  Given a generically finite local extension of valuation rings $V \subset W$,
the question of whether $W$ is the localization of a finitely generated
$V$-algebra is significant for approaches to the problem of local
uniformization of valuations using ramification theory. Hagen Knaf proposed a
characterization of when $W$ is essentially of finite type over $V$ in terms of
classical invariants of the extension of associated valuations. Knaf's
conjecture has been verified in important special cases by Cutkosky and
Novacoski using local uniformization of Abhyankar valuations and resolution of
singularities of excellent surfaces in arbitrary characteristic, and by
Cutkosky for valuation rings of function fields of characteristic $0$ using
embedded resolution of singularities. In this paper we prove Knaf's conjecture
in full generality.
