Let $R=k[M]$ be a monoid algebra, where $M$ is a fine and saturated monoid. If $k$ is a field, it is a theorem of Hochster that $R$ is Cohen-Macaulay. What if $k$ is a Dedekind domain? Is $R$ Cohen-Macaulay in this case?
Now I have a finite group $G$ acting on this ring. Again Hochster shows that if $k$ is a field, the ring of invariants $R^G$ is Cohen-Macaulay. Can this be generalized to the case $k$ is a Dedekind domain?
If anyone knows the answer, could you also provide a precise reference?
