Lifting Chains of Prime Ideals

We give an elementary proof that for a ring homomorphism A → B satisfying the property that every ideal in A is contracted from B the following property holds: for every chain of prime ideals p0 ⊂ . . . ⊂ pr in A there exists a chain of prime ideals q0 ⊂ . . . ⊂ qr in B such that qi ∩ A = pi. Mathematical Subject Classification (1991): 13B24. Let A and B be commutative rings and let φ : A → B be a ring homomorphism. This induces a continouus mapping φ : Spec B → Spec A by sending a prime ideal q ⊂ B to φ(q). Properties of the ring homomorphism are then often reflected by topological properties of φ. For example, if A → B is integral, then “going up” holds, and if A → B is flat, then “going down” holds (see [4, Proposition 4.15 and Lemma 10.11]. If moreover φ : Spec B → Spec A is surjective and going up or going down holds, then also the following property holds: for every given chain of prime ideals p0 ⊂ . . . ⊂ pr in A there exists a chain of prime ideals q0 ⊂ . . . ⊂ qr in B lying over it. In this note we give a direct and elementary proof showing that this chain lifting property holds also under the condition that every ideal in A is contracted from B, i.e. I = φ(IB) holds for every ideal I ⊆ A. This result can be found for pure homomorphisms in Picavet’s paper (see [11][Proposition 60 and Theorem 37]) and is proved using valuation theory. Our direct method allows to find explicitely chains of prime ideals and characterizes which prime ideals q0 over p0 may be extended to a chain. We start with the following lemma. Lemma 1. Let B be a commutative ring, let a0, . . . , ar be ideals and F0, . . . , Fr multiplicatively closed systems. Define inductively (set Sr+1 = {1}) for i = r, . . . , 0 the following multiplicatively closed sets Si = {s ∈ B : (s, ai) ∩ Fi · Si+1 6= ∅} . Then the following are equivalent. (i) 0 6∈ S0. (ii) ai ∩ Fi · Si+1 = ∅ for i = 0, . . . , r. (iii) There exists a chain of prime ideals q0 ⊆ . . . ⊆ qr such that ai ⊆ qi and qi ∩ Fi · Si+1 = ∅. (iv) There exists a chain of prime ideals q0 ⊆ . . . ⊆ qr such that ai ⊆ qi and qi ∩ Fi = ∅. Proof. It is clear that the Si are multiplicatively closed and that Si+1 ⊆ Si. (i) ⇔ (ii). If 0 ∈ S0, then a0 ∩ F0 · Si+1 6= ∅, and if ai ∩ Fi · Si+1 6= ∅ for some i, then 0 ∈ Si and thus also 0 ∈ S0. We show (ii) ⇒ (iii) by induction. Since a0 ∩F0S1 = ∅, there exists ([2, Ch.2 §5, Corollary 2]) a prime ideal q0 such that a0 ⊆ q0 and q0 ∩ F0S1 = ∅.