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 CohenMacaulay if $R$ is CohenMacaulay?

$\begingroup$ I don't understand the question. Are you not assuming that $R$ is local? In that case, take $G$ to be trivial... $\endgroup$ – Qiaochu Yuan Jun 7 '16 at 5:58

$\begingroup$ @QiaochuYuan ah sorry my bad, you're right. Fixed. $\endgroup$ – user64623 Jun 7 '16 at 6:02

1$\begingroup$ The map $R^G \subseteq R$ is finite and so it follows that $R$ local implies that $R^G$ local. The rest is then the content of the HochsterRoberts Theorem. $\endgroup$ – Ben Lim Jun 7 '16 at 6:05
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 (HochsterEagon Theorem, see Prop. 12 of "HochsterEagon: CohenMacaulay Rings, Invariant Theory, and the Generic Perfection of Determinantal Loci".