  In this paper we consider the question of when the associated graded ring
along a valuation, ${\rm gr}_{\nu^*}(S)$, is a finite ${\rm
gr}_{\nu^*}(R)$-module, where $S$ is a normal local ring which lies over a
normal local ring $R$ and $\nu^*$ is a valuation of the quotient field of $S$
which dominates $S$. We obtain a very general result in equicharacteristic zero
in Theorem 1.5. We also obtain general results for unramified extensions of
excellent local rings in Proposition 1.7.
