Remark 3

Suppose that ρ : A → B is a local homomorphism of local rings and that there exists a B-module N flat over A and such that N ⊗_B κ(q) ≠ 0 for every prime ideal q of B. Then N is faithfully flat over A and ρ therefore has the property (PM): indeed we have N/m_B N = N ⊗_B κ(m_B) ≠ 0, hence N ≠ m_B N and fortiori N ≠ m_A N, and the conclusion follows from prop. 1 of I, § 3, No. 1.