Integral Extensions, Primes over P

Primes over P Cannot Contain One Another

Let S be integral over R and let P be a prime ideal in R. Suppose there are two prime ideals, Q₁ and Q₂, lying over P, such that Q₁ contains Q₂. Localize about P, so that S_P is integral over R_P. (I'm abusing the notation here; S_P means the fractions of S by R-P.) By prime correspondence the prime ideals Q₁ and Q₂ become distinct prime ideals in S_P with one containing the other. In the last section we showed that intersection and localization commute. Thus both ideals intersect R_P in P_P, which is a maximal ideal. Maximal ideals correspond, so both prime ideals in S_P are maximal, even though one properly contains the other. This is a contradiction, hence Q₁ cannot contain Q₂. Primes over P cannot contain one another.

An ascending or descending chain of prime ideals in S contracts to an ascending or descending chain of prime ideals in R. If not, then a prime P in R has two primes Q₁ and Q₂ in S, lying over P, with one containing the other.

What about chains that go beyond infinity? If the union of an ascending chain of prime ideals in S is prime, or is contained in a larger prime ideal, this prime ideal in S contracts to a prime ideal in R that is the union of, or contains the union of, the foregoing prime ideals in R. A similar result holds for the intersection of a descending chain of prime ideals. Use transfinite induction to show that ascending or descending chains of prime ideals (beyond infinity) in S, contract to same in R.

This does not imply chains in R lift to chains in S. Yes, P₁ might contain P₂ in R, and there exists Q₁ and Q₂ lying over P₁ and P₂, but that doesn't mean Q₁ contains Q₂.

Finitely Many Primes over P

Let S/R be an integral extension, where S is a finitely generated R module. We will show there are finitely many primes lying over P.

Localize about P, so that R_P becomes a local ring. The primes in S, lying over P, persist in the ring S_P, and remain distinct. Furthermore, they lie over a maximal ideal in the local ring, namely P_P, hence they are maximal. A finitely generated module over a local ring is semilocal, and has finitely many maximal ideals. Hence the fiber of P is finite.