A local proof

D. J. Bernstein

19951124

The object of this sheet is to prove the following fact: if F ⊆ K are number fields, with rings of integers R and A respectively, then ∑_{Q} e(Q : P)f(Q : P) = dim_{F} K for any nonzero prime ideal P of R. Here Q runs through the prime ideals of A lying over P. (We have already seen several special cases of this fact.)

The local Krull-Akizuki situation. Let R ⊆ A, F ⊆ K be in the Krull-Akizuki situation, and let P be a nonzero prime ideal of R. Recall that ∑_{Q} e(Q : P)f(Q : P) = len_{R}(A/P A).

One way to understand this situation is to localize. Define R′ = (R − P)^{−1}R, A′ = (R − P)^{−1}A, P′ = (R − P)^{−1}P, and Q′ = (R − P)^{−1}Q. Then R′ ⊆ A′, F ⊆ K are in the Krull-Akizuki situation, Q′ is a nonzero prime ideal of A′, P′ is a nonzero prime ideal of R′, and Q′ lies over P′.

Why localize? Answer: R′ is a local ring, with unique maximal ideal P′. Local rings are much easier to understand than general rings. This sheet proves the ef fact by working locally. (Note that one can go further, by localizing at Q.)

Here are two more examples of theorems easily proven via localization. First, let K be a number field, A any order in K. Then pZ ramifies in A if and only if p divides disc A. (We have already seen this in the special case A = Z[α]. The most important special case, when A is the ring of integers of K, is a fundamental theorem of Dedekind on discriminants.)

Second, if K = F and A is finitely generated over R, then there is a nonzero A-ideal I ⊆ R; and for any such ideal there is an exact sequence

1 → R^{∗} → A^{∗} → (A/I)^{∗}/(R/I)^{∗} → Pic R → Pic A → 1.

Application: If K is a number field and A is its ring of integers, we already know how to compute A^{∗} and Pic A. From this exact sequence we can compute Pic R and R^{∗} for any order R in K.

Theorem 1. Let R be a local domain with invertible maximal ideal P. Then P is principal.

Proof. By assumption P J = R for some fractional ideal J. In particular 1 ∈ P J, so 1 = ∑ a_{k}b_{k} with a_{k} ∈ P and b_{k} ∈ J. If a_{k}b_{k} ∈ P for all k then 1 = ∑ a_{k}b_{k} ∈ P, contradiction. Hence a_{k}b_{k} ∈ R − P = R^{∗} for some k. Thus P = a_{k}b_{k}P ⊆ a_{k}JP = a_{k}R ⊆ P R = P, so P = a_{k}R.

Theorem 2. Let R ⊆ A, F ⊆ K be in the Krull-Akizuki situation. Assume that R is local, with invertible maximal ideal P. Assume that A is finitely generated over R. Then len_{R}(A/P A) = dim_{F} AF.

Proof. By Theorem 1, P is principal; say P = aR. By Theorem V1 from the Krull-Akizuki sheet, len_{R}(A/P A) = (dim_{F} AF) len_{R}(R/P) = dim_{F} AF.

Theorem 3. Let R ⊆ A, F ⊆ K be in the Krull-Akizuki situation. Let P be an invertible prime ideal of R. Assume that A is finitely generated over R. Then len_{R}(A/P A) = dim_{F} AF.

Proof. Localize. Outline: P′ is invertible, and A′ is finitely generated over R′, so len_{R′}(A′/P′A′) = dim_{F} A′F by Theorem 2. But dim_{F} A′F = dim_{F} AF and len_{R}(A/P A) = len_{R′}(A′/P′A′).

Theorem 4. Let F ⊆ K be an extension of number fields. Let R be the ring of integers of F. Let A be an order in K containing R. Let P be a nonzero prime ideal of R. Then len_{R}(A/P A) = dim_{F} K.

Proof. Since A is an order, it is finitely generated over Z, and hence over R; also K = AF. Since R is Dedekind, P is invertible. Finally, by Theorem 3, len_{R}(A/P A) = dim_{F} AF = dim_{F} K.