Corollary 4

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

Commutative Algebra, chapter VI, § 10. Extensions of a valuation to a transcendental extension

Suppose that v is improper (in which case $k = K$ ). Then

h(\Gamma') + d(k'/K) \leq r(\Gamma') + D(k'/K) \leq d(K'/K).

If, in particular, v' is of height 1, then

d(k'/K) \leq d(K'/K) - 1;

further, if $K'$ is a finitely generated extension of $K$ and there is equality in (12), then $v'$ is a discrete valuation and $k'$ is a finitely generated extension of $K$ .

It is a series of special cases of Corollaries 1, 2, 3.

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

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