Corollary

Tag 04UE, permanent. Cite it as [Bourbaki, Tag 04UE].

Commutative Algebra, chapter VI, § 4. The height of a valuation

Let K be a field, $(v_\alpha)_{\alpha \in I}$ a family of valuations of height 1 on K and A the intersection of the rings of the v. Then A is a completely integrally closed domain.

A completely integrally closed domain is not always an intersection of valuation rings of height 1 (Exercise 6).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Theorem 2, § 1

Cites

Exercise 6