Corollary 5.43.95B3D7BB20499

Proof. On a smooth curve, local modifications are isomorphisms. By theorem 5.37.3, the base inclusions are therefore equivalent to $\nu (v,s)\geq \nu (w,s)$ for every $s\in C$. Let $\pi $ be any local modification of a neighborhood in $X$ with smooth source, and let $y$ be a point of its source. In a coordinate at $s=p(\pi (y))$, the germ $p\circ \pi -s$ is nonconstant. Indeed, a modification is an isomorphism on a dense open set, and a nonconstant holomorphic map cannot be constant on an open set. Its order $e_ y$ is positive and finite. The preceding lemma gives

The base inequalities thus imply all inequalities upstairs, and the same valuative theorem gives the ideal inclusions. Conversely the upstairs inclusions give the inequalities for the identity modification at any $x\in p^{-1}(s)$. Dividing by $\operatorname {ord}_ x(p-s)>0$ recovers the base inequalities. Apply this argument in both directions for equality. $\square$