Proposition 4

Tag 04U4, permanent. Cite it as [Bourbaki, Tag 04U4].

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

The mapping $B \mapsto H_B$ is an increasing bijection of the set of subrings of $K$ containing $A$ onto the set of isolated subgroups of $\Gamma_A$ .

Given $H$ , $v_B$ is defined up to equivalence and hence $B$ is determined uniquely. On the other hand, let $H$ be an isolated subgroup of $\Gamma_A$ ; considering $\Gamma_A / H$ as a totally ordered group (Proposition 3), the composite mapping

K^* \xrightarrow{v_A} \Gamma_A \longrightarrow \Gamma_A / H

is a valuation on $K$ whose ring contains $A$ .

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

Cited by

§ 4. The height of a valuation

Cites

Proposition 3