Finite Generation of Extensions of Associated Graded Rings Along a Valuation

In this paper we consider the question of when the associated graded ring along a valuation, ${\mathrm{g}\mathrm{r}}_{{\nu }^{\ast }}(S)$, is a finite ${\rm gr}_{\nu^*}®$-module,where$S$ is a normal local ring which lies over a normal local ring $R$ and ${\nu }^{\ast }$ 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.