Example of a reflexive canonical module

Question

I want an example of a Cohen-Macaulay local ring $R$ with the canonical module $\omega_R$ such that $\omega_R$ is reflexive (that is $\omega_R \cong {\rm Hom}_R ({\rm Hom}_R(\omega_R,R),R)$).

Answer 1

As a simple example, take $R=k[[x^3,x^2y,xy^2,y^3]]$. It is a two dimensional CM local ring with an isolated singularity. It is also not Gorenstein. So, $\omega_R$ has depth 2 and the natural map $\omega_R\to\omega_R^{**}$ is injective and the cokernel is finite length, since $R$ has an isolated singularity. This forces the cokernel to be zero by depth considerations. Thus $\omega_R$ is reflexive. This can easily be generalized, for example to most CM local rings of dimension at least 2 with isolated singularity.