Corollary

Let A be a semi-local noetherian ring whose completion is reduced. Then the integral closure A' of A in its total ring of fractions R is a finite A-algebra.

Suppose first that A is local and complete, and let p₁, ..., pₙ be the minimal (distinct) prime ideals of A; for i = 1, ..., n, let Kᵢ denote the field of fractions of A/pᵢ and Aᵢ' the integral closure of A/pᵢ. Since A is reduced, R is the product of the rings Kᵢ and A' the product of the rings Aᵢ' (V, § 1, No. 2, cor. 1 to prop. 9). Since the local rings A/pᵢ are integral domains and complete, they are Japanese (Theorem 2), so that each Aᵢ' is a finite A-algebra, and A' is a finite A-algebra.

If A is semi-local and complete, it is isomorphic to a finite product of complete local rings (III, § 2, No. 13, corollary to prop. 19), and we conclude at once from the preceding.

Let us pass to the general case, and note that the completion  of A is a semi-local, complete, noetherian ring faithfully flat over A (III, loc. cit., § 3, No. 4, corollary to prop. 8 and § 3, No. 5, prop. 9). Let S be the set of non-zero-divisors of A; one has R = S⁻¹A. Since  is flat over A, the elements of S are non-zero-divisors in Â, and S⁻¹Â is identified with a subring of the total ring of fractions T of Â. Again because  is flat over A, the ring A' ⊗_A  is identified with a subring of R ⊗_A  = S⁻¹Â, hence also with a subring of T integral over Â. By the first part of the proof, A' ⊗_A  is therefore a finitely generated Â-module; hence A' is a finitely generated A-module (I, § 3, No. 6, prop. 11).

Recall (A, V, p. 114, def. 1) that an algebra E over a field K is said to be separable if the ring L ⊗_K E is reduced for every extension L of K; it is enough that this be so for every finite extension of K. The following proposition generalizes Theorem 2: