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

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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