Suppose that $R = S/I = k[x_1, \dots, x_n]/I$ is a (normal) domain of finite type over a field (or any semi-local ring $k$ with a dualizing complex). In this case, I can define $\omega_R = \textrm{Ext}^{n - \dim R}_{S}(R, S)$.

$R$ is called *quasi-Gorenstein* if $\omega_R$ is locally free of rank 1. Presumably there are easy examples where $\omega_R$ is locally free but not free, does anyone know one off the top of their head?

**EDIT:** Of course, It would be even better to have a Gorenstein (ie, quasi-Gorenstein + CM) example too.

Obviously I can choose a canonical module up to tensoring with a locally free at some level, but for many purposes the $\omega_R$ I defined above is better to work with.