Proposition 1

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

Commutative Algebra, chapter VI, § 7. Approximation theorem

Let $K$ be a field, $(\mathbf{A}_i)_{1 \leq i \leq n}$ ajinitefamily of valuation rings of $K$ and $B = \bigcap_{i=1}^n \mathbf{A}_i$ . We write $p_i = B \cap m(\mathbf{A}_i)$ . Then $A_i = B_{p_i}$ for all $i$ and the jield of ructions of $B$ is $K$ .

Clearly $B_{p_i} \subset A_i$ . To prove the converse inclusion we need the following lemma:

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

Cited by

Exercise 2, § 7
Corollary 1, § 7
Proposition 2, § 7
Proposition 3, § 7
Theorem 1 (Approximation Theorem for Valuations), § 7

Cites

Nothing.