Proposition 1

Let ρ : A → B be a homomorphism of rings satisfying the condition (PM).

a) Let h : A → A′ be a homomorphism of rings. Then the homomorphism ρ′ : A′ → A′ ⊗_A B deduced from ρ satisfies the condition (PM).

b) Let q be a prime ideal of B and p = ρ^{-1}(q). The canonical homomorphism ρ_q : A_p → B_q satisfies the condition (PM).

c) Let q be a prime ideal of B and p = ρ^{-1}(q). For every prime ideal p′ of A contained in p, there exists a prime ideal q′ of B lying above p′ and contained in q.

Let N be a B-module faithfully flat over A and such that N ⊗_B κ(q) ≠ 0 for every prime ideal q of B.

Prove a). The A′-module N′ = A′ ⊗_A N is faithfully flat (I, § 3, No. 3, prop. 5); let q′ be a prime ideal of B′ = A′ ⊗_A B and q its inverse image in B. We have isomorphisms

as we have N ⊗_B κ(q) ≠ 0, we also have N′ ⊗_{B′} κ(q′) ≠ 0.

Prove b). According to props. 13 and 14 of II, § 3, No. 4, the A_p-module N_q is flat. On the other hand, let b′ be a prime ideal of B_q; it is of the form bB_q where b is a prime ideal of B contained in q (II, § 3, No. 1, prop. 3); we have N ⊗_B κ(b) ≠ 0 in view of the assumption made on N, and since N_q ⊗_{B_q} κ(b′) is isomorphic to N ⊗_B κ(b′), we have N_q ⊗_{B_q} κ(b′) ≠ 0. Remark 3 allows us to conclude.

Prove c). The local homomorphism ρ_q : A_p → B_q deduced from ρ satisfies the condition (PM) according to b). The mapping Spec(B_q) → Spec(A_p) is therefore surjective (Remark 2), which was to be proved.