The Stacks project

Remark 31.12.3. If $X$ is a scheme of finite type over a field, then sometimes a different notion of reflexive modules is used (see for example [bottom of page 5 and Definition 1.1.9, HL]). This other notion uses $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits $ into a dualizing complex $\omega _ X^\bullet $ instead of into $\mathcal{O}_ X$ and should probably have a different name because it can be different when $X$ is not Gorenstein. For example, if $X = \mathop{\mathrm{Spec}}(k[t^3, t^4, t^5])$, then a computation shows the dualizing sheaf $\omega _ X$ is not reflexive in our sense, but it is reflexive in the other sense as $\omega _ X \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\omega _ X, \omega _ X), \omega _ X)$ is an isomorphism.


Comments (0)

There are also:

  • 2 comment(s) on Section 31.12: Reflexive modules

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Your email address will not be published.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 0AY1. Beware of the difference between the letter 'O' and the digit '0'.