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.

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