Automorphism group acts transitively on fibers of spectrum over fixed-point subring

This fact is an application of the following pivotal fact/result/idea: prime avoidance lemma
View other applications of prime avoidance lemma OR Read a survey article on applying prime avoidance lemma

Statement

Let $G$ be a finite group acting as automorphisms of a commutative unital ring $B$, and let $A$ be the subring of $B$ comprising those elements that are fixed under every element of $G$ (i.e. $B^{G}=A$). Consider the map at the level of spectra:

$Spec(B)\to Spec(A)$

There is a natural action of $G$ on the fibers over any point in $A$. The action is transitive.

Proof

Proof outline

• Consider the $G$-orbit of one element in the fiber
• Prove that any element in the fiber must, as a prime ideal, be contained in the union of all the prime ideals in that $G$-orbit. (This is the step where the main work is done)
• Apply the prime avoidance lemma