Let $R$ be a local Noetherian ring which contains the field $\mathbb{Q}$ of rational numbers, let $G$ be a finite group acting on $R$, and let $R^G \subseteq R$ be the fixed points for the action of $G$. How do I see that $R^G$ is also a local Noetherian ring, which is Cohen-Macaulay if $R$ is Cohen-Macaulay?



Fixed points for action of finite group acting on Noetherian ring is a local Noetherian ring
In general, the ring of invariants of a finite group acting on a Noetherian ring need not be Noetherian. Counterexamples were given by Nagata. But if $|G|$ is inverible in $R$, as in your case, then $R^G$ is indeed Noetherian. See for example "Nagata, Masayoshi: Some questions on rational actions of groups". Moreover, under these circumstances if $R$ is CM then also $R^G$ is CM (Hochster-Eagon Theorem, see Prop. 12 of "Hochster-Eagon: Cohen-Macaulay Rings, Invariant Theory, and the Generic Perfection of Determinantal Loci".


