Ring Automorphisms, G Acts Transitively on Prime Ideals

G Acts Transitively on Prime Ideals

If c is a ring automorphism, verify that c maps an ideal to an ideal, and c maps the product of two ideals onto the product of their images. It follows that c maps a prime ideal onto a prime ideal.

An ideal is proper, or maximal, or trivial, iff the same is true of its image under c.

Let A be the subring of B fixed by G, hence B/A is an integral extension. The primes of B lie over the primes of A. Since A is fixed, a prime ideal Q, lying over P, has to move, via c, to a prime ideal lying over P. In other words, G permutes the prime ideals lying over P.

Let B be an integral domain. Let Q₁ and Q₂ be two different prime ideals lying over P. We want to move Q₁ onto Q₂.

Let u be an element of Q₁ that is not in A. Let v be the product of all the conjugates of u. (This is where we need B to be an integral domain, to take advantage of the previous theorem.) Now v is the last coefficient of the monic polynomial P(u), and it lies in A. It also lies in Q₁, hence in P. Since Q₂ is prime, one of the conjugates of u lies in Q₂. Let c(u) lie in Q₂.

Let d be the inverse of c. Now u is contained in d(Q₂). This can be done for any u in Q₁. Therefore Q₁ lies in the finite union of all the images of Q₂ under G. These images are prime, hence Q₁ is contained in the finite union of prime ideals. It follows that Q₁ is contained in one of these prime ideals.

Let Q₁ lie in d(Q₂). Since one prime ideal cannot contain another, Q₁ = d(Q₂). Reverse this, and Q₂ = c(Q₁). Since Q₂ was arbitrary, G maps Q₁ to every prime ideal lying over P. In other words, G acts transitively on the fiber of P.

The subgroup of G that stabilizes Q₁, times the orbit, gives the order of G. Thus the number of primes lying over P divides |G|.

Prime Spectrum

Inclusion of A into B induces a continuous map from spec B into spec A. Since every prime P has a prime lying over it, this map is onto. The fiber of P becomes the preimage of P under the continuous map.

Let G define an equivalence relation on the prime ideals of B, which are the points of spec B. Since G acts transitively, it equates all the primes in the fiber of P.

Since B is integral over A, the map from spec B onto spec A is bicontinuous. This turns spec A into a quotient space of spec B, where G provides the equivalence relation.