Exercise 5, § 3

Let ρ : A → B be a local homomorphism of complete Noetherian local rings, such that [κ(B):κ(A)] < ∞. Put n = dim(B/m_A B); prove that there exists an A-local homomorphism u : A[[T_1, ..., T_n]] → B which makes B a finite A[[T_1, ..., T_n]]-algebra. (Choose in B a maximal transversal sequence of B/m_A B and use III, § 3, Exercise 18.)