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

From Commalg

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

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

## Proof

### Proof outline

- Consider the -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 -orbit. (This is the step where the main work is done)
- Apply the prime avoidance lemma