Going down for fixed-point subring under finite automorphism group

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This fact is an application of the following pivotal fact/result/idea: going up theorem
View other applications of going up theorem OR Read a survey article on applying going up theorem

Statement

Suppose B is a commutative unital ring and G is a finite subgroup of the automorphism group of B. Suppose A = B^Gis the fixed-point subring. Then, the extension B of A is a going down extension; in other words, we have going down for prime ideals. Explicitly:

If P_1 \supset P_2 are prime ideals of A and Q_1 is a prime ideal of B contracting to P_1 then there exists a prime ideal Q_2 of B contracting to P_2, such that Q_2 \subset Q_1.

Definitions used

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Facts used

Proof

Proof outline

  • Show that B is an integral extension of A
  • Use the fact that integral extensions give surjective maps on spectra,to find a prime ideal Q_2' lying over P_2
  • Use going up to find a prime ideal Q_1' lying over P_1 and containing Q_2'
  • Use the transitivity of action of G on the fiber over P_1 to find an automorphism that sends Q_1' to Q_1
  • The image of Q_2' under that automorphism is the required Q_2