and the definitions (in particular Cohomology of Schemes, Definition 30.11.1). $\square$

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Your email address will not be published.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