Corollary 1

Let ρ : Λ → B be a local homomorphism of noetherian local rings, M a nonzero finitely generated A-module and N a nonzero finitely generated B-module which is a flat A-module. In order that the B-module M ⊗_A N be a Macaulay module, it is necessary and sufficient that the A-module M be a Macaulay module and that the B/m_A B-module N/m_A N be a Macaulay module.

In fact, N is a faithfully flat A-module since N/m_A N is nonzero (I, § 3, No. 1, definition 1).