Proposition 3

Let A be a semi-local noetherian integral domain, K its field of fractions. If the K-algebra K ⊗_A Â is separable, the ring A is Japanese.

Let L be a finite extension of K and B the integral closure of A in L. Let F be a finite subset of B such that L = K[F] (V, § 1, No. 5, cor. 2 to prop. 16); let C denote the (finite) A-algebra generated by F. Since L is the field of fractions of C, the ring B is the integral closure of C (V, § 1, No. 1, prop. 6) and it is enough to prove that B is a finite C-algebra. Now, C is a semi-local noetherian ring (IV, § 2, No. 5, cor. 3 to prop. 9 ); its completion is identified with C ⊗_A Â (III, § 3, No. 4, th. 3 (ii)), hence also with a subring of the reduced ring L ⊗_A Â = L ⊗_K (K ⊗_A Â) and therefore is reduced. Proposition 3 therefore follows from the corollary to Theorem 2.