According to Lang the valuation $v$ of the field $K$ is well-behaved, if for every finite extension $E/K$ the equation

$
[E:K] =\sum\limits_{w|v} [E_w:K_v]
$

holds, where the summation runs over all extension $w$ of $v$ to $E$, and $K_v$, $E_w$ are the completions of the fields $K$, $E$ with respect to the valuations $v$, $w$ respectively.

For a discrete valuation $v$ the completion $K_v$ is equal to the field of fractions of the $M_v$-adical completion $\widehat{O_v}$ of the local ring $O_v$, where $M_v$ is the maximal ideal of $O_v$.

The discrete valuation $v$ we are discussing here by assumption is a localization of an integral, finitely generated $k$-algebra($k$ the field over which the variety $V$ lives). Such an algebra has a finite normalisation in every finite extension of their field of fractions. This property is inherited by localisations of the algebra, thus $O_v$ has this property too: the normalization $O_v(E)$ of $O_v$ in a finite extension $E$ of the field of fractions $K$ is a finitely generated, torsion-free $O_v$-module. It is well-known that such modules over a factorial ring are free - and $O_v$ is factorial. The rank of $O_v(E)$ msut be $[E:K]$ - just localise at $0$.

There is a bijection between the valuation rings $O_w$ of the extensions $w$ of $v$ to $E$ and the localisations of $O_v(E)$ at maximal ideals $M$.

The product of all these maximal ideals is some ideal $I$. The $I$-adical completion $\widehat{O_v(E)}$ of $O(E)$ satisfies:

$
\widehat{O_v(E)} = \prod\limits_{w|v}\widehat{O_w}
$

(Matsumura, Thm. 8.15).

Since a power of $I$ lies in the ideal $M_vO_v(E)$ and a power of $M_vO_v(E)$ lies in $I$, the completions of $O_v(E)$ with respect to these two ideals coincide.

The completion of $O_v(E)$ with respect to $M_vO_v(E)$ on the other hand equals the tensor product $
O(E)\otimes_{O_v}\widehat{O_v} $. Since the extension $\widehat{O_v}/O_v$ is faithfully flat this tensor product is a free $\widehat{O_v}$-module of rank $[E:K]$.

Altogether we see now:

$
\prod\limits_{w|v}\widehat{O_w}
$

is a free $\widehat{O_v}$-module of rank $[E:K]$, from which we get

$
\prod\limits_{w|v}E_w
$

is a free $K_v$-module of rank $[E:K]$.

Hagen