Proposition 4

Let ρ : A → B be a local homomorphism of local noetherian rings, and let N be a finitely generated B-module, flat over A, and such that N ⊗A κ_A is a B-module of finite length. If M is a nonzero finitely generated A-module and q an ideal of A distinct from A and such that M/qM is of finite length, then (M ⊗_A N)/(qB)(M ⊗_A N) is a B-module of finite length, and one has*

Let L be an A-module of finite length r. Then L possesses a Jordan-Hölder sequence of length r, with quotients isomorphic to κ_A; since N is flat over A, the B-module L ⊗_A N possesses a composition sequence of length r, with quotients isomorphic to N ⊗_A κ_A, hence is of length r · long_B(N ⊗_A κ_A). Since the B-module (M ⊗_A N)/(qB)^n(M ⊗_A N) is isomorphic to (M/q^nM) ⊗_A N for every $n \in \mathbf{N}$, the proposition follows from the definition of multiplicities.