The Stacks project

Example 15.23.17. The results above and below suggest reflexivity is related to the $(S_2)$ condition; here is an example to prevent too optimistic conjectures. Let $k$ be a field. Let $R$ be the $k$-subalgebra of $k[x, y]$ generated by $1, y, x^2, xy, x^3$. Then $R$ is not $(S_2)$. So $R$ as an $R$-module is an example of a reflexive $R$-module which is not $(S_2)$. Let $M = k[x, y]$ viewed as an $R$-module. Then $M$ is a reflexive $R$-module because

\[ \mathop{\mathrm{Hom}}\nolimits _ R(M, R) = \mathfrak m = (y, x^2, xy, x^3) \quad \text{and}\quad \mathop{\mathrm{Hom}}\nolimits _ R(\mathfrak m, R) = M \]

and $M$ is $(S_2)$ as an $R$-module (computations omitted). Thus $R$ is a Noetherian domain possessing a reflexive $(S_2)$ module but $R$ is not $(S_2)$ itself.

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Your email address will not be published.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