Going down for fixed-point subring under finite automorphism group

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 $A=B^{G}$is 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

• Ring is integral extension of fixed-point subring under finite automorphism group
• Integral extension implies surjective map on spectra
• Automorphism group acts transitively on fibers of spectrum over fixed-point subring
• Going up theorem

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 $Q_{2}$