Let $S$ be a [numerical semigroup][1] and $k[S]$ is the associated semigroup ring. I would like to compute canonical module $\omega$ of $k[S].$

***I want to show that $\omega=k[t^{-n}:n\in\mathbb Z\setminus S]$.***

I have shown that $H^1_{m}(k[S])=k[t^{n}:n\in\mathbb Z\setminus S]$ where $m$ is the maximal homogeneous ideal of $k[S].$ Using duality I tried to compute canonical ideal but I am not able to do it.


  [1]: https://en.wikipedia.org/wiki/Numerical_semigroup