# 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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## 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 $Q_2$