### **SETUP:**  Let $R$ be a finitely generated, noetherian, integral domain of characteristic $0$.  Let $G$ be a finite group that acts on $R$ by ring automorphisms.

## **QUESTION:**  *Is $R$ a finitely generated module over the fixed subring $R^G$?*

**REMARK:**  Since $R$ has characteristic $0$, K R Nagarajan's 1968 counter-example does not apply.  Since the order of $G$ is not a unit in $R$, G M Bergman's 1971 positive result does not apply, either.  My secondary source for this information is page 13 of an [e-presentation of Dobbs--Shapiro][1].

**COROLLARY 1:**  *$R^G$ is noetherian,* since $R$ is, by the Eakin–Nagata theorem.

**COROLLARY 2:**  *$R$ is integral over $R^G$,* by the so-called Determinantal Trick.  Since $R$ is a domain, then this is true independently, by Exercise 13.2 in Eisenbud's book on commutative algebra.  In particular, $\dim(R^G)=\dim(R)$ by the Cohen–Seidenberg theorems [Eisenbud: 4.15, 4.18].

**MOTIVATION:**  I am a topologist, and the class of examples of the Question that occur for me are the group rings $R=\mathbb{Z}[A]$, where $A$ is a finite-rank free-abelian group, and where $G$ acts on $A$ by group automorphisms.  *Is the Question true at least in this setting?*  (Note that $R$ is not a hereditary ring.)

**SIMILAR POST:** [Question #241618][2] 


  [1]: http://math.gmu.edu/~jshapiro/MAAC.pdf
  [2]: http://mathoverflow.net/questions/241618/fixed-points-for-action-of-finite-group-acting-on-noetherian-ring-is-a-local-noe